Riemannian and Hermitian Metrics

Abstract

Riemannian Geometry is a multidimensional generalization of the intrinsic geometry of 2D surfaces in the Euclidean space $\mathbb{E}^{3}$.

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Riemannian Geometry is a multidimensional generalization of the intrinsic geometry of 2D surfaces in the Euclidean space $\mathbb{E}^{3}$. It studies real smooth manifolds equipped with Riemannian metrics, i.e., collections of positive-definite symmetric bilinear forms ((g _ij )) on their tangent spaces which vary smoothly from point to point. The geometry of such (Riemannian) manifolds is based on the line element $\mathit{ds}^{2} =\sum _{i,j}g_{\mathit{ij}}\mathit{dx}_{i}\mathit{dx}_{j}$. This gives, in particular, local notions of angle, length of curve, and volume.

From these notions some other global quantities can be derived, by integrating local contributions. Thus, the value ds is interpreted as the length of the vector $(\mathit{dx}_{1},\ldots,\mathit{dx}_{n})$, and it is called the infinitesimal distance . The arc length of a curve γ is expressed by $\int _{\gamma }\sqrt{\sum _{i,j } g_{\mathit{ij } } \mathit{dx } _{i } \mathit{dx } _{j}}$, and then the intrinsic metric on a Riemannian manifold is defined as the infimum of lengths of curves joining two given points of the manifold.

Therefore, a Riemannian metric is not an ordinary metric, but it induces an ordinary metric, in fact, the intrinsic metric, called Riemannian distance, on any connected Riemannian manifold. A Riemannian metric is an infinitesimal form of the corresponding Riemannian distance.

As particular special cases of Riemannian Geometry, there occur Euclidean Geometry as well as the two standard types, Elliptic Geometry and Hyperbolic Geometry, of non-Euclidean Geometry. If the bilinear forms ((g _ij )) are nondegenerate but indefinite, one obtains pseudo-Riemannian Geometry. In the case of dimension four (and signature (1, 3)) it is the main object of the General Theory of Relativity.

If $\mathit{ds} = F(x_{1},\ldots,x_{n},\mathit{dx}_{1},\ldots,\mathit{dx}_{n})$, where F is a real positive-definite convex function which cannot be given as the square root of a symmetric bilinear form (as in the Riemannian case), one obtains the Finsler Geometry generalizing Riemannian Geometry.

Hermitian Geometry studies complex manifolds equipped with Hermitian metrics, i.e., collections of positive-definite symmetric sesquilinear forms (or $\frac{3}{2}$ -linear forms) since they are linear in one argument and antilinear in the other) on their tangent spaces, which vary smoothly from point to point. It is a complex analog of Riemannian Geometry.

A special class of Hermitian metrics form Kähler metrics which have a closed fundamental form w. A generalization of Hermitian metrics give complex Finsler metrics which cannot be written as a bilinear symmetric positive-definite sesqulinear form.

1 Riemannian Metrics and Generalizations

A real n-manifold M ⁿ with boundary is (cf. Chap. 2) a Hausdorff space in which every point has an open neighborhood homeomorphic to either an open subset of $\mathbb{E}^{n}$, or an open subset of the closed half of $\mathbb{E}^{n}$. The set of points which have an open neighborhood homeomorphic to $\mathbb{E}^{n}$ is called the interior (of the manifold); it is always nonempty.

The complement of the interior is called the boundary (of the manifold); it is an (n − 1)-dimensional manifold. If it is empty, one obtains a real n-manifold without boundary. Such manifold is called closed if it is compact, and open, otherwise.

An open set of M ⁿ together with a homeomorphism between the open set and an open set of $\mathbb{E}^{n}$ is called a coordinate chart. A collection of charts which cover M ⁿ is an atlas on M ⁿ. The homeomorphisms of two overlapping charts provide a transition mapping from a subset of $\mathbb{E}^{n}$ to some other subset of $\mathbb{E}^{n}$.

If all these mappings are continuously differentiable, then M ⁿ is a differentiable manifold. If they are k times (infinitely often) continuously differentiable, then the manifold is a C ^k manifold (respectively, a smooth manifold, or C ^∞ manifold).

An atlas of a manifold is called oriented if the Jacobians of the coordinate transformations between any two charts are positive at every point. An orientable manifold is a manifold admitting an oriented atlas.

Manifolds inherit many local properties of the Euclidean space: they are locally path-connected, locally compact, and locally metrizable. Every smooth Riemannian manifold embeds isometrically (Nash, 1956) in some finite-dimensional Euclidean space.

Associated with every point on a differentiable manifold is a tangent space and its dual, a cotangent space. Formally, let M ⁿ be a C ^k manifold, k ≥ 1, and p a point of M ⁿ. Fix a chart $\varphi: U \rightarrow \mathbb{E}^{n}$, where U is an open subset of M ⁿ containing p. Suppose that two curves γ ¹: (−1, 1) → M ⁿ and γ ²: (−1, 1) → M ⁿ with $\gamma ^{1}(0) =\gamma ^{2}(0) = p$ are given such that $\varphi \cdot \gamma ^{1}$ and $\varphi \cdot \gamma ^{2}$ are both differentiable at 0.

Then γ ¹ and γ ² are called tangent at 0 if $(\varphi \cdot \gamma ^{1})^{'}(0) =$ $(\varphi \cdot \gamma ^{2})^{'}(0)$. If the functions $\varphi \cdot \gamma ^{i}: (-1,1) \rightarrow \mathbb{E}^{n}$, i = 1, 2, are given by n real-valued component functions $(\varphi \cdot \gamma ^{i})_{1}(t),\ldots,(\varphi \cdot \gamma ^{i})_{n}(t)$, the condition above means that their Jacobians $\left (\frac{d(\varphi \cdot \gamma ^{i})_{ 1}(t)} {\mathit{dt}},\ldots, \frac{d(\varphi \cdot \gamma ^{i})_{ n}(t)} {\mathit{dt}} \right )$ coincide at 0. This is an equivalence relation, and the equivalence class γ ^′(0) of the curve γ is called a tangent vector of M ⁿ at p.

The tangent space T _p (M ⁿ) of M ⁿ at p is defined as the set of all tangent vectors at p. The function $(d\varphi )_{p}: T_{p}(M^{n}) \rightarrow \mathbb{E}^{n}$ defined by $(d\varphi )_{p}(\gamma ^{'}(0)) = (\varphi \cdot \gamma )^{'}(0)$, is bijective and can be used to transfer the vector space operations from $\mathbb{E}^{n}$ over to T _p (M ⁿ).

All the tangent spaces T _p (M ⁿ), p ∈ M ⁿ, when “glued together”, form the tangent bundle T(M ⁿ) of M ⁿ. Any element of T(M ⁿ) is a pair (p, v), where v ∈ T _p (M ⁿ).

If for an open neighborhood U of p the function $\varphi: U \rightarrow \mathbb{R}^{n}$ is a coordinate chart, then the preimage V of U in T(M ⁿ) admits a mapping $\psi: V \rightarrow \mathbb{R}^{n} \times \ \mathbb{R}^{n}$ defined by $\psi (p,v) = (\varphi (p),d\varphi (p))$. It defines the structure of a smooth 2n-dimensional manifold on T(M ⁿ). The cotangent bundle T ^∗(M ⁿ) of M ⁿ is obtained in similar manner using cotangent spaces T _p ^∗(M ⁿ), p ∈ M ⁿ.

A vector field on a manifold M ⁿ is a section of its tangent bundle T(M ⁿ), i.e., a smooth function f: M ⁿ → T(M ⁿ) which assigns to every point p ∈ M ⁿ a vector v ∈ T _p (M ⁿ).

A connection (or covariant derivative) is a way of specifying a derivative of a vector field along another vector field on a manifold.

Formally, the covariant derivative ∇ of a vector u (defined at a point p ∈ M ⁿ) in the direction of the vector v (defined at the same point p) is a rule that defines a third vector at p, called ∇ _v u which has the properties of a derivative. A Riemannian metric uniquely defines a special covariant derivative called the Levi-Civita connection. It is the torsion-free connection ∇ of the tangent bundle, preserving the given Riemannian metric.

The Riemann curvature tensor R is the standard way to express the curvature of Riemannian manifolds. The Riemann curvature tensor can be given in terms of the Levi-Civita connection ∇ by the following formula:

$$\displaystyle{R(u,v)w = \nabla _{u}\nabla _{v}w -\nabla _{v}\nabla _{u}w -\nabla _{[u,v]}w,}$$

where R(u, v) is a linear transformation of the tangent space of the manifold M ⁿ; it is linear in each argument. If $u = \frac{\partial } {\partial x_{i}}$ and $v = \frac{\partial } {\partial x_{j}}$ are coordinate vector fields, then [u, v] = 0, and the formula simplifies to $R(u,v)w = \nabla _{u}\nabla _{v}w -\nabla _{v}\nabla _{u}w$, i.e., the curvature tensor measures anti-commutativity of the covariant derivative. The linear transformation w → R(u, v)w is also called the curvature transformation.

The Ricci curvature tensor (or Ricci curvature) Ric is obtained as the trace of the full curvature tensor R. It can be thought of as a Laplacian of the Riemannian metric tensor in the case of Riemannian manifolds. Ricci curvature is a linear operator on the tangent space at a point. Given an orthonormal basis (e _i ) _i in the tangent space T _p (M ⁿ), we have

$$\displaystyle{\mathit{Ric}(u) =\sum _{i}R(u,e_{i})e_{i}.}$$

The value of Ric(u) does not depend on the choice of an orthonormal basis. Starting with dimension four, the Ricci curvature does not describe the curvature tensor completely.

The Ricci scalar (or scalar curvature) Sc of a Riemannian manifold M ⁿ is the full trace of the curvature tensor; given an orthonormal basis (e _i ) _i at p ∈ M ⁿ, we have

$$\displaystyle{\mathit{Sc} =\sum _{i,j}\langle R(e_{i},e_{j})e_{j},e_{i}\rangle =\sum _{i}\langle \mathit{Ric}(e_{i}),e_{i}\rangle.}$$

The sectional curvature K(σ) of a Riemannian manifold M ⁿ is defined as the Gauss curvature of an σ-section at a point p ∈ M ⁿ, where a σ-section is a locally-defined piece of surface which has the 2-plane σ as a tangent plane at p, obtained from geodesics which start at p in the directions of the image of σ under the exponential mapping.

• Metric tensor

  The metric (or basic, fundamental) tensor is a symmetric tensor of rank 2, that is used to measure distances and angles in a real n-dimensional differentiable manifold M ⁿ. Once a local coordinate system (x _i ) _i is chosen, the metric tensor appears as a real symmetric n × n matrix ((g _ij )).

  The assignment of a metric tensor on M ⁿ introduces a scalar product (i.e., symmetric bilinear, but in general not positive-definite, form) $\langle,\rangle _{p}$ on the tangent space T _p (M ⁿ) at any p ∈ M ⁿ defined by

  $$\displaystyle{\langle x,y\rangle _{p} = g_{p}(x,y) =\sum _{i,j}g_{\mathit{ij}}(p)x_{i}y_{j},}$$

  where $x = (x_{1},\ldots,x_{n})$, $y = (y_{1},\ldots,y_{n}) \in T_{p}(M^{n})$. The collection of all these scalar products is called the metric g with the metric tensor ((g _ij )). The length ds of the vector $(\mathit{dx}_{1},\ldots,\mathit{dx}_{n})$ is expressed by the quadratic differential form

  $$\displaystyle{\mathit{ds}^{2} =\sum _{ i,j}g_{\mathit{ij}}\mathit{dx}_{i}\mathit{dx}_{j},}$$

  which is called the line element (or first fundamental form) of the metric g.

  The length of a curve γ is expressed by the formula $\int _{\gamma }\sqrt{\sum _{i,j } g_{\mathit{ij } } \mathit{dx } _{i } \mathit{dx } _{j}}$. In general it may be real, purely imaginary, or zero (an isotropic curve).

  Let p, q and r be the numbers of positive, negative and zero eigenvalues of the matrix ((g _ij )) of the metric g; so, $p + q + r = n$. The metric signature (or, simply, signature) of g is the pair (p, q). A nondegenerated metric (i.e., one with r = 0) is Riemannian or pseudo-Riemannian if its signature is positive-definite (q = 0) or indefinite (pq > 0), respectively.

  The nonmetricity tensor is the covariant derivative of a metric tensor. It is 0 for Riemannian metrics but can be ≠ 0 for pseudo-Riemannian ones.

• Nondegenerate metric

  A nondegenerate metric is a metric g with the metric tensor ((g _ij )), for which the metric discriminant det((g _ij )) ≠ 0. All Riemannian and pseudo-Riemannian metrics are nondegenerate.

  A degenerate metric is a metric g with det((g _ij )) = 0 (cf. semi-Riemannian metric and semi-pseudo-Riemannian metric). A manifold with a degenerate metric is called an isotropic manifold.

• Diagonal metric

  A diagonal metric is a metric g with a metric tensor ((g _ij )) which is zero for i ≠ j. The Euclidean metric is a diagonal metric, as its metric tensor has the form $g_{\mathit{ii}} = 1,g_{\mathit{ij}} = 0$ for i ≠ j.

• Riemannian metric

  Consider a real n-dimensional differentiable manifold M ⁿ in which each tangent space is equipped with an inner product (i.e., a symmetric positive-definite bilinear form) which varies smoothly from point to point.

  A Riemannian metric on M ⁿ is a collection of inner products $\langle,\rangle _{p}$ on the tangent spaces T _p (M ⁿ), one for each p ∈ M ⁿ.

  Every inner product $\langle,\rangle _{p}$ is completely defined by inner products $\langle e_{i},e_{j}\rangle _{p} = g_{\mathit{ij}}(p)$ of elements $e_{1},\ldots,e_{n}$ of a standard basis in $\mathbb{E}^{n}$, i.e., by the real symmetric and positive-definite n × n matrix ((g _ij )) = ((g _ij (p))), called a metric tensor. In fact, $\langle x,y\rangle _{p} =\sum _{i,j}g_{\mathit{ij}}(p)x_{i}y_{j}$, where $x = (x_{1},\ldots,x_{n})$ and $y = (y_{1},\ldots,y_{n}) \in T_{p}(M^{n})$. The smooth function g completely determines the Riemannian metric.

  A Riemannian metric on M ⁿ is not an ordinary metric on M ⁿ. However, for a connected manifold M ⁿ, every Riemannian metric on M ⁿ induces an ordinary metric on M ⁿ, in fact, the intrinsic metric of M ⁿ,

  For any points p, q ∈ M ⁿ the Riemannian distance between them is defined as

  $$\displaystyle{\inf _{\gamma }\int _{0}^{1}\langle \frac{d\gamma } {\mathit{dt}}, \frac{d\gamma } {\mathit{dt}}\rangle ^{\frac{1} {2} }\mathit{dt} =\inf _{\gamma }\int _{0}^{1}\sqrt{\sum _{i,j } g_{\mathit{ij } } \frac{\mathit{dx } _{i } } {\mathit{dt}} \frac{\mathit{dx}_{j}} {\mathit{dt}}} \mathit{dt},}$$

  where the infimum is over all rectifiable curves γ: [0, 1] → M ⁿ, connecting p and q.

  A Riemannian manifold (or Riemannian space) is a real n-dimensional differentiable manifold M ⁿ equipped with a Riemannian metric. The theory of Riemannian spaces is called Riemannian Geometry. The simplest examples of Riemannian spaces are Euclidean spaces, hyperbolic spaces, and elliptic spaces.

• Conformal metric

  A conformal structure on a vector space V is a class of pairwise-homothetic Euclidean metrics on V. Any Euclidean metric d _E on V defines a conformal structure {λ d _E : λ > 0}.

  A conformal structure on a manifold is a field of conformal structures on the tangent spaces or, equivalently, a class of conformally equivalent Riemannian metrics. Two Riemannian metrics g and h on a smooth manifold M ⁿ are called conformally equivalent if g = f ⋅ h for some positive function f on M ⁿ, called a conformal factor.

  A conformal metric is a Riemannian metric that represents the conformal structure. Cf. conformally invariant metric in Chap. 8.

• Conformal space

  The conformal space (or inversive space) is the Euclidean space $\mathbb{E}^{n}$ extended by an ideal point (at infinity). Under conformal transformations, i.e., continuous transformations preserving local angles, the ideal point can be taken to be an ordinary point. Therefore, in a conformal space a sphere is indistinguishable from a plane: a plane is a sphere passing through the ideal point.

  Conformal spaces are considered in Conformal (or angle-preserving, Möbius) Geometry in which properties of figures are studied that are invariant under conformal transformations. It is the set of transformations that map spheres into spheres, i.e., generated by the Euclidean transformations together with inversions which in coordinate form are conjugate to $x_{i} \rightarrow \frac{r^{2}x_{ i}} {\sum _{j}x_{j}^{2}}$, where r is the radius of the inversion. An inversion in a sphere becomes an everywhere well defined automorphism of period two. Any angle inverts into an equal angle.

  The 2D conformal space is the Riemann sphere, on which the conformal transformations are given by the Möbius transformations $z \rightarrow \frac{\mathit{az}+b} {\mathit{cz}+d}$, ad −bc ≠ 0.

  In general, a conformal mapping between two Riemannian manifolds is a diffeomorphism between them such that the pulled back metric is conformally equivalent to the original one. A conformal Euclidean space is a Riemannian space admitting a conformal mapping onto an Euclidean space.

  In the General Theory of Relativity, conformal transformations are considered on the Minkowski space $\mathbb{R}^{1,3}$ extended by two ideal points.

• Space of constant curvature

  A space of constant curvature is a Riemannian space M ⁿ for which the sectional curvature K(σ) is constant in all 2D directions σ.

  A space form is a connected complete space of constant curvature k. Examples of a flat space form, i.e., with k = 0, are the Euclidean space and flat torus. The sphere and hyperbolic space are space forms with k > 0 and k < 0, respectively.

• Generalized Riemannian space

  A generalized Riemannian space is a metric space with the intrinsic metric, subject to certain restrictions on the curvature. Such spaces include spaces of bounded curvature, Riemannian spaces, etc. They are defined and investigated on the basis of their metric alone, without coordinates.

  A space of bounded curvature ( ≤ k and ≥ k ^′) is defined by the condition: for any sequence of geodesic triangles T _n contracting to a point, we have

  $$\displaystyle{k \geq \overline{\lim }\frac{\overline{\delta }(T_{n})} {\sigma (T_{n}^{0})} \geq \underline{\lim } \frac{\overline{\delta }(T_{n})} {\sigma (T_{n}^{0})} \geq k^{'},}$$

  where a geodesic triangle T = xyz is the triplet of geodesic segments [x, y], [y, z], [z, x] (the sides of T) connecting in pairs three different points x, y, z, $\overline{\delta }(T) =\alpha +\beta +\gamma -\pi$ is the excess of the geodesic triangle T, and σ(T ⁰) is the area of a Euclidean triangle T ⁰ with the sides of the same lengths. The intrinsic metric on the space of bounded curvature is called a metric of bounded curvature.

  Such a space turns out to be Riemannian under two additional conditions: local compactness of the space (this ensures the condition of local existence of geodesics), and local extendability of geodesics. If in this case k = k ^′, it is a Riemannian space of constant curvature k (cf. space of geodesics in Chap. 6).

  A space of curvature ≤ k is defined by the condition $\overline{\lim } \frac{\overline{\delta }(T_{n})} {\sigma (T_{n}^{0})} \leq k$. In such a space any point has a neighborhood in which the sum $\alpha +\beta +\gamma$ of the angles of a geodesic triangle T does not exceed the sum $\alpha _{k} +\beta _{k} +\gamma _{k}$ of the angles of a triangle T ^k with sides of the same lengths in a space of constant curvature k. The intrinsic metric of such space is called a k -concave metric.

  A space of curvature ≥ k is defined by the condition $\underline{\lim } \frac{\overline{\delta }(T_{n})} {\sigma (T_{n}^{0})} \geq k$. In such a space any point has a neighborhood in which $\alpha +\beta +\gamma \geq \alpha _{k} +\beta _{k} +\gamma _{k}$ for triangles T and T ^k. The intrinsic metric of such space is called a K -concave metric.

  An Alexandrov metric space is a generalized Riemannian space with upper, lower or integral curvature bounds. Cf. a CAT( κ ₁ ) space in Chap. 6.

• Complete Riemannian metric

  A Riemannian metric g on a manifold M ⁿ is called complete if M ⁿ forms a complete metric space with respect to g.

  Any Riemannian metric on a compact manifold is complete.

• Ricci-flat metric

  A Ricci-flat metric is a Riemannian metric with vanished Ricci curvature tensor.

  A Ricci-flat manifold is a Riemannian manifold equipped with a Ricci-flat metric. Ricci-flat manifolds represent vacuum solutions to the Einstein field equation, and are special cases of Kähler–Einstein manifolds. Important Ricci-flat manifolds are Calabi–Yau manifolds, and hyper-Kähler manifolds.

• Osserman metric

  An Osserman metric is a Riemannian metric for which the Riemannian curvature tensor R is Osserman, i.e., the eigenvalues of the Jacobi operator $\mathcal{J} (x): y \rightarrow R(y,x)x$ are constant on the unit sphere S ⁿ⁻¹ in $\mathbb{E}^{n}$ (they are independent of the unit vectors x).

• G-invariant Riemannian metric

  Given a Lie group (G, ⋅ , id) of transformations, a Riemannian metric g on a differentiable manifold M ⁿ is called G -invariant, if it does not change under any x ∈ G. The group (G, ⋅ , id) is called the group of motions (or group of isometries) of the Riemannian space (M ⁿ, g). Cf. G -invariant metric in Chap. 10.

• Ivanov–Petrova metric

  Let R be the Riemannian curvature tensor of a Riemannian manifold M ⁿ, and let {x, y} be an orthogonal basis for an oriented 2-plane π in the tangent space T _p (M ⁿ) at a point p of M ⁿ.

  The Ivanov–Petrova metric is a Riemannian metric on M ⁿ for which the eigenvalues of the antisymmetric curvature operator $\mathcal{R}(\pi ) = R(x,y)$ [IvSt95] depend only on the point p of a Riemannian manifold M ⁿ, but not upon the plane π.

• Zoll metric

  A Zoll metric is a Riemannian metric on a smooth manifold M ⁿ whose geodesics are all simple closed curves of an equal length. A 2D sphere S ² admits many such metrics, besides the obvious metrics of constant curvature. In terms of cylindrical coordinates (z, θ) (z ∈ [−1, 1], θ ∈ [0, 2π]), the line element

  $$\displaystyle{\mathit{ds}^{2} = \frac{(1 + f(z))^{2}} {1 - z^{2}} \mathit{dz}^{2} + (1 - z^{2})d\theta ^{2}}$$

  defines a Zoll metric on S ² for any smooth odd function $f: [-1,1] \rightarrow (-1,1)$ which vanishes at the endpoints of the interval.

• Berger metric

  The Berger metric is a Riemannian metric on the Berger sphere (i.e., the three-sphere S ³ squashed in one direction) defined by the line element

  $$\displaystyle{\mathit{ds}^{2} = d\theta ^{2} +\sin ^{2}\theta d\phi ^{2} +\cos ^{2}\alpha (d\psi +\cos \theta d\phi )^{2},}$$

  where α is a constant, and θ, ϕ, ψ are Euler angles.

• Cycloidal metric

  The cycloidal metric is a Riemannian metric on the half-plane $\mathbb{R}_{+}^{2} =\{ x \in \mathbb{R}^{2}: x_{2} > 0\}$ defined by the line element

  $$\displaystyle{\mathit{ds}^{2} = \frac{\mathit{dx}_{1}^{2} + \mathit{dx}_{ 2}^{2}} {2x_{2}}.}$$

  It is called cycloidal because its geodesics are cycloid curves. The corresponding distance d(x, y) between two points $x,y \in \mathbb{R}_{+}^{2}$ is equivalent to the distance

  $$\displaystyle{\rho (x,y) = \frac{\vert x_{1} - y_{1}\vert + \vert x_{2} - y_{2}\vert } {\sqrt{x_{1}} + \sqrt{x_{2}} + \sqrt{\vert x_{2 } - y_{2 } \vert }}}$$

  in the sense that d ≤ C ρ, and ρ ≤ Cd for some positive constant C.

• Klein metric

  The Klein metric is a Riemannian metric on the open unit ball $B^{n} =\{ x \in \mathbb{R}^{n}: \vert \vert x\vert \vert _{2} < 1\}$ in $\mathbb{R}^{n}$ defined by

  $$\displaystyle{\frac{\sqrt{\vert \vert y\vert \vert _{2 }^{2 } - (\vert \vert x\vert \vert _{2 }^{2 }\vert \vert y\vert \vert _{2 }^{2 } -\langle x, y\rangle ^{2 } )}} {1 -\vert \vert x\vert \vert _{2}^{2}} }$$

  for any x ∈ B ⁿ and y ∈ T _x (B ⁿ), where | | . | | ₂ is the Euclidean norm on $\mathbb{R}^{n}$, and $\langle,\rangle$ is the ordinary inner product on $\mathbb{R}^{n}$.

  The Klein metric is the hyperbolic case $a = -1$ of the general form

  $$\displaystyle{\frac{\sqrt{(1 + a\vert \vert x\vert \vert ^{2 } )\vert \vert y\vert \vert ^{2 } - a\langle x, y\rangle ^{2}}} {1 + a\vert \vert x\vert \vert ^{2}},}$$

  while a = 0, 1 correspond to the Euclidean and spherical cases.

• Carnot–Carathéodory metric

  A distribution (or polarization) on a manifold M ⁿ is a subbundle of the tangent bundle T(M ⁿ) of M ⁿ. Given a distribution H(M ⁿ), a vector field in H(M ⁿ) is called horizontal. A curve γ on M ⁿ is called horizontal (or distinguished, admissible) with respect to H(M ⁿ) if γ ^′(t) ∈ H _γ(t)(M ⁿ) for any t.

  A distribution H(M ⁿ) is called completely nonintegrable if the Lie brackets of H(M ⁿ), i.e., [⋯ , [H(M ⁿ), H(M ⁿ)]], span the tangent bundle T(M ⁿ), i.e., for all p ∈ M ⁿ any tangent vector v from T _p (M ⁿ) can be presented as a linear combination of vectors of the following types: u, [u, w], [u, [w, t]], $[u,[w,[t,s]]],\ldots \in T_{p}(M^{n})$, where all vector fields $u,w,t,s,\ldots$ are horizontal.

  The Carnot–Carathéodory metric (or CC metric , sub-Riemannian metric , control metric) is a metric on a manifold M ⁿ with a completely nonintegrable horizontal distribution H(M ⁿ) defined as the section g _C of positive-definite scalar products on H(M ⁿ). The distance d _C (p, q) between any points p, q ∈ M ⁿ is defined as the infimum of the g _C -lengths of the horizontal curves joining p and q.

  A sub-Riemannian manifold (or polarized manifold) is a manifold M ⁿ equipped with a Carnot–Carathéodory metric. It is a generalization of a Riemannian manifold. Roughly, in order to measure distances in a sub-Riemannian manifold, one is allowed to go only along curves tangent to horizontal spaces.

• Pseudo-Riemannian metric

  Consider a real n-dimensional differentiable manifold M ⁿ in which every tangent space T _p (M ⁿ), p ∈ M ⁿ, is equipped with a scalar product which varies smoothly from point to point and is nondegenerate, but indefinite.

  A pseudo-Riemannian metric on M ⁿ is a collection of scalar products $\langle,\rangle _{p}$ on the tangent spaces T _p (M ⁿ), p ∈ M ⁿ, one for each p ∈ M ⁿ.

  Every scalar product $\langle,\rangle _{p}$ is completely defined by scalar products $\langle e_{i},e_{j}\rangle _{p} = g_{\mathit{ij}}(p)$ of elements $e_{1},\ldots,e_{n}$ of a standard basis in $\mathbb{E}^{n}$, i.e., by the real symmetric indefinite n × n matrix ((g _ij )) = ((g _ij (p))), called a metric tensor (cf. Riemannian metric in which case this tensor is not only nondegenerate but, moreover, positive-definite).

  In fact, $\langle x,y\rangle _{p} =\sum _{i,j}g_{\mathit{ij}}(p)x_{i}y_{j}$, where $x = (x_{1},\ldots,x_{n})$ and $y = (y_{1},\ldots,y_{n}) \in T_{p}(M^{n})$. The smooth function g determines the pseudo-Riemannian metric.

  The length ds of the vector $(\mathit{dx}_{1},\ldots,\mathit{dx}_{n})$ is given by the quadratic differential form

  $$\displaystyle{\mathit{ds}^{2} =\sum _{ i,j}g_{\mathit{ij}}\mathit{dx}_{i}\mathit{dx}_{j}.}$$

  The length of a curve γ: [0, 1] → M ⁿ is expressed by the formula

  $$\displaystyle{\int _{\gamma }\sqrt{\sum _{i,j } g_{\mathit{ij } } \mathit{dx } _{i } \mathit{dx } _{j}} =\int _{ 0}^{1}\sqrt{\sum _{ i,j}g_{\mathit{ij}}\frac{\mathit{dx}_{i}} {\mathit{dt}} \frac{\mathit{dx}_{j}} {\mathit{dt}}} \mathit{dt}.}$$

  In general it may be real, purely imaginary or zero (an isotropic curve).

  A pseudo-Riemannian metric on M ⁿ is a metric with a fixed, but indefinite signature (p, q), $p + q = n$. A pseudo-Riemannian metric is nondegenerate, i.e., its metric discriminant det((g _ij )) ≠ 0. Therefore, it is a nondegenerate indefinite metric.

  A pseudo-Riemannian manifold (or pseudo-Riemannian space) is a real n-dimensional differentiable manifold M ⁿ equipped with a pseudo-Riemannian metric. The theory of pseudo-Riemannian spaces is called Pseudo-Riemannian Geometry.

• Pseudo-Euclidean distance

  The model space of a pseudo-Riemannian space of signature (p, q) is the pseudo-Euclidean space $\mathbb{R}^{p,q}$, $p + q = n$ which is a real n-dimensional vector space $\mathbb{R}^{n}$ equipped with the metric tensor ((g _ij )) of signature (p, q) defined, for i ≠ j, by $g_{11} =\ldots = g_{\mathit{pp}} = 1$, $g_{p+1,p+1} =\ldots = g_{\mathit{nn}} = -1$, g _ij  = 0.

  The line element of the corresponding metric is given by

  $$\displaystyle{\mathit{ds}^{2} = \mathit{dx}_{ 1}^{2} +\ldots +\mathit{dx}_{ p}^{2} -\mathit{dx}_{ p+1}^{2} -\ldots -\mathit{dx}_{ n}^{2}.}$$

  The pseudo-Euclidean distance of signature $(p,q = n - p)$ on $\mathbb{R}^{n}$ is defined by

  $$\displaystyle{d_{\mathit{pE}}^{2}(x,y) = D(x,y) =\sum _{ i=1}^{p}(x_{ i} - y_{i})^{2} -\sum _{ i=p+1}^{n}(x_{ i} - y_{i})^{2}.}$$

  Such a pseudo-Euclidean space can be seen as $\mathbb{R}^{p} \times i\mathbb{R}^{q}$, where $i = \sqrt{-1}$.

  The pseudo-Euclidean space with (p, q) = (1, 3) is used as flat space-time model of Special Relativity; cf. Minkowski metric in Chap. 26.

  The points correspond to events; the line spanned by x and y is space-like if D(x, y) > 0 and time-like if D(x, y) < 0. If D(x, y) > 0, then $\sqrt{D(x, y)}$ is Euclidean distance and if D(x, y) < 0, then $\sqrt{\vert D(x, y)\vert }$ is the lifetime of a particle (from x to y).

  The pseudo-Euclidean distance of signature $(p,q = n - p)$ is the case A = diag(a _i ) with a _i  = 1 for 1 ≤ i ≤ p and $a_{i} = -1$ for p + 1 ≤ i ≤ n, of the weighted Euclidean distance $\sqrt{\sum _{1\leq i\leq n } a_{i } (x_{i } - y_{i } )^{2}}$ in Chap. 17.

• Blaschke metric

  The Blaschke metric on a nondegenerate hypersurface is a pseudo-Riemannian metric, associated to the affine normal of the immersion $\phi: M^{n} \rightarrow \mathbb{R}^{n+1}$, where M ⁿ is an n-dimensional manifold, and $\mathbb{R}^{n+1}$ is considered as an affine space.

• Semi-Riemannian metric

  A semi-Riemannian metric on a real n-dimensional differentiable manifold M ⁿ is a degenerate Riemannian metric, i.e., a collection of positive-semidefinite scalar products $\langle x,y\rangle _{p} =\sum _{i,j}g_{\mathit{ij}}(p)x_{i}y_{j}$ on the tangent spaces T _p (M ⁿ), p ∈ M ⁿ; the metric discriminant det((g _ij )) = 0.

  A semi-Riemannian manifold (or semi-Riemannian space) is a real n-dimensional differentiable manifold M ⁿ equipped with a semi-Riemannian metric.

  The model space of a semi-Riemannian manifold is the semi-Euclidean space R _d ⁿ, d ≥ 1 (sometimes denoted also by $\mathbb{R}_{n-d}^{n}$), i.e., a real n-dimensional vector space $\mathbb{R}^{n}$ equipped with a semi-Riemannian metric.

  It means that there exists a scalar product of vectors such that, relative to a suitably chosen basis, the scalar product $\langle x,x\rangle$ has the form $\langle x,x\rangle =\sum _{ i=1}^{n-d}x_{i}^{2}$. The number d ≥ 1 is called the defect (or deficiency) of the space.

• Grushin metric

  The Grushin metric is a semi-Riemannian metric on $\mathbb{R}^{2}$ defined by the line element

  $$\displaystyle{\mathit{ds}^{2} = \mathit{dx}_{ 1}^{2} + \frac{\mathit{dx}_{2}^{2}} {x_{1}^{2}}.}$$

• Agmon distance

  The Agmon metric attached to an energy E and a potential V is defined as

  $$\displaystyle{\mathit{ds}^{2} =\max \{ 0,V (x) - E_{ 0}(h)\}\mathit{dx}^{2},}$$

  where dx ² is the standard metric on $\mathbb{R}^{d}$. Then the Agmon distance on $\mathbb{R}^{d}$ is the corresponding Riemannian distance defined, for any $x,y \in \mathbb{R}^{d}$, by

  $$\displaystyle{\inf _{\gamma }\{\int _{0}^{1}\sqrt{\max \{V (\gamma (s)) - E_{ 0}(h),0\}} \cdot \vert \gamma ^{'}(s)\vert \mathit{ds}:\gamma (0) = x,\gamma (1) = y,\gamma \in C^{1}\}.}$$

• Semi-pseudo-Riemannian metric

  A semi-pseudo-Riemannian metric on a real n-dimensional differentiable manifold M ⁿ is a degenerate pseudo-Riemannian metric, i.e., a collection of degenerate indefinite scalar products $\langle x,y\rangle _{p} =\sum _{i,j}g_{\mathit{ij}}(p)x_{i}y_{j}$ on the tangent spaces T _p (M ⁿ), p ∈ M ⁿ; the metric discriminant det((g _ij )) = 0. In fact, a semi-pseudo-Riemannian metric is a degenerate indefinite metric.

  A semi-pseudo-Riemannian manifold (or semi-pseudo-Riemannian space) is a real n-dimensional differentiable manifold M ⁿ equipped with a semi-pseudo-Riemannian metric. The model space of such manifold is the semi-pseudo-Euclidean space $\mathbb{R}_{_{m_{ 1},\ldots,m_{r-1}}^{l_{1},\ldots,l_{r}}}^{n}$, i.e., a vector space $\mathbb{R}^{n}$ equipped with a semi-pseudo-Riemannian metric.

  It means that there exist r scalar products $\langle x,y\rangle _{a} =\sum \epsilon _{i_{a}}x_{i_{a}}y_{i_{a}}$, where $a =\ 1,\ldots r$, $0 = m_{0} < m_{1} <\ldots < m_{r} = n$, $i_{a} = m_{a-1} + 1,\ldots m_{a}$, $\epsilon _{i_{a}} = \pm 1$, and − 1 occurs l _a times among the numbers $\epsilon _{i_{a}}$. The product $\langle x,y\rangle _{a}$ is defined for those vectors for which all coordinates x _i , i ≤ m _a−1 or i > m _a + 1 are zero.

  The first scalar square of an arbitrary vector x is a degenerate quadratic form $\langle x,x\rangle _{1} = -\sum _{i=1}^{l_{1}}x_{i}^{2} +\sum _{ j=l_{ 1}+1}^{n-d}x_{ j}^{2}$. The number l ₁ ≥ 0 is called the index, and the number $d = n - m_{1}$ is called the defect of the space. If $l_{1} =\ldots = l_{r} = 0$, we obtain a semi-Euclidean space. The spaces $\mathbb{R}_{_{m}^{}}^{n}$ and $\mathbb{R}_{_{m}^{k,l}}^{n}$ are called quasi-Euclidean spaces.

  The semi-pseudo-non-Euclidean space $\mathbb{S}_{_{m_{ 1},\ldots,m_{r-1}}^{l_{1},\ldots,l_{r}}}^{n}$ is a hypersphere in $\mathbb{R}_{_{m_{ 1},\ldots,m_{r-1}}^{l_{1},\ldots,l_{r}}}^{n+1}$ with identified antipodal points. It is called semielliptic (or semi-non-Euclidean) space if $l_{1} =\ldots = l_{r} = 0$ and a semihyperbolic space if there exist l _i ≠ 0.

• Finsler metric

  Consider a real n-dimensional differentiable manifold M ⁿ in which every tangent space T _p (M ⁿ), p ∈ M ⁿ, is equipped with a Banach norm | | . | | such that the Banach norm as a function of position is smooth, and the matrix ((g _ij )),

  $$\displaystyle{g_{\mathit{ij}} = g_{\mathit{ij}}(p,x) = \frac{1} {2} \frac{\partial ^{2}\vert \vert x\vert \vert ^{2}} {\partial x_{i}\partial x_{j}},}$$

  is positive-definite for any p ∈ M ⁿ and any x ∈ T _p (M ⁿ).

  A Finsler metric on M ⁿ is a collection of Banach norms | | . | | on the tangent spaces T _p (M ⁿ), one for each p ∈ M ⁿ. Its line element has the form

  $$\displaystyle{\mathit{ds}^{2} =\sum _{ i,j}g_{\mathit{ij}}\mathit{dx}_{i}\mathit{dx}_{j}.}$$

  The Finsler metric can be given by fundamental function, i.e., a real positive-definite convex function F(p, x) of p ∈ M ⁿ and x ∈ T _p (M ⁿ) acting at the point p. F(p, x) is positively homogeneous of degree one in x: F(p, λ x) = λ F(p, x) for every λ > 0. Then F(p, x) is the length of the vector x.

  The Finsler metric tensor has the form $((g_{\mathit{ij}})) = ((\frac{1} {2} \frac{\partial ^{2}F^{2}(p,x)} {\partial x_{i}\partial x_{j}} ))$. The length of a curve γ: [0, 1] → M ⁿ is given by $\int _{0}^{1}F(p, \frac{\mathit{dp}} {\mathit{dt}} )\mathit{dt}$. For each fixed p the Finsler metric tensor is Riemannian in the variables x.

  The Finsler metric is a generalization of the Riemannian metric, where the general definition of the length | | x | | of a vector x ∈ T _p (M ⁿ) is not necessarily given in the form of the square root of a symmetric bilinear form as in the Riemannian case.

  A Finsler manifold (or Finsler space) is a real differentiable n-manifold M ⁿ equipped with a Finsler metric. The theory of such spaces is Finsler Geometry.

  The difference between a Riemannian space and a Finsler space is that the former behaves locally like a Euclidean space, and the latter locally like a Minkowskian space or, analytically, the difference is that to an ellipsoid in the Riemannian case there corresponds an arbitrary convex surface which has the origin as the center.

  A pseudo-Finsler metric F is defined by weakening the definition of a Finsler metric): ((g _ij )) should be nondegenerate and of constant signature (not necessarily positive-definite) and hence F could be negative. The pseudo-Finsler metric is a generalization of the pseudo-Riemannian metric.

• (α, β)-metric

  Let $\alpha (x,y) = \sqrt{\alpha _{\mathit{ij } } (x)y^{i } y^{j}}$ be a Riemannian metric and β(x, y) = b _i (x)y ⁱ be a 1-form on a n-dimensional manifold M ⁿ. Let $s = \frac{\beta }{\alpha }$ and ϕ(s) is an C ^∞-positive function on some symmetric interval (−r, r) with $r > \frac{\beta }{\alpha }$ for all (x, y) in the tangent bundle $\mathit{TM} = \cup _{x\in M}T_{x}(M^{n})$ of the tangent spaces T _x (M ⁿ). Then F = α ϕ(s) is a Finsler metric (Matsumoto, 1972) called an (α, β)-metric. The main examples of (α, β)-metrics follow.

  The Kropina metric is the case $\phi (s) = \frac{1} {s}$, i.e., $F = \frac{\alpha ^{2}} {\beta }$.

  The generalized Kropina metric is the case ϕ(s) = s ^m, i.e., $F = \beta ^{m}\alpha ^{1-m}$.

  The Randers metric (1941) is the case $\phi (s) = 1 + s$, i.e., $F =\alpha +\beta$.

  The Matsumoto slope metric is the case $\phi (s) = \frac{1} {1-s}$, i.e., $F = \frac{\alpha ^{2}} {\alpha -\beta }$.

  The Riemann-type (α, β)-metric is the case $\phi (s) = \sqrt{1 + s^{2}}$, i.e., $F = \alpha ^{2} +\ \beta ^{2}$.

  Park and Lee, 1998, considered the case $\phi (s) = 1 + s^{2}$, i.e., $F =\alpha +\frac{\beta ^{2}} {\alpha }$.

• Shen metric

  Given a vector $a \in \mathbb{R}^{n}$, | | a | | ₂ < 1, the Shen metric (2003) is a Finsler metric on the open unit ball $B^{n} =\{ x \in \mathbb{R}^{n}: \vert \vert x\vert \vert _{2} < 1\}$ in $\mathbb{R}^{n}$ defined by

  $$\displaystyle{\frac{\sqrt{\vert \vert y\vert \vert _{2 }^{2 } - (\vert \vert x\vert \vert _{2 }^{2 }\vert \vert y\vert \vert _{2 }^{2 } -\langle x, y\rangle ^{2 } )} +\langle x,y\rangle } {1 -\vert \vert x\vert \vert _{2}^{2}} + \frac{\langle a,y\rangle } {1 +\langle a,x\rangle }}$$

  for any x ∈ B ⁿ and y ∈ T _x (B ⁿ), where | | . | | ₂ is the Euclidean norm on $\mathbb{R}^{n}$, and $\langle,\rangle$ is the ordinary inner product on $\mathbb{R}^{n}$. It is a Randers metric and a projective metric. Cf. Klein metric and Berwald metric.

• Berwald metric

  The Berwald metric (1929) is a Finsler metric F _Be on the open unit ball $B^{n} =\{ x \in \mathbb{R}^{n}: \vert \vert x\vert \vert _{2} < 1\}$ in $\mathbb{R}^{n}$ defined, for any x ∈ B ⁿ and y ∈ T _x (B ⁿ), by

  $$\displaystyle{ \frac{\left (\sqrt{\vert \vert y\vert \vert _{2 }^{2 } - (\vert \vert x\vert \vert _{2 }^{2 }\vert \vert y\vert \vert _{2 }^{2 } -\langle x, y\rangle ^{2 } )} +\langle x,y\rangle \right )^{2}} {(1 -\vert \vert x\vert \vert _{2}^{2})^{2}\sqrt{\vert \vert y\vert \vert _{2 }^{2 } - (\vert \vert x\vert \vert _{2 }^{2 }\vert \vert y\vert \vert _{2 }^{2 } -\langle x, y\rangle ^{2 } )}},}$$

  where | | . | | ₂ is the Euclidean norm on $\mathbb{R}^{n}$, and $\langle,\rangle$ is the inner product on $\mathbb{R}^{n}$. It is a projective metric and an (α, β)-metric with $\phi (s) = (1 + s)^{2}$, i.e., $F = \frac{(\alpha +\beta )^{2}} {\alpha }$.

  The Riemannian metrics are special Berwald metrics. Every Berwald metric is affinely equivalent to a Riemannian metric.

  In general, every Finsler metric on a manifold M ⁿ induces a spray (second-order homogeneous ordinary differential equation) $y_{i} \frac{\partial } {\partial x_{i}} - 2G^{i} \frac{\partial } {\partial y_{i}}$ which determines the geodesics. A Finsler metric is a Berwald metric if the spray coefficients G ⁱ = G ⁱ(x, y) are quadratic in y ∈ T _x (M ⁿ) at any point x ∈ M ⁿ, i.e., $G^{i} = \frac{1} {2}\Gamma _{\mathit{jk}}^{i}(x)y^{j}y^{k}$.

  A Finsler metric is a more general Landsberg metric $\Gamma _{\mathit{jk}}^{i}\!=\frac{1} {2}\partial _{y^{j}}\partial _{y^{k}}(\Gamma _{\mathit{jk}}^{i}(x)y^{j}y^{k})$. The Landsberg metric is the one for which the Landsberg curvature (the covariant derivative of the Cartan torsion along a geodesic) is zero.

• Douglas metric

  A Douglas metric a Finsler metric for which the spray coefficients G ⁱ = G ⁱ(x, y) have the following form:

  $$\displaystyle{G^{i} = \frac{1} {2}\Gamma _{\mathit{jk}}^{i}(x)y_{ i}y_{k} + P(x,y)y_{i}.}$$

  Every Finsler metric which is projectively equivalent to a Berwald metric is a Douglas metric. Every Berwald metric is a Douglas metric. Every known Douglas metric is (locally) projectively equivalent to a Berwald metric.

• Bryant metric

  Let α be an angle with $\vert \alpha \vert < \frac{\pi } {2}$. Let, for any $x,y \in \mathbb{R}^{n}$, $A = \vert \vert y\vert \vert _{2}^{4}\sin ^{2}2\alpha + \left (\vert \vert y\vert \vert _{2}^{2}\cos 2\alpha + \vert \vert x\vert \vert _{2}^{2}\vert \vert y\vert \vert _{2}^{2} -\langle x,y\rangle ^{2}\right )^{2}$, $B = \vert \vert y\vert \vert _{2}^{2}\cos 2\alpha + \vert \vert x\vert \vert _{2}^{2}\vert \vert y\vert \vert _{2}^{2} -\langle x,y\rangle ^{2}$, $C =\langle x,y\rangle \sin 2\alpha$, $D = \vert \vert x\vert \vert _{2}^{4} + 2\vert \vert x\vert \vert _{2}^{2}\cos 2\alpha + 1$. Then we get a Finsler metric

  $$\displaystyle{\sqrt{\frac{\sqrt{A} + B} {2D} + \left (\frac{C} {D}\right )^{2}} + \frac{C} {D}.}$$

  On the 2D unit sphere S ², it is the Bryant metric (1996).

• m -th root pseudo-Finsler metric

  An m -th root pseudo-Finsler metric is (Shimada, 1979) a pseudo-Finsler metric defined (with $a_{i_{1}\ldots i_{m}}$ symmetric in all its indices) by

  $$\displaystyle{F(x,y) = (a_{i_{1}\ldots i_{m}}(x)y^{i_{1}\ldots i_{m} })^{ \frac{1} {m} }.}$$

  For m = 2, it is a pseudo-Riemannian metric. The 3rd and 4th root pseudo-Finsler metrics are called cubic metric and quartic metric, respectively.

• Antonelli–Shimada metric

  The Antonelli–Shimada metric (or ecological Finsler metric) is an m -th root pseudo-Finsler metric defined, via linearly independent 1-forms a ⁱ, by

  $$\displaystyle{F(x,y) = \left (\sum _{i=1}^{n}(a^{i})^{m}\right )^{ \frac{1} {m} }.}$$

  The Uchijo metric is defined, for a constant k, by

  $$\displaystyle{F(x,y) = \left (\sum _{i=1}^{n}(a^{i})^{2}\right )^{\frac{1} {2} } + \mathit{ka}^{1}.}$$

• Berwald–Moör metric

  The Berwald–Moör metric is an m -th root pseudo-Finsler metric, defined by

  $$\displaystyle{F(x,y) = (y^{1}\ldots y^{n})^{ \frac{1} {n} }.}$$

  More general Asanov metric is defined, via linearly independent 1-forms a ⁱ, by

  $$\displaystyle{F(x,y) = (a^{1}\ldots a^{n})^{ \frac{1} {n} }.}$$

  The Berwald–Moör metrics with n = 4 and n = 6 are applied in Relativity Theory and Diffusion Imaging, respectively. The pseudo-Finsler spaces which are locally isomorphic to the 4th root Berwald–Moör metric, are expected to be more general and productive space-time models than usual pseudo-Riemannian spaces, which are locally isomorphic to the Minkowski metric.

• Kawaguchi metric

  The Kawaguchi metric is a metric on a smooth n-dimensional manifold M ⁿ, given by the arc element ds of a regular curve x = x(t), t ∈ [t ₀, t ₁] via the formula

  $$\displaystyle{\mathit{ds} = F(x, \frac{\mathit{dx}} {\mathit{dt}},\ldots, \frac{d^{k}x} {\mathit{dt}^{k}} )\mathit{dt},}$$

  where the metric function F satisfies Zermelo’s conditions: $\sum _{s=1}^{k}sx^{(s)}F_{(s)i}=\ F$, $\sum _{s=r}^{k}(_{k}^{s})x^{(s-r+1)i}F_{(s)i} = 0$, $x^{(s)i} = \frac{d^{s}x^{i}} {\mathit{dt}^{s}}$, $F_{(s)i} = \frac{\partial F} {\partial x^{(s)i}}$, and $r = 2,\ldots,k$.

  These conditions ensure that the arc element ds is independent of the parametrization of the curve x = x(t).

  A Kawaguchi manifold (or Kawaguchi space) is a smooth manifold equipped with a Kawaguchi metric. It is a generalization of a Finsler manifold.

• Lagrange metric

  Consider a real n-dimensional manifold M ⁿ. A set of symmetric nondegenerated matrices ((g _ij (p, x))) define a generalized Lagrange metric on M ⁿ if a change of coordinates (p, x) → (q, y), such that $q_{i} = q_{i}(p_{1},\ldots,p_{n})$, y _i  = (∂ _j q _i )x _j and rank (∂ _j q _i ) = n, implies $g_{\mathit{ij}}(p,x) = (\partial _{i}q_{i})(\partial _{j}q_{j})g_{\mathit{ij}}(q,y)$.

  A generalized Lagrange metric is called a Lagrange metric if there exists a Lagrangian, i.e., a smooth function L(p, x) such that it holds

  $$\displaystyle{g_{\mathit{ij}}(p,x) = \frac{1} {2} \frac{\partial ^{2}L(p,x)} {\partial x_{i}\partial x_{j}}.}$$

  Every Finsler metric is a Lagrange metric with L = F ².

• DeWitt supermetric

  The DeWitt supermetric (or Wheeler–DeWitt supermetric ) G = ((G _ijkl )) calculates distances between metrics on a given manifold, and it is a generalization of a Riemannian (or pseudo-Riemannian) metric g = ((g _ij )).

  For example, for a given connected smooth 3-dimensional manifold M ³, consider the space $\mathcal{M}(M^{3})$ of all Riemannian (or pseudo-Riemannian) metrics on M ³. Identifying points of $\mathcal{M}(M^{3})$ that are related by a diffeomorphism of M ³, one obtains the space Geom(M ³) of 3-geometries (of fixed topology), points of which are the classes of diffeomorphically equivalent metrics. The space Geom(M ³) is called a superspace. It plays an important role in several formulations of Quantum Gravity.

  A supermetric , i.e., a “metric on metrics”, is a metric on $\mathcal{M}(M^{3})$ (or on Geom(M ³)) which is used for measuring distances between metrics on M ³ (or between their equivalence classes). Given $g = ((g_{\mathit{ij}})) \in \mathcal{M}(M^{3})$, we obtain

  $$\displaystyle{\vert \vert \delta g\vert \vert ^{2} =\int _{ M^{3}}d^{3}\mathit{xG}^{\mathit{ijkl}}(x)\delta g_{\mathit{ ij}}(x)\delta g_{\mathit{kl}}(x),}$$

  where G ^ijkl is the inverse of the DeWitt supermetric

  $$\displaystyle{G_{\mathit{ijkl}} = \frac{1} {2\sqrt{\mathit{det } ((g_{\mathit{ij } } ))}}(g_{\mathit{ik}}g_{\mathit{jl}} + g_{\mathit{il}}g_{\mathit{jk}} -\lambda g_{\mathit{ij}}g_{\mathit{kl}}).}$$

  The value λ parametrizes the distance between metrics in $\mathcal{M}(M^{3})$, and may take any real value except $\lambda = \frac{2} {3}$, for which the supermetric is singular.

• Lund–Regge supermetric

  The Lund–Regge supermetric (or simplicial supermetric ) is an analog of the DeWitt supermetric, used to measure the distances between simplicial 3-geometries in a simplicial configuration space.

  More exactly, given a closed simplicial 3D manifold M ³ consisting of several tetrahedra (i.e., 3-simplices), a simplicial geometry on M ³ is fixed by an assignment of values to the squared edge lengths of M ³, and a flat Riemannian Geometry to the interior of each tetrahedron consistent with those values.

  The squared edge lengths should be positive and constrained by the triangle inequalities and their analogs for the tetrahedra, i.e., all squared measures (lengths, areas, volumes) must be nonnegative (cf. tetrahedron inequality in Chap. 3).

  The set $\mathcal{T} (M^{3})$ of all simplicial geometries on M ³ is called a simplicial configuration space. The Lund–Regge supermetric ((G _mn )) on $\mathcal{T} (M^{3})$ is induced from the DeWitt supermetric on $\mathcal{M}(M^{3})$, using for representations of points in $\mathcal{T} (M^{3})$ such metrics in $\mathcal{M}(M^{3})$ which are piecewise flat in the tetrahedra.

• Space of Lorentz metrics

  Let M ⁿ be an n-dimensional compact manifold, and $\mathcal{L}(M^{n})$ the set of all Lorentz metrics (i.e., the pseudo-Riemannian metrics of signature (n − 1, 1)) on M ⁿ.

  Given a Riemannian metric g on M ⁿ, one can identify the vector space S ²(M ⁿ) of all symmetric 2-tensors with the vector space of endomorphisms of the tangent to M ⁿ which are symmetric with respect to g. In fact, if $\tilde{h}$ is the endomorphism associated to a tensor h, then the distance on S ²(M ⁿ) is given by

  $$\displaystyle{d_{g}(h,t) =\sup _{x\in M^{n}}\sqrt{\mathit{tr } (\tilde{h}_{x } -\tilde{ t}_{x } )^{2}}.}$$

  The set $\mathcal{L}(M^{n})$ taken with the distance d _g is an open subset of S ²(M ⁿ) called the space of Lorentz metrics. Cf. manifold triangulation metric in Chap. 9.

• Perelman supermetric proof

  The Thurston’s Geometrization Conjecture is that, after two well-known splittings, any 3D manifold admits, as remaining components, only one of eight Thurston model geometries. If true, this conjecture implies the validity of the famous Poincaré Conjecture of 1904, that any 3-manifold, in which every simple closed curve can be deformed continuously to a point, is homeomorphic to the 3-sphere.

  In 2002, Perelman gave a gapless “sketch of an eclectic proof” of Thurston’s conjecture using a kind of supermetric approach to the space of all Riemannian metrics on a given smooth 3-manifold. In a Ricci flow the distances decrease in directions of positive curvature since the metric is time-dependent. Perelman’s modification of the standard Ricci flow permitted systematic elimination of arising singularities.

2 Riemannian Metrics in Information Theory

Some special Riemannian metrics are commonly used in Information Theory. A list of such metrics is given below.

• Thermodynamic metrics

  Given the space of all extensive (additive in magnitude, mechanically conserved) thermodynamic variables of a system (energy, entropy, amounts of materials), a thermodynamic metric is a Riemannian metric on the manifold of equilibrium states defined as the 2nd derivative of one extensive quantity, usually entropy or energy, with respect to the other extensive quantities. This information geometric approach provides a geometric description of thermodynamic systems in equilibrium.

  The Ruppeiner metric (Ruppeiner, 1979) is defined by the line element $\mathit{ds}_{R}^{2} = g_{\mathit{ij}}^{R}\mathit{dx}^{i}\mathit{dx}^{j},$ where the matrix ((g _ij ^R)) of the symmetric metric tensor is a negative Hessian (the matrix of 2nd order partial derivatives) of the entropy function S:

  $$\displaystyle{g_{\mathit{ij}}^{R} = -\partial _{ i}\partial _{j}S(M,N^{a}).}$$

  Here M is the internal energy (which is the mass in black hole applications) of the system and N ^a refer to other extensive parameters such as charge, angular momentum, volume, etc. This metric is flat if and only if the statistical mechanical system is noninteracting, while curvature singularities are a signal of critical behavior, or, more precisely, of divergent correlation lengths (cf. Chap. 24).

  The Weinhold metric (Weinhold, 1975) is defined by $g_{\mathit{ij}}^{W} = \partial _{i}\partial _{j}M(S,N^{a})$.

  The Ruppeiner and Weinhold metrics are conformally equivalent (cf. conformal metric) via $\mathit{ds}^{2} = g_{\mathit{ij}}^{R}\mathit{dM}^{i}\mathit{dM}^{j} = \frac{1} {T}g_{\mathit{ij}}^{W}\mathit{dS}^{i}\mathit{dS}^{j}$, where T is the temperature.

  The thermodynamic length in Chap. 24 is a path function that measures the distance along a path in the state space.

• Fisher information metric

  In Statistics, Probability, and Information Geometry, the Fisher information metric is a Riemannian metric for a statistical differential manifold (see, for example, [Amar85, Frie98]). Formally, let p _θ  = p(x, θ) be a family of densities, indexed by n parameters $\theta = (\theta _{1},\ldots,\theta _{n})$ which form the parameter manifold P.

  The Fisher information metric g = g _θ on P is a Riemannian metric, defined by the Fisher information matrix ((I(θ) _ij )), where

  $$\displaystyle{I(\theta )_{\mathit{ij}} = \mathbb{E}_{\theta }\left [\frac{\partial \ln p_{\theta }} {\partial \theta _{i}} \cdot \frac{\partial \ln p_{\theta }} {\partial \theta _{j}} \right ] =\int \frac{\partial \ln p(x,\theta )} {\partial \theta _{i}} \frac{\partial \ln p(x,\theta )} {\partial \theta _{j}} p(x,\theta )\mathit{dx}.}$$

  It is a symmetric bilinear form which gives a classical measure (Rao measure) for the statistical distinguishability of distribution parameters.

  Putting $i(x,\theta ) = -\ln p(x,\theta )$, one obtains an equivalent formula

  $$\displaystyle{I(\theta )_{\mathit{ij}} = \mathbb{E}_{\theta }\left [\frac{\partial ^{2}i(x,\theta )} {\partial \theta _{i}\partial \theta _{j}} \right ] =\int \frac{\partial ^{2}i(x,\theta )} {\partial \theta _{i}\partial \theta _{j}} p(x,\theta )\mathit{dx}.}$$

  In a coordinate-free language, we get

  $$\displaystyle{I(\theta )(u,v) = \mathbb{E}_{\theta }\left [u(\ln p_{\theta }) \cdot v(\ln p_{\theta })\right ],}$$

  where u and v are vectors tangent to the parameter manifold P, and $u(\ln p_{\theta }) = \frac{d} {\mathit{dt}}\ln p_{\theta +\mathit{tu}\vert t=0}$ is the derivative of lnp _θ along the direction u.

  A manifold of densities M is the image of the parameter manifold P under the mapping θ → p _θ with certain regularity conditions. A vector u tangent to this manifold is of the form $u = \frac{d} {\mathit{dt}}p_{\theta +\mathit{tu}\vert t=0}$, and the Fisher information metric g =  g _p on M, obtained from the metric g _θ on P, can be written as

  $$\displaystyle{g_{p}(u,v) = \mathbb{E}_{p}\left [\frac{u} {p} \cdot \frac{v} {p}\right ].}$$

• Fisher–Rao metric

  Let $\mathcal{P}_{n} =\{ p \in \mathbb{R}^{n}:\sum _{ i=1}^{n}p_{i} = 1,p_{i} > 0\}$ be the simplex of strictly positive probability vectors. An element $p \in \mathcal{P}_{n}$ is a density of the n-point set $\{1,\ldots,n\}$ with p(i) = p _i . An element u of the tangent space $T_{p}(\mathcal{P}_{n}) =\{ u \in \mathbb{R}^{n}:\sum _{ i=1}^{n}u_{i} = 0\}$ at a point $p \in \mathcal{P}_{n}$ is a function on $\{1,\ldots,n\}$ with u(i) = u _i .

  The Fisher–Rao metric g _p on $\mathcal{P}_{n}$ is a Riemannian metric defined by

  $$\displaystyle{g_{p}(u,v) =\sum _{ i=1}^{n}\frac{u_{i}v_{i}} {p_{i}} }$$

  for any $u,v \in T_{p}(\mathcal{P}_{n})$, i.e., it is the Fisher information metric on $\mathcal{P}_{n}$.

  The Fisher–Rao metric is the unique (up to a constant factor) Riemannian metric on $\mathcal{P}_{n}$, contracting under stochastic maps [Chen72].

  This metric is isometric, by $p \rightarrow 2(\sqrt{p_{1}},\ldots,\sqrt{p_{n}})$, with the standard metric on an open subset of the sphere of radius two in $\mathbb{R}^{n}$. This identification allows one to obtain on $\mathcal{P}_{n}$ the geodesic distance, called the Rao distance , by

  $$\displaystyle{2\arccos (\sum _{i}p_{i}^{1/2}q_{ i}^{1/2}).}$$

  The Fisher–Rao metric can be extended to the set $\mathcal{M}_{n} =\{ p \in \mathbb{R}^{n},p_{i} > 0\}$ of all finite strictly positive measures on the set $\{1,\ldots,n\}$. In this case, the geodesic distance on $\mathcal{M}_{n}$ can be written as

  $$\displaystyle{2\left (\sum _{i}(\sqrt{p_{i}} -\sqrt{q_{i}})^{2}\right )^{1/2}}$$

  for any $p,q \in \mathcal{M}_{n}$ (cf. Hellinger metric in Chap. 14).

• Monotone metrics

  Let M _n be the set of all complex n × n matrices. Let $\mathcal{M}\subset M_{n}$ be the manifold of all such positive-definite matrices. Let $\mathcal{D}\subset \mathcal{M}$, $\mathcal{D} =\{\rho \in \mathcal{M}: \mathrm{Tr}\rho = 1\}$, be the submanifold of all density matrices. It is the space of faithful states of an n-level quantum system; cf. distances between quantum states in Chap. 24.

  The tangent space of $\mathcal{M}$ at $\rho \in \mathcal{M}$ is $T_{\rho }(\mathcal{M}) =\{ x \in M_{n}: x = x^{{\ast}}\}$, i.e., the set of all n × n Hermitian matrices. The tangent space $T_{\rho }(\mathcal{D})$ at $\rho \in \mathcal{D}$ is the subspace of traceless (i.e., with trace 0) matrices in $T_{\rho }(\mathcal{M})$.

  A Riemannian metric λ on $\mathcal{M}$ is called monotone metric if the inequality

  $$\displaystyle{\lambda _{h(\rho )}(h(u),h(u)) \leq \lambda _{\rho }(u,u)}$$

  holds for any $\rho \in \mathcal{M}$, any $u \in T_{\rho }(\mathcal{M})$, and any stochastic, i.e., completely positive trace preserving mapping h.

  It was proved in [Petz96] that λ is monotone if and only if it can be written as

  $$\displaystyle{\lambda _{\rho }(u,v) = \mathrm{Tr}\,\,\mathit{uJ}_{\rho }(v),}$$

  where J _ρ is an operator of the form $J_{\rho } = \frac{1} {f(L_{\rho }/R_{\rho })R_{\rho }}$. Here L _ρ and R _ρ are the left and the right multiplication operators, and $f: (0,\infty ) \rightarrow \mathbb{R}$ is an operator monotone function which is symmetric, i.e., $f(t) = \mathit{tf }(t^{-1})$, and normalized, i.e., f(1) = 1. Then $J_{\rho }(v) =\rho ^{-1}v$ if v and ρ are commute, i.e., any monotone metric is equal to the Fisher information metric on commutative submanifolds.

  The Bures metric (or statistical metric) is the smallest monotone metric, obtained for $f(t) = \frac{1+t} {2}$. In this case J _ρ (v) = g, $\rho g + g\rho = 2v$, is the symmetric logarithmic derivative. For any $\rho _{1},\rho _{2} \in \mathcal{M}$ the geodesic distance defined by the Bures metric, (cf. Bures length in Chap. 24) can be written as

  $$\displaystyle{2\sqrt{\mathrm{Tr } (\rho _{1 } ) + \mathrm{Tr } (\rho _{2 } ) - 2\mathrm{Tr } (\sqrt{\sqrt{\rho _{1 }} \rho _{2 } \sqrt{\rho _{1 }}} )}.}$$

  On the submanifold $\mathcal{D} =\{\rho \in \mathcal{M}: \mathrm{Tr}\rho = 1\}$ of density matrices it has the form

  $$\displaystyle{2\arccos \mathrm{Tr}(\sqrt{\sqrt{\rho _{1 }} \rho _{2 } \sqrt{\rho _{1}}}).}$$

  The right logarithmic derivative metric (or RLD-metric) is the greatest monotone metric, corresponding to the function $f(t) = \frac{2t} {1+t}$. In this case $J_{\rho }(v) = \frac{1} {2}(\rho ^{-1}v + v\rho ^{-1})$ is the right logarithmic derivative.

  The Bogolubov–Kubo–Mori metric (or BKM-metric) is obtained for $f(x) = \frac{x-1} {\ln x}$. It can be written as $\lambda _{\rho }(u,v) = \frac{\partial ^{2}} {\partial s\partial t}\mathrm{Tr}(\rho +\mathit{su})\ln (\rho +\mathit{tv})\vert _{s,t=0}$.

• Wigner–Yanase–Dyson metrics

  The Wigner–Yanase–Dyson metrics (or WYD-metrics) form a family of metrics on the manifold $\mathcal{M}$ of all complex positive-definite n × n matrices defined by

  $$\displaystyle{\lambda _{\rho }^{\alpha }(u,v) = \frac{\partial ^{2}} {\partial t\partial s}\mathrm{Tr}f_{\alpha }(\rho +\mathit{tu})f_{-\alpha }(\rho +\mathit{sv})\vert _{s,t=0},}$$

  where $f_{\alpha }(x) = \frac{2} {1-\alpha }x^{\frac{1-\alpha } {2} }$, if α ≠ 1, and is lnx, if α = 1. These metrics are monotone for α ∈ [−3, 3]. For α = ±1 one obtains the Bogolubov–Kubo–Mori metric; for α = ±3 one obtains the right logarithmic derivative metric.

  The Wigner–Yanase metric (or WY-metric) is λ _ρ ⁰, the smallest metric in the family. It can be written as $\lambda _{\rho }(u,v) = 4\mathrm{Tr}\,\,u(\sqrt{L_{\rho }} + \sqrt{R_{\rho }})^{2}(v).$

• Connes metric

  Roughly, the Connes metric is a generalization (from the space of all probability measures of a set X, to the state space of any unital C ^∗ -algebra) of the transportation distance (Chap. 14) defined via Lipschitz seminorm.

  Let M ⁿ be a smooth n-dimensional manifold. Let A = C ^∞(M ⁿ) be the (commutative) algebra of smooth complex-valued functions on M ⁿ, represented as multiplication operators on the Hilbert space H = L ²(M ⁿ, S) of square integrable sections of the spinor bundle on M ⁿ by (f ξ)(p) = f(p)ξ(p) for all f ∈ A and for all ξ ∈ H.

  Let D be the Dirac operator. Let the commutator [D, f] for f ∈ A be the Clifford multiplication by the gradient ∇f, so that its operator norm | | . | | in H is given by $\vert \vert [D,f]\vert \vert =\sup _{p\in M^{n}}\vert \vert \nabla f\vert \vert$.

  The Connes metric is the intrinsic metric on M ⁿ, defined by

  $$\displaystyle{\sup _{f\in A,\vert \vert \,[D,f]\,\vert \vert \leq 1}\vert f(p) - f(q)\vert.}$$

  This definition can also be applied to discrete spaces, and even generalized to C ^∗ -algebras; cf. Rieffel metric space. In particular, for a labeled connected locally finite graph G = (V, E) with the vertex-set $V =\{ v_{1},\ldots,v_{n},\ldots \}$, the Connes metric on V is defined, for any v _i , v _j  ∈ V, by $\sup _{\vert \vert \,[D,f]\,\vert \vert =\vert \vert df\vert \vert \leq 1}\vert f_{v_{i}} - f_{v_{j}}\vert$, where $\{f =\sum f_{v_{i}}v_{i}:\sum \vert f_{v_{i}}\vert ^{2} < \infty \}$ is the set of formal sums f, forming a Hilbert space, and | | [D, f] | | is $\sup _{i}(\sum _{k=1}^{\mathit{deg}(v_{i})}(f_{v_{ k}} - f_{v_{i}})^{2})^{\frac{1} {2} }$.

• Rieffel metric space

  Let V be a normed space (or, more generally, a locally convex topological vector space, cf. Chap. 2), and let V ′ be its continuous dual space, i.e., the set of all continuous linear functionals f on V. The weak- ^∗ topology on V ′ is defined as the weakest (i.e., with the fewest open sets) topology on V ′ such that, for every x ∈ V, the map $F_{x}: V ' \rightarrow \mathbb{R}$ defined by F _x (f) = f(x) for all f ∈ V ′, remains continuous.

  An order-unit space is a partially ordered real (complex) vector space (A, ⪯ ) with a special distinguished element e (order unit) satisfying the following properties:

  • 1. For any a ∈ A, there exists $r \in \mathbb{R}$ with a ⪯ re;

  • 2. If a ∈ A and a ⪯ re for all positive $r \in \mathbb{R}$, then a ⪯0 (Archimedean property).

  The main example of an order-unit space is the vector space of all self-adjoint elements in a unital C ^∗ -algebra with the identity element being the order unit. Here a C ^∗ -algebra is a Banach algebra over $\mathbb{C}$ equipped with a special involution. It is called unital if it has a unit (multiplicative identity element); such C ^∗-algebras are also called, roughly, compact noncommutative topological spaces.

  Main example of a unital C ^∗-algebra is the complex algebra of linear operators on a complex Hilbert space which is topologically closed in the norm topology of operators, and is closed under the operation of taking adjoints of operators.

  The state space of an order-unit space (A, ⪯, e) is the set $S(A) =\{ f \in A'_{+}: \vert \vert f\vert \vert = 1\}$ of states, i.e., continuous linear functionals f with $\vert \vert f\vert \vert = f(e) = 1$.

  A Rieffel (or compact quantum as in Rieffel, 1999) metric space is a pair (A, | | . | |  _Lip ), where (A, ⪯, e) is an order-unit space, and | | . | |  _Lip is a [0, +∞]-valued seminorm on A (generalizing the Lipschitz seminorm) for which it hold:

  • 1. For a ∈ A, | | a | |  _Lip  = 0 holds if and only if $a \in \mathbb{R}e$;

  • 2. the metric $d_{\mathit{Lip}}(f,g) =\sup _{a\in A:\vert \vert a\vert \vert _{\mathit{Lip}}\leq 1}\vert f(a) - g(a)\vert$ generates on the state space S(A) its weak-^∗ topology.

  So, (S(A), d _Lip ) is a usual metric space. If the order-unit space (A, ⪯, e) is a C ^∗-algebra, then d _Lip is the Connes metric, and if, moreover, the C ^∗-algebra is noncommutative, (S(A), d _Lip ) is called a noncommutative metric space .

  The term quantum is due to the belief that the Planck-scale geometry of space-time comes from such C ^∗-algebras; cf. quantum space-time in Chap. 24.

  Kuperberg and Weaver, 2010, proposed a new definition of quantum metric space, in the setting of von Neumann algebras.

3 Hermitian Metrics and Generalizations

A vector bundle is a geometrical construct where to every point of a topological space M we attach a vector space so that all those vector spaces “glued together” form another topological space E. A continuous mapping π: E → M is called a projection E on M. For every p ∈ M, the vector space π ⁻¹(p) is called a fiber of the vector bundle.

A real (complex) vector bundle is a vector bundle π: E → M whose fibers π ⁻¹(p), p ∈ M, are real (complex) vector spaces.

In a real vector bundle, for every p ∈ M, the fiber π ⁻¹(p) locally looks like the vector space $\mathbb{R}^{n}$, i.e., there is an open neighborhood U of p, a natural number n, and a homeomorphism $\varphi: U \times \mathbb{R}^{n} \rightarrow \pi ^{-1}(U)$ such that, for all x ∈ U and $v \in \mathbb{R}^{n}$, one has $\pi (\varphi (x,v)) = v$, and the mapping $v \rightarrow \varphi (x,v)$ yields an isomorphism between $\mathbb{R}^{n}$ and π ⁻¹(x). The set U, together with $\varphi$, is called a local trivialization of the bundle.

If there exists a “global trivialization”, then a real vector bundle $\pi: M \times \mathbb{R}^{n} \rightarrow M$ is called trivial. Similarly, in a complex vector bundle, for every p ∈ M, the fiber π ⁻¹(p) locally looks like the vector space $\mathbb{C}^{n}$. The basic example of such bundle is the trivial bundle $\pi: U \times \mathbb{C}^{n} \rightarrow U$, where U is an open subset of $\mathbb{R}^{k}$.

Important special cases of a real vector bundle are the tangent bundle T(M ⁿ) and the cotangent bundle T ^∗(M ⁿ) of a real n-dimensional manifold $M_{\mathbb{R}}^{n} = M^{n}$. Important special cases of a complex vector bundle are the tangent bundle and the cotangent bundle of a complex $n$ -dimensional manifold.

Namely, a complex n-dimensional manifold $M_{\mathbb{C}}^{n}$ is a topological space in which every point has an open neighborhood homeomorphic to an open set of the n-dimensional complex vector space $\mathbb{C}^{n}$, and there is an atlas of charts such that the change of coordinates between charts is analytic. The (complex) tangent bundle $T_{\mathbb{C}}(M_{\mathbb{C}}^{n})$ of a complex manifold $M_{\mathbb{C}}^{n}$ is a vector bundle of all (complex) tangent spaces of $M_{\mathbb{C}}^{n}$ at every point $p \in M_{\mathbb{C}}^{n}$. It can be obtained as a complexification $T_{\mathbb{R}}(M_{\mathbb{R}}^{n}) \otimes \mathbb{C} = T(M^{n}) \otimes \mathbb{C}$ of the corresponding real tangent bundle, and is called the complexified tangent bundle of $M_{\mathbb{C}}^{n}$.

The complexified cotangent bundle of $M_{\mathbb{C}}^{n}$ is obtained similarly as $T^{{\ast}}(M^{n}) \otimes \mathbb{C}$. Any complex n-dimensional manifold $M_{\mathbb{C}}^{n} = M^{n}$ can be regarded as a real 2n-dimensional manifold equipped with a complex structure on each tangent space.

A complex structure on a real vector space V is the structure of a complex vector space on V that is compatible with the original real structure. It is completely determined by the operator of multiplication by the number i, the role of which can be taken by an arbitrary linear transformation J: V → V, $J^{2} = -\mathit{id}$, where id is the identity mapping.

A connection (or covariant derivative) is a way of specifying a derivative of a vector field along another vector field in a vector bundle. A metric connection is a linear connection in a vector bundle π: E → M, equipped with a bilinear form in the fibers, for which parallel displacement along an arbitrary piecewise-smooth curve in M preserves the form, that is, the scalar product of two vectors remains constant under parallel displacement.

In the case of a nondegenerate symmetric bilinear form, the metric connection is called the Euclidean connection. In the case of nondegenerate antisymmetric bilinear form, the metric connection is called the symplectic connection.

• Bundle metric

  A bundle metric is a metric on a vector bundle.

• Hermitian metric

  A Hermitian metric on a complex vector bundle π: E → M is a collection of Hermitian inner products (i.e., positive-definite symmetric sesquilinear forms) on every fiber $E_{p} =\pi ^{-1}(p)$, p ∈ M, that varies smoothly with the point p in M. Any complex vector bundle has a Hermitian metric.

  The basic example of a vector bundle is the trivial bundle $\pi: U \times \mathbb{C}^{n} \rightarrow U$, where U is an open set in $\mathbb{R}^{k}$. In this case a Hermitian inner product on $\mathbb{C}^{n}$, and hence, a Hermitian metric on the bundle $\pi: U \times \mathbb{C}^{n} \rightarrow U$, is defined by

  $$\displaystyle{\langle u,v\rangle = u^{T}H\overline{v},}$$

  where H is a positive-definite Hermitian matrix, i.e., a complex n × n matrix such that $H^{{\ast}} = \overline{H}^{T} = H$, and $\overline{v}^{T}\mathit{Hv} > 0$ for all $v \in \mathbb{C}^{n}\setminus \{0\}$. In the simplest case, one has $\langle u,v\rangle =\sum _{ i=1}^{n}u_{i}\overline{v}_{i}$.

  An important special case is a Hermitian metric h on a complex manifold M ⁿ, i.e., on the complexified tangent bundle $T(M^{n}) \otimes \mathbb{C}$ of M ⁿ. This is the Hermitian analog of a Riemannian metric. In this case $h = g + \mathit{iw}$, and its real part g is a Riemannian metric, while its imaginary part w is a nondegenerate antisymmetric bilinear form, called a fundamental form. Here g(J(x), J(y)) = g(x, y), w(J(x), J(y)) = w(x, y), and w(x, y) = g(x, J(y)), where the operator J is an operator of complex structure on M ⁿ; as a rule, J(x) = ix. Any of the forms g, w determines h uniquely.

  The term Hermitian metric can also refer to the corresponding Riemannian metric g, which gives M ⁿ a Hermitian structure.

  On a complex manifold, a Hermitian metric h can be expressed in local coordinates by a Hermitian symmetric tensor ((h _ij )):

  $$\displaystyle{h =\sum _{i,j}h_{\mathit{ij}}\mathit{dz}_{i} \otimes d\overline{z}_{j},}$$

  where ((h _ij )) is a positive-definite Hermitian matrix. The associated fundamental form w is then written as $w = \frac{i} {2}\sum _{i,j}h_{\mathit{ij}}\mathit{dz}_{i} \wedge d\overline{z}_{j}$. A Hermitian manifold (or Hermitian space) is a complex manifold equipped with a Hermitian metric.

• Kähler metric

  A Kähler metric (or Kählerian metric) is a Hermitian metric $h = g + \mathit{iw}$ on a complex manifold M ⁿ whose fundamental form w is closed, i.e., dw = 0 holds. A Kähler manifold is a complex manifold equipped with a Kähler metric.

  If h is expressed in local coordinates, i.e., $h =\sum _{i,j}h_{\mathit{ij}}\mathit{dz}_{i} \otimes d\overline{z}_{j}$, then the associated fundamental form w can be written as $w = \frac{i} {2}\sum _{i,j}h_{\mathit{ij}}\mathit{dz}_{i} \wedge d\overline{z}_{j}$, where ∧ is the wedge product which is antisymmetric, i.e., $\mathit{dx} \wedge \mathit{dy} = -\mathit{dy} \wedge \mathit{dx}$ (hence, dx ∧dx = 0).

  In fact, w is a differential 2-form on M ⁿ, i.e., a tensor of rank 2 that is antisymmetric under exchange of any pair of indices: $w =\sum _{i,j}f_{\mathit{ij}}\mathit{dx}^{i} \wedge \mathit{dx}^{j}$, where f _ij is a function on M ⁿ. The exterior derivative dw of w is defined by $\mathit{dw} =\sum _{i,j}\sum _{k}\frac{\partial f_{\mathit{ij}}} {\partial x_{k}} \mathit{dx}_{k} \wedge \mathit{dx}_{i} \wedge \mathit{dx}_{k}$. If dw = 0, then w is a symplectic (i.e., closed nondegenerate) differential 2-form. Such differential 2-forms are called Kähler forms.

  The metric on a Kähler manifold locally satisfies $h_{\mathit{ij}} = \frac{\partial ^{2}K} {\partial z_{i}\partial \overline{z}_{j}}.$ for some function K, called the Kähler potential. The term Kähler metric can also refer to the corresponding Riemannian metric g, which gives M ⁿ a Kähler structure. Then a Kähler manifold is defined as a complex manifold which carries a Riemannian metric and a Kähler form on the underlying real manifold.

• Hessian metric

  Given a smooth f on an open subset of a real vector space, the associated Hessian metric is defined by

  $$\displaystyle{g_{\mathit{ij}} = \frac{\partial ^{2}f} {\partial x_{i}\partial x_{j}}.}$$

  A Hessian metric is also called an affine Kähler metric since a Kähler metric on a complex manifold has an analogous description as $\frac{\partial ^{2}f} {\partial z_{i}\partial \overline{z_{j}}}$.

• Calabi–Yau metric

  The Calabi–Yau metric is a Kähler metric which is Ricci-flat.

  A Calabi–Yau manifold (or Calabi–Yau space) is a simply connected complex manifold equipped with a Calabi–Yau metric. It can be considered as a 2n-dimensional (6D being particularly interesting) smooth manifold with holonomy group (i.e., the set of linear transformations of tangent vectors arising from parallel transport along closed loops) in the special unitary group.

• Kähler–Einstein metric

  A Kähler–Einstein metric is a Kähler metric on a complex manifold M ⁿ whose Ricci curvature tensor is proportional to the metric tensor. This proportionality is an analog of the Einstein field equation in the General Theory of Relativity.

  A Kähler–Einstein manifold (or Einstein manifold) is a complex manifold equipped with a Kähler–Einstein metric. In this case the Ricci curvature tensor, seen as an operator on the tangent space, is just multiplication by a constant.

  Such a metric exists on any domain $D \subset \mathbb{C}^{n}$ that is bounded and pseudo-convex. It can be given by the line element

  $$\displaystyle{\mathit{ds}^{2} =\sum _{ i,j}\frac{\partial ^{2}u(z)} {\partial z_{i}\partial \overline{z}_{j}}\mathit{dz}_{i}d\overline{z}_{j},}$$

  where u is a solution to the boundary value problem: $\mathit{det}( \frac{\partial ^{2}u} {\partial z_{i}\partial \overline{z}_{j}}) = e^{2u}$ on D, and u = ∞ on ∂ D. The Kähler–Einstein metric is a complete metric. On the unit disk $\Delta =\{ z \in \mathbb{C}: \vert z\vert < 1\}$ it is coincides with the Poincaré metric.

  Let h be the Einstein metric on a smooth compact manifold M ⁿ⁻¹ without boundary, having scalar curvature $(n - 1)(n - 2)$. A generalized Delaunay metric on $\mathbb{R} \times M^{n-1}$ is (Delay, 2010) of the form $g = u^{ \frac{4} {n-2} }(\mathit{dy}^{2} + h)$, where u = u(y) > 0 is a periodic solution of $u'' -\frac{(n-2)^{2}} {4} u + \frac{n(n-2)} {4} u^{\frac{n+2} {n-2} } = 0$.

  There is one parameter family of constant positive curvature conformal metrics on $\mathbb{R} \times \mathbb{S}^{n-1}$, referred to as Delaunay metric ; cf. Kottler metric in Chap. 26.

• Hodge metric

  The Hodge metric is a Kähler metric whose fundamental form w defines an integral cohomology class or, equivalently, has integral periods.

  A Hodge manifold (or Hodge variety) is a complex manifold equipped with a Hodge metric. A compact complex manifold is a Hodge manifold if and only if it is isomorphic to a smooth algebraic subvariety of some complex projective space.

• Fubini–Study metric

  The Fubini–Study metric (or Cayley–Fubini–Study metric) is a Kähler metric on a complex projective space $\mathbb{C}P^{n}$ defined by a Hermitian inner product $\langle,\rangle$ in $\mathbb{C}^{n+1}$. It is given by the line element

  $$\displaystyle{\mathit{ds}^{2} = \frac{\langle x,x\rangle \langle \mathit{dx},\mathit{dx}\rangle -\langle x,d\overline{x}\rangle \langle \overline{x},\mathit{dx}\rangle } {\langle x,x\rangle ^{2}}.}$$

  The Fubini–Study distance between points (x ₁: …: x _n+1) and $(y_{1}:\ldots: y_{n+1}) \in \mathbb{C}P^{n}$, where $x = (x_{1},\ldots,x_{n+1})$ and $y = (y_{1},\ldots,y_{n+1}) \in \mathbb{C}^{n+1}\setminus \{0\}$, is equal to

  $$\displaystyle{\arccos \frac{\vert \langle x,y\rangle \vert } {\sqrt{\langle x, x\rangle \langle y, y\rangle }}.}$$

  The Fubini–Study metric is a Hodge metric. The space $\mathbb{C}P^{n}$ endowed with this metric is called a Hermitian elliptic space (cf. Hermitian elliptic metric).

• Bergman metric

  The Bergman metric is a Kähler metric on a bounded domain $D \subset \mathbb{C}^{n}$ defined, for the Bergman kernel K(z, u), by the line element

  $$\displaystyle{\mathit{ds}^{2} =\sum _{ i,j}\frac{\partial ^{2}\ln K(z,z)} {\partial z_{i}\partial \overline{z}_{j}} \mathit{dz}_{i}d\overline{z}_{j}.}$$

  It is a biholomorhically invariant metric on D, and it is complete if D is homogeneous. For the unit disk $\Delta =\{ z \in \mathbb{C}: \vert z\vert < 1\}$ the Bergman metric coincides with the Poincaré metric; cf. also Bergman p -metric in Chap. 13.

  The set of all analytic functions f ≠ 0 of class L ₂(D) with respect to the Lebesgue measure, forms the Hilbert space L _2, a (D) ⊂ L ₂(D) with an orthonormal basis (ϕ _i ) _i . The Bergman kernel is a function in the domain $D \times D \subset \mathbb{C}^{2n}$, defined by $K_{D}(z,u) = K(z,u) =\sum _{ i=1}^{\infty }\phi _{i}(z)\overline{\phi _{i}(u)}$.

  The Skwarczynski distance is defined by

  $$\displaystyle{(1 \frac{\vert K(z,u)\vert } {\sqrt{K(z, z)}\sqrt{K(u, u)}})^{\frac{1} {2} }.}$$

• Hyper-Kähler metric

  A hyper-Kähler metric is a Riemannian metric g on a 4n-dimensional Riemannian manifold which is compatible with a quaternionic structure on the tangent bundle of the manifold.

  Thus, the metric g is Kählerian with respect to 3 Kähler structures (I, w _I , g), (J, w _J , g), and (K, w _K , g), corresponding to the complex structures, as endomorphisms of the tangent bundle, which satisfy the quaternionic relationship

  $$\displaystyle{I^{2} = J^{2} = K^{2} = \mathit{IJK} = -\mathit{JIK} = -1.}$$

  A hyper-Kähler manifold is a Riemannian manifold equipped with a hyper-Kähler metric. manifolds are Ricci-flat. Compact 4D hyper-Kähler manifolds are called K ₃ -surfaces; they are studied in Algebraic Geometry.

• Calabi metric

  The Calabi metric is a hyper-Kähler metric on the cotangent bundle $T^{{\ast}}(\mathbb{C}P^{n+1})$ of a complex projective space $\mathbb{C}P^{n+1}$.

  For $n = 4k + 4$, this metric can be given by the line element

  $$\displaystyle\begin{array}{rcl} \mathit{ds}^{2}& =& \frac{\mathit{dr}^{2}} {1 - r^{-4}} + \frac{1} {4}r^{2}(1 - r^{-4})\lambda ^{2} + r^{2}(\nu _{ 1}^{2} +\nu _{ 2}^{2}) {}\\ & & \quad + \frac{1} {2}(r^{2} - 1)(\sigma _{ 1\alpha }^{2} +\sigma _{ 2\alpha }^{2}) + \frac{1} {2}(r^{2} + 1)(\Sigma _{ 1\alpha }^{2} + \Sigma _{ 2\alpha }^{2}), {}\\ \end{array}$$

  where $(\lambda,\nu _{1},\nu _{2},\sigma _{1\alpha },\sigma _{2\alpha },\Sigma _{1\alpha },\Sigma _{2\alpha })$, with α running over k values, are left-invariant one-forms (i.e., linear real-valued functions) on the coset $\mathit{SU}(k +\ 2)/U(k)$. Here U(k) is the unitary group consisting of complex k × k unitary matrices, and SU(k) is the special unitary group of complex k × k unitary matrices with determinant 1.

  For k = 0, the Calabi metric coincides with the Eguchi–Hanson metric.

• Stenzel metric

  The Stenzel metric is a hyper-Kähler metric on the cotangent bundle T ^∗(S ⁿ⁺¹) of a sphere S ⁿ⁺¹.

• SO(3)-invariant metric

  An SO(3)-invariant metric is a 4D 4-dimensional hyper-Kähler metric with the line element given, in the Bianchi type IX formalism (cf. Bianchi metrics in Chap. 26) by

  $$\displaystyle{\mathit{ds}^{2} = f^{2}(t)\mathit{dt}^{2} + a^{2}(t)\sigma _{ 1}^{2} + b^{2}(t)\sigma _{ 2}^{2} + c^{2}(t)\sigma _{ 3}^{2},}$$

  where the invariant one-forms σ ₁, σ ₂, σ ₃ of SO(3) are expressed in terms of Euler angles θ, ψ, ϕ as $\sigma _{1} = \frac{1} {2}(\sin \psi d\theta -\sin \theta \cos \psi d\phi )$, $\sigma _{2} = -\frac{1} {2}(\cos \psi d\theta +\sin \theta \sin \psi d\phi )$, $\sigma _{3} = \frac{1} {2}(d\psi +\cos \theta d\phi )$, and the normalization has been chosen so that $\sigma _{i} \wedge \sigma _{j} = \frac{1} {2}\epsilon _{\mathit{ijk}}d\sigma _{k}$. The coordinate t of the metric can always be chosen so that $f(t) = \frac{1} {2}\mathit{abc}$, using a suitable reparametrization.

• Atiyah–Hitchin metric

  The Atiyah–Hitchin metric is a complete regular SO(3)-invariant metric with the line element

  $$\displaystyle{\mathit{ds}^{2} = \frac{1} {4}a^{2}b^{2}c^{2}\left ( \frac{\mathit{dk}} {k(1 - k^{2})K^{2}}\right )^{2} + a^{2}(k)\sigma _{ 1}^{2} + b^{2}(k)\sigma _{ 2}^{2} + c^{2}(k)\sigma _{ 3}^{2},}$$

  where a, b, c are functions of k, $\mathit{ab} = -K(k)(E(k) - K(k))$, $\mathit{bc} = -K(k)(E(k) - (1 - k^{2})K(k))$, $\mathit{ac} = -K(k)E(k)$, and K(k), E(k) are the complete elliptic integrals, respectively, of the first and second kind, with 0 < k < 1. The coordinate t is given by the change of variables $t = -\frac{2K(1-k^{2})} {\pi K(k)}$ up to an additive constant.

• Taub–NUT metric

  The Taub–NUT metric (cf. also Chap. 26) is a complete regular SO(3)-invariant metric with the line element

  $$\displaystyle{\mathit{ds}^{2} = \frac{1} {4} \frac{r + m} {r - m}\mathit{dr}^{2} + (r^{2} - m^{2})(\sigma _{ 1}^{2} +\sigma _{ 2}^{2}) + 4m^{2}\frac{r - m} {r + m}\sigma _{3}^{2},}$$

  where m is the relevant moduli parameter, and the coordinate r is related to t by $r = m + \frac{1} {2\mathit{mt}}$. NUT manifold was discovered in Ehlers, 1957, and rediscovered in Newman–Tamburino–Unti, 1963; it is closely related to the metric in Taub, 1951.

• Eguchi–Hanson metric

  The Eguchi–Hanson metric is a complete regular SO(3)-invariant metric with the line element

  $$\displaystyle{\mathit{ds}^{2} = \frac{\mathit{dr}^{2}} {1 -\left (\frac{a} {r}\right )^{4}} + r^{2}\left (\sigma _{ 1}^{2} +\sigma _{ 2}^{2} + \left (1 -\left (\frac{a} {r}\right )^{4}\right )\sigma _{ 3}^{2}\right ),}$$

  where a is the moduli parameter, and the coordinate r is $a\sqrt{\coth (a^{2 } t)}$.

  The Eguchi–Hanson metric coincides with the 4D Calabi metric.

• Complex Finsler metric

  A complex Finsler metric is an upper semicontinuous function $F: T(M^{n}) \rightarrow \ \mathbb{R}_{+}$ on a complex manifold M ⁿ with the analytic tangent bundle T(M ⁿ) satisfying the following conditions:

  • 1. F ² is smooth on $\check{M^{n}}$, where $\check{M^{n}}$ is the complement in T(M ⁿ) of the zero section;

  • 2. F(p, x) > 0 for all p ∈ M ⁿ and $x \in \check{ M_{p}^{n}}$;

  • 3. F(p, λ x) =  | λ | F(p, x) for all p ∈ M ⁿ, x ∈ T _p (M ⁿ), and $\lambda \in \mathbb{C}$.

  The function G = F ² can be locally expressed in terms of the coordinates $(p_{1},\ldots,p_{n},$ $x_{1},\ldots,x_{n})$; the Finsler metric tensor of the complex Finsler metric is given by the matrix$((G_{\mathit{ij}})) = ((\frac{1} {2} \frac{\partial ^{2}F^{2}} {\partial x_{i}\partial \overline{x}_{j}}$)), called the Levi matrix. If the matrix ((G _ij )) is positive-definite, the complex Finsler metric F is called strongly pseudo-convex.

• Distance-decreasing semimetric

  Let d be a semimetric which can be defined on some class $\mathcal{M}$ of complex manifolds containing the unit disk $\Delta =\{ z \in \mathbb{C}: \vert z\vert < 1\}$. It is called distance-decreasing if, for any analytic mapping f: M ₁ → M ₂ with $M_{1},M_{2} \in \mathcal{M}$, the inequality d(f(p), f(q)) ≤ d(p, q) holds for all $p,q \in M_{_{1}}$.

  The Carathéodory semimetric F _C , Sibony semimetric F _S , Azukawa semimetric F _A and Kobayashi semimetric F _K are distance-decreasing with F _C and F _K being the smallest and the greatest distance-decreasing semimetrics. They are generalizations of the Poincaré metric to higher-dimensional domains, since F _C  = F _K is the Poincaré metric on the unit disk $\Delta$, and $F_{C} = F_{K} \equiv 0$ on $\mathbb{C}^{n}$.

  It holds $F_{C}(z,u) \leq F_{S}(z,u) \leq F_{A}(z,u) \leq F_{K}(z,u)$ for all z ∈ D and $u \in \mathbb{C}^{n}$. If D is convex, then all these metrics coincide.

• Biholomorphically invariant semimetric

  A biholomorphism is a bijective holomorphic (complex differentiable in a neighborhood of every point in its domain) function whose inverse is also holomorphic.

  A semimetric $F(z,u): D \times \mathbb{C}^{n} \rightarrow [0,\infty ]$ on a domain D in $\mathbb{C}^{n}$ is called biholomorphically invariant if F(z, u) =  | λ | F(z, u) for all $\lambda \in \mathbb{C}$, and F(z, u) = F(f(z), f′(z)u) for any biholomorphism f: D → D′.

  Invariant metrics, including the Carathéodory, Kobayashi, Sibony, Azukawa, Bergman, and Kähler–Einstein metrics, play an important role in Complex Function Theory, Complex Dynamics and Convex Geometry. The first four metrics are used mostly because they are distance-decreasing. But they are almost never Hermitian. On the other hand, the Bergman metric and the Kähler–Einstein metric are Hermitian (in fact, Kählerian), but, in general, not distance-decreasing.

  The Wu metric (Cheung and Kim, 1996) is an invariant non-Kähler Hermitian metric on a complex manifold M ⁿ which is distance-decreasing, up to a fixed constant factor, for any holomorphic mapping between two complex manifolds.

• Kobayashi metric

  Let D be a domain in $\mathbb{C}^{n}$. Let $\mathcal{O}(\Delta,D)$ be the set of all analytic mappings $f: \Delta \rightarrow D$, where $\Delta =\{ z \in \mathbb{C}: \vert z\vert < 1\}$ is the unit disk.

  The Kobayashi metric (or Kobayashi–Royden metric ) F _K is a complex Finsler metric defined, for all z ∈ D and $u \in \mathbb{C}^{n}$, by

  $$\displaystyle{F_{K}(z,u) =\inf \{\alpha > 0: \exists f \in \mathcal{O}(\Delta,D),f(0) = z,\alpha f^{'}(0) = u\}.}$$

  Given a complex manifold M ⁿ, the Kobayashi semimetric F _K is defined by

  $$\displaystyle{F_{K}(p,u) =\inf \{\alpha > 0: \exists f \in \mathcal{O}(\Delta,M^{n}),f(0) = p,\alpha f^{'}(0) = u\}}$$

  for all p ∈ M ⁿ and u ∈ T _p (M ⁿ).

  F _K (p, u) is a seminorm of the tangent vector u, called the Kobayashi seminorm. F _K is a metric if M ⁿ is taut, i.e., $\mathcal{O}(\Delta,M^{n})$ is a normal family (every sequence has a subsequence which either converge or diverge compactly).

  The Kobayashi semimetric is an infinitesimal form of the Kobayashi semidistance (or Kobayashi pseudo-distance, 1967) $K_{M^{n}}$ on M ⁿ, defined as follows. Given p, q ∈ M ⁿ, a chain of disks α from p to q is a collection of points $p = p^{0},p^{1},\ldots,p^{k} = q$ of M ⁿ, pairs of points $a^{1},b^{1};\ldots;a^{k},b^{k}$ of the unit disk $\Delta$, and analytic mappings $f_{1},\ldots f_{k}$ from $\Delta$ into M ⁿ, such that $f_{j}(a^{j}) = p^{j-1}$ and f _j (b ^j) = p ^j for all j.

  The length l(α) of a chain α is the sum $d_{P}(a^{1},b^{1}) +\ldots +d_{P}(a^{k},b^{k})$, where d _P is the Poincaré metric. The Kobayashi semimetric $K_{M^{n}}$ on M ⁿ is defined by

  $$\displaystyle{K_{M^{n}}(p,q) =\inf _{\alpha }l(\alpha ),}$$

  where the infimum is taken over all lengths l(α) of chains of disks α from p to q.

  Given a complex manifold M ⁿ, the Kobayashi–Busemann semimetric on M ⁿ is the double dual of the Kobayashi semimetric. It is a metric if M ⁿ is taut.

• Carathéodory metric

  Let D be a domain in $\mathbb{C}^{n}$. Let $\mathcal{O}(D,\Delta )$ be the set of all analytic mappings $f: D \rightarrow \Delta$, where $\Delta =\{ z \in \mathbb{C}: \vert z\vert < 1\}$ is the unit disk.

  The Carathéodory metric F _C is a complex Finsler metric defined by

  $$\displaystyle{F_{C}(z,u) =\sup \{ \vert f^{'}(z)u\vert: f \in \mathcal{O}(D,\Delta )\}}$$

  for any z ∈ D and $u \in \mathbb{C}^{n}$.

  Given a complex manifold M ⁿ, the Carathéodory semimetric F _C is defined by

  $$\displaystyle{F_{C}(p,u) =\sup \{ \vert f^{'}(p)u\vert: f \in \mathcal{O}(M^{n},\Delta )\}}$$

  for all p ∈ M ⁿ and u ∈ T _p (M ⁿ). F _C is a metric if M ⁿ is taut, i.e., every sequence in $\mathcal{O}(\Delta,M^{n})$ has a subsequence which either converge or diverge compactly.

  The Carathéodory semidistance (or Carathéodory pseudo-distance, 1926) $C_{M^{n}}$ is a semimetric on a complex manifold M ⁿ, defined by

  $$\displaystyle{C_{M^{n}}(p,q) =\sup \{ d_{P}(f(p),f(q)): f \in \mathcal{O}(M^{n},\Delta )\},}$$

  where d _P is the Poincaré metric.

  In general, the integrated semimetric of the infinitesimal Carathéodory semimetric is internal for the Carathéodory semidistance, but does not equal to it.

• Azukawa semimetric

  Let D be a domain in $\mathbb{C}^{n}$. Let K _D (z) be the set of all logarithmically plurisubharmonic functions f: D → [0, 1] such that there exist M, r > 0 with f(u) ≤ M | | u − z | | ₂ for all u ∈ B(z, r) ⊂ D; here | | . | | ₂ is the l ₂ -norm on $\mathbb{C}^{n}$, and $B(z,r) =\{ x \in \mathbb{C}^{n}: \vert \vert z - x\vert \vert _{2} < r\}$. Let g _D (z, u) be $\sup \{f(u): f \in K_{D}(z)\}$.

  The Azukawa semimetric F _A is a complex Finsler metric defined by

  $$\displaystyle{F_{A}(z,u) = \overline{\lim }_{\lambda \rightarrow 0} \frac{1} {\vert \lambda \vert }g_{D}(z,z +\lambda u)}$$

  for all z ∈ D and $u \in \mathbb{C}^{n}$.

  The Azukawa metric is an infinitesimal form of the Azukawa semidistance.

• Sibony semimetric

  Let D be a domain in $\mathbb{C}^{n}$. Let K _D (z) be the set of all logarithmically plurisubharmonic functions f: D → [0, 1) such that there exist M, r > 0 with f(u) ≤ M | | u − z | | ₂ for all $u \in B(z,r) =\{ x \in \mathbb{C}^{n}: \vert \vert z - x\vert \vert _{2} < r\} \subset D$. Let C _loc ²(z) be the set of all functions of class C ² on some open neighborhood of z.

  The Sibony semimetric F _S is a complex Finsler semimetric defined by

  $$\displaystyle{F_{S}(z,u) =\sup _{f\in K_{D}(z)\cap C_{\mathit{loc}}^{2}(z)}\sqrt{\sum _{i,j } \frac{\partial ^{2 } f} {\partial z_{i}\partial \overline{z}_{j}}(z)u_{i}\overline{u}_{j}}}$$

  for all z ∈ D and $u \in \mathbb{C}^{n}$.

  The Sibony semimetric is an infinitesimal form of the Sibony semidistance.

• Teichmüller metric

  A Riemann surface R is a one-dimensional complex manifold. Two Riemann surfaces R ₁ and R ₂ are called conformally equivalent if there exists a bijective analytic function (i.e., a conformal homeomorphism) from R ₁ into R ₂. More precisely, consider a fixed closed Riemann surface R ₀ of a given genus g ≥ 2.

  For a closed Riemann surface R of genus g, one can construct a pair (R, f), where f: R ₀ → R is a homeomorphism. Two pairs (R, f) and (R ₁, f ₁) are called conformally equivalent if there exists a conformal homeomorphism h: R → R ₁ such that the mapping $(f_{1})^{-1} \cdot h \cdot f: R_{0} \rightarrow R_{0}$ is homotopic to the identity.

  An abstract Riemann surface R ^∗ = (R, f)^∗ is the equivalence class of all Riemann surfaces, conformally equivalent to R. The set of all equivalence classes is called the Teichmüller space T(R ₀) of the surface R ₀.

  For closed surfaces R ₀ of given genus g, the spaces T(R ₀) are isometrically isomorphic, and one can speak of the Teichmüller space T _g of surfaces of genus g. T _g is a complex manifold. If R ₀ is obtained from a compact surface of genus g ≥ 2 by removing n points, then the complex dimension of T _g is $3g - 3 + n$.

  The Teichmüller metric is a metric on T _g defined by

  $$\displaystyle{\frac{1} {2}\inf _{h}\ln K(h)}$$

  for any R ₁ ^∗, R ₂ ^∗ ∈ T _g , where h: R ₁ → R ₂ is a quasi-conformal homeomorphism, homotopic to the identity, and K(h) is the maximal dilation of h. In fact, there exists a unique extremal mapping, called the Teichmüller mapping which minimizes the maximal dilation of all such h, and the distance between R ₁ ^∗ and R ₂ ^∗ is equal to $\frac{1} {2}\ln K$, where the constant K is the dilation of the Teichmüller mapping.

  In terms of the extremal length $\mathit{ext}_{R^{{\ast}}}(\gamma )$, the distance between R ₁ ^∗ and R ₂ ^∗ is

  $$\displaystyle{\frac{1} {2}\ln \sup _{\gamma }\frac{\mathit{ext}_{R_{1}^{{\ast}}}(\gamma )} {\mathit{ext}_{R_{2}^{{\ast}}}(\gamma )},}$$

  where the supremum is taken over all simple closed curves on R ₀.

  The Teichmüller space T _g , with the Teichmüller metric on it, is a geodesic metric space (moreover, a straight G -space) but it is neither Gromov hyperbolic, nor a Busemann convex metric space.

  The Thurston quasi-metric on the Teichmüller space T _g is defined by

  $$\displaystyle{\frac{1} {2}\inf _{h}\ln \vert \vert h\vert \vert _{\mathit{Lip}}}$$

  for any R ₁ ^∗, R ₂ ^∗ ∈ T _g , where h: R ₁ → R ₂ is a quasi-conformal homeomorphism, homotopic to the identity, and | | . | |  _Lip is the Lipschitz norm on the set of all injective functions f: X → Y defined by $\vert \vert f\vert \vert _{\mathit{Lip}} =\sup _{x,y\in X,x\neq y}\frac{d_{Y }(f(x),f(y))} {d_{X}(x,y)}$.

  The moduli space R _g of conformal classes of Riemann surfaces of genus g is obtained by factorization of T _g by some countable group of automorphisms of it, called the modular group. The Zamolodchikov metric, defined (1986) in terms of exactly marginal operators, is a natural metric on the conformal moduli spaces.

  Liu, Sun and Yau, 2005, showed that all known complete metrics on the Teichmüller space and moduli space (including Teichmüller metric, Bergman metric, Cheng–Yau–Mok Kähler–Einstein metric, Carathéodory metric, McMullen metric) are equivalent since they are quasi-isometric (cf. Chap. 1) to the Ricci metric and the perturbed Ricci metric introduced by them.

• Weil–Petersson metric

  The Weil–Petersson metric is a Kähler metric on the Teichmüller space T _g, n of abstract Riemann surfaces of genus g with n punctures and negative Euler characteristic. This metric has negative Ricci curvature; it is geodesically convex (cf. Chap. 1) and not complete.

  The Weil–Peterson metric is Gromov hyperbolic if and only if (Brock and Farb, 2006) the complex dimension $3g - 3 + n$ of T _g, n is at most two.

• Gibbons–Manton metric

  The Gibbons–Manton metric is a 4n-dimensional hyper-Kähler metric on the moduli space of n-monopoles which admits an isometric action of the n-dimensional torus T ⁿ. It is a hyper-Kähler quotient of a flat quaternionic vector space.

• Metrics on determinant lines

  Let M ⁿ be an n-dimensional compact smooth manifold, and let F be a flat vector bundle over M ⁿ. Let $H^{\bullet }(M^{n},F) = \oplus _{i=0}^{n}H^{i}(M^{n},F)$ be the de Rham cohomology of M ⁿ with coefficients in F. Given an n-dimensional vector space V, the determinant line det V of V is defined as the top exterior power of V, i.e., det V = ∧ⁿ V. Given a finite-dimensional graded vector space $V = \oplus _{i=0}^{n}V _{i}$, the determinant line of V is defined as the tensor product $\mathit{det}\,\,V = \otimes _{i=0}^{n}(\mathit{det}V _{i})^{(-1)^{i} }$.

  Thus, the determinant line det H ^•(M ⁿ, F) of the cogomology H ^•(M ⁿ, F) can be written as $\mathit{det}H^{\bullet }(M^{n},F) = \otimes _{i=0}^{n}(\mathit{det}H^{i}(M^{n},F))^{(-1)^{i} }$.

  The Reidemeister metric is a metric on det H ^•(M ⁿ, F), defined by a given smooth triangulation of M ⁿ, and the classical Reidemeister–Franz torsion.

  Let g ^F and $g^{T(M^{n}) }$ be smooth metrics on the vector bundle F and tangent bundle T(M ⁿ), respectively. These metrics induce a canonical L ₂ -metric $h^{H^{\bullet }(M^{n},F) }$ on H ^•(M ⁿ, F). The Ray–Singler metric on det H ^•(M ⁿ, F) is defined as the product of the metric induced on det H ^•(M ⁿ, F) by $h^{H^{\bullet }(M^{n},F) }$ with the Ray–Singler analytic torsion. The Milnor metric on det H ^•(M ⁿ, F) can be defined in a similar manner using the Milnor analytic torsion. If g ^F is flat, the above two metrics coincide with the Reidemeister metric. Using a co-Euler structure, one can define a modified Ray–Singler metric on det H ^•(M ⁿ, F).

  The Poincaré–Reidemeister metric is a metric on the cohomological determinant line det H ^•(M ⁿ, F) of a closed connected oriented odd-dimensional manifold M ⁿ. It can be constructed using a combination of the Reidemeister torsion with the Poincaré duality. Equivalently, one can define the Poincaré–Reidemeister scalar product on det H ^•(M ⁿ, F) which completely determines the Poincaré–Reidemeister metric but contains an additional sign or phase information.

  The Quillen metric is a metric on the inverse of the cohomological determinant line of a compact Hermitian one-dimensional complex manifold. It can be defined as the product of the L ₂-metric with the Ray-Singler analytic torsion.

• Kähler supermetric

  The Kähler supermetric is a generalization of the Kähler metric for the case of a supermanifold. A supermanifold is a generalization of the usual manifold with fermonic as well as bosonic coordinates. The bosonic coordinates are ordinary numbers, whereas the fermonic coordinates are Grassmann numbers.

  Here the term supermetric differs from the one used in this chapter.

• Hofer metric

  A symplectic manifold (M ⁿ, w), n = 2k, is a smooth even-dimensional manifold M ⁿ equipped with a symplectic form, i.e., a closed nondegenerate 2-form, w.

  A Lagrangian manifold is a k-dimensional smooth submanifold L ^k of a symplectic manifold (M ⁿ, w), n = 2k, such that the form w vanishes identically on L ^k, i.e., for any p ∈ L ^k and any x, y ∈ T _p (L ^k), one has w(x, y) = 0.

  Let $\mathit{L}(M^{n},\Delta )$ be the set of all Lagrangian submanifolds of a closed symplectic manifold (M ⁿ, w), diffeomorphic to a given Lagrangian submanifold $\Delta$. A smooth family α = { L _t } _t , t ∈ [0, 1], of Lagrangian submanifolds $L_{t} \in \mathit{L}(M^{n},\Delta )$ is called an exact path connecting L ₀ and L ₁, if there exists a smooth mapping $\Psi: \Delta \times [0,1] \rightarrow M^{n}$ such that, for every t ∈ [0, 1], one has $\Psi (\Delta \times \{ t\}) = L_{t}$, and $\Psi {\ast} w = \mathit{dH}_{t} \wedge \mathit{dt}$ for some smooth function $H: \Delta \times [0,1] \rightarrow \mathbb{R}$. The Hofer length l(α) of an exact path α is defined by $l(\alpha ) =\int _{ 0}^{1}\{\max _{p\in \Delta }H(p,t) -\min _{p\in \Delta }H(p,t)\}\mathit{dt}$.

  The Hofer metric on the set $\mathit{L}(M^{n},\Delta )$ is defined by

  $$\displaystyle{\inf _{\alpha }l(\alpha )}$$

  for any $L_{0},L_{1} \in \mathit{L}(M^{n},\Delta )$, where the infimum is taken over all exact paths on $\mathit{L}(M^{n},\Delta )$, that connect L ₀ and L ₁.

  The Hofer metric can be defined similarly on the group Ham(M ⁿ, w) of Hamiltonian diffeomorphisms of a closed symplectic manifold (M ⁿ, w), whose elements are time-one mappings of Hamiltonian flows ϕ _t ^H: it is inf _α l(α), where the infimum is taken over all smooth paths α = {ϕ _t ^H}, t ∈ [0, 1], connecting ϕ and ψ.

• Sasakian metric

  A Sasakian metric is a metric on a contact manifold, naturally adapted to the contact structure.

  A contact manifold equipped with a Sasakian metric is called a Sasakian space, and it is an odd-dimensional analog of a Kähler manifold. The scalar curvature of a Sasakian metric which is also Einstein metric, is positive.

• Cartan metric

  A Killing form (or Cartan–Killing form) on a finite-dimensional Lie algebra $\Omega$ over a field $\mathbb{F}$ is a symmetric bilinear form

  $$\displaystyle{B(x,y) = \mathrm{Tr}(\mathit{ad}_{x} \cdot \mathit{ad}_{y}),}$$

  where Tr denotes the trace of a linear operator, and ad _x is the image of x under the adjoint representation of $\Omega$, i.e., the linear operator on the vector space $\Omega$ defined by the rule z → [x, z], where [, ] is the Lie bracket.

  Let $e_{1},\ldots e_{n}$ be a basis for the Lie algebra $\Omega$, and $[e_{i},e_{j}] =\sum _{ k=1}^{n}\gamma _{\mathit{ij}}^{k}e_{k}$, where γ _ij ^k are corresponding structure constants. Then the Killing form is given by

  $$\displaystyle{B(x_{i},x_{j}) = g_{\mathit{ij}} =\sum _{ k,l=1}^{n}\gamma _{ \mathit{il}}^{k}\gamma _{ \mathit{ik}}^{l}.}$$

  In Theoretical Physics, the metric tensor ((g _ij )) is called a Cartan metric.