Abstract
Given an isometry invariant valuation on a complex space form we compute its value on the tubes of sufficiently small radii around a set of positive reach. This generalizes classical formulas of Weyl, Gray and others about the volume of tubes. We also develop a general framework on tube formulas for valuations in Riemannian manifolds.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
For a compact convex set \(A\subset {\mathbb {R}}^m\), the Steiner formula computes the volume of the set \(A_t\) consisting of points at distance smaller than t from A as follows
Here the functionals \(\mu _i\) are the so-called intrinsic volumes, and the normalizing constant \(\omega _k\) is the volume of the k-dimensional unit ball. By Hadwiger’s characterization theorem, the intrinsic volumes span the space of valuations (finitely additive functionals on convex bodies) that are continuous and invariant under rigid motions.
The famous tube formula of H. Weyl [28] is the assertion that (1) holds true for \(A\subset {\mathbb {R}}^m\) a smooth compact submanifold and \(t\ge 0\) small enough, with the additional insight that the coefficients \(\mu _i(A)\) depend only on the induced Riemannian structure of A. Even more generally, Federer extended the validity of (1) to the class of compact sets of positive reach. Later on, the same formula has been proven to hold for bigger classes of sets (see e.g. [14, 16]). As for the coefficients \(\mu _i\), the current perspective is to view them as smooth valuations in the sense of Alesker’s theory of valuations on manifolds (see [4]).
Already in Weyl’s original work, the tube formula was extended to the sphere and to hyperbolic space. In that case, instead of a polynomial on the radius t one has a polynomial in certain functions \(\sin _\lambda (t),\cos _\lambda (t)\) whose definition we recall in (52). Later, Gray and Vanhecke computed the volume of tubes around submanifolds of rank one symmetric spaces (cf. [19, 20]).
All these classical tube formulas are most naturally expressed in the language of valuations on manifolds. Furthermore, this theory has allowed for the determination of kinematic formulas (a far-reaching generalization of tube formulas) in isotropic spaces. These spaces are Riemannian manifolds under the action of a group of isometries that is transitive on the sphere bundle. For instance, in [10] and [11] the kinematic formulas of complex complex space forms (i.e. complex euclidean, projective and hyperbolic spaces) were obtained, and Gray’s tube formulas on such spaces were recovered.
Tube formulas, however, exist also for other valuations than the volume, and these do not follow from the kinematic formulas. For instance, differentiating the Steiner formula one easily obtains
In real space forms (i.e. the sphere and hyperbolic space), Santaló obtained similar tube formulas for all isometry invariant valuations (see [26]). For rank one symmetric spaces, the tube formulas of a certain class of valuations (integrated mean curvatures) were found in [19], still with a differential-geometric viewpoint. There are however many invariant valuations on these spaces that were not considered.
In this paper we prove the existence of tube formulas for any smooth valuation in a Riemannian manifold. Then we develop a method to determine these formulas for the invariant valuations of an isotropic space. Using this method we compute all tube formulas explicitly in the case of complex space forms. In fact, our approach also reveals some interesting aspects in the case of real space forms.
Let us briefly describe our results. First, given a Riemannian manifold M we construct a family \({\textbf{T}}_t\) of tubular operators on the space \({\mathcal {V}}(M)\) of smooth valuations of M such that for any \(\mu \in {\mathcal {V}}(M)\) and every compact set of positive reach \(A\subset M\) one has
for \(t\ge 0\) small enough (see Definition 4.1 and Corollary 4.7). Differentiating \({\textbf{T}}_t\) at \(t=0\) yields an operator \(\partial :{\mathcal {V}}(M)\rightarrow {\mathcal {V}}(M)\). If G is a group of isometries of M acting transitively on the sphere bundle SM, the subspace \({\mathcal {V}}(M)^G\) of G-invariant valuations is finite dimensional, and the determination of the tube operators \({\textbf{T}}_t\) reduces to the computation of the flow generated by \(\partial \).
Once this general framework is established we concentrate on the complex space forms \({\mathbb {C}}P^{n}_{\lambda }\). For \(\lambda =0\) this refers to complex euclidean space \({\mathbb {C}}^n\) under the group of complex isometries, and for \(\lambda \ne 0\) this is the n-dimensional complex projective or hyperbolic space of constant holomorphic curvature \(4\lambda \), under the full group of isometries G. We simply denote \(\mathcal V_{\lambda ,{\mathbb {C}}}^n:= {\mathcal {V}}({\mathbb {C}}P^n_\lambda )^G\).
For \(\lambda =0\), we will readily obtain the tube formulas \({\textbf{T}}_t\mu \) of all translation-invariant and U(n)-invariant continuous valuations \(\mu \) thanks to the existence of an \(\mathfrak {sl}_2\)-module structure on the space \({\text {Val}}^{U(n)}\) of such valuations. This structure, discovered by Bernig and Fu in [10], is induced by two natural operators \(\Lambda ,L\), the first of which is a normalization of \(\partial \).
Remarkably, it turns out that also for \(\lambda \ne 0\) the derivation operator \(\partial \) is closely related to the operators \(\Lambda ,L\) of the flat space. Indeed, in Theorem 4.11 we find an isomorphism \(\Phi _\lambda :{\text {Val}}^{U(n)}\rightarrow {\mathcal {V}}_{\lambda ,{\mathbb {C}}}^n\) such that
Using the decomposition of \({\text {Val}}^{U(n)}\) into \(\mathfrak {sl}_2\)-irreducible components, the computation of the tubular operator boils down to the solution of a Cauchy problem in some abstract model spaces, yielding our main result.
Theorem
There exists a basis \(\{\sigma _{k,r}^\lambda \}\) of the space \({\mathcal {V}}_{\lambda ,{\mathbb {C}}}^n\) of invariant valuations of \({\mathbb {C}}P^{n}_{\lambda }\) such that
where
We describe the basis \(\sigma _{k,r}^\lambda \) explicitly in terms of the previously known valuations \(\tau _{k,p}^\lambda \) of [11]. The tube formulas for the \(\tau _{k,p}^\lambda \) can be easily obtained from the previous ones, as we also provide the expression of these valuations in terms of the \(\sigma _{k,r}^\lambda \).
Curiously, the expressions (4) are extremely similar to those obtained by Santaló in the real space form \({\mathbb {S}}^{m}_{\lambda }\) of constant curvature \(\lambda \). Indeed, for a certain basis \(\{\sigma _i\}_{i=0}^m\) of the space \(\mathcal V_{\lambda ,{\mathbb {R}}}^m\) of isometry invariant valuations of \({\mathbb {S}}^{m}_{\lambda }\) one has
The tube formula for \(\sigma _m={\text {vol}}\) is however quite different. As an explanation for these similarities, we show in Theorem 4.12 the existence of a phenomenon similar (but not completely analogous) to (3).
The paper concludes with a detailed study of the spectrum and the eigenspaces of the derivative operator \(\partial \) in \(\mathcal V_{\lambda ,{\mathbb {C}}}^n\) and \({\mathcal {V}}_{\lambda ,{\mathbb {R}}}^m\). In particular, we compute the kernel of \(\partial \) in \({\mathcal {V}}_{\lambda ,{\mathbb {C}}}^n\); i.e. we determine the invariant valuations of \({\mathbb {C}}P_\lambda ^n\) for which the tube formulas are constant. We also identify the images \(\partial ({\mathcal {V}}_{\lambda ,{\mathbb {C}}}^n)\) and \(\partial (\mathcal V_{\lambda ,{\mathbb {R}}}^m)\), and we compute the preimage by \(\partial \) of any element belonging to these subspaces.
2 Background
2.1 Valuations
Let V be a finite-dimensional real vector space, and let \({\mathcal {K}}(V)\) be the space of convex compact subsets of V, endowed with the Hausdorff metric. A valuation on V is a map \(\varphi :{\mathcal {K}} (V) \rightarrow {\mathbb {C}}\) such that
for \(A,B,A\cup B\in {\mathcal {K}}(V)\). The space of translation-invariant, continuous valuations on V is denoted by \({\text {Val}}(V)\).
The notion of valuation was extended to smooth manifolds by Alesker (cf. [2,3,4, 6]). For simplicity we will focus on the case of a Riemannian manifold \(M^n\). It is also natural to consider here the class of compact sets of positive reach in M, which we denote \({\mathcal {R}}(M)\). The definition and some basic properties of such sets are recalled in Sect. 4.2.
Let SM be the sphere bundle of M consisting of unit tangent vectors, and let \(\pi :SM\rightarrow M\) be the canonical projection.
Definition 2.1
(Smooth valuation) A smooth valuation on M is a functional \(\varphi :{\mathcal {R}}(M) \rightarrow {\mathbb {C}}\) of the form
where \(\omega \in \Omega ^{n-1}(SM)\) and \(\eta \in \Omega ^{n}(M)\), are complex-valued differential forms, and N(A) is the normal cycle of A (cf. e.g. [14]). We will denote \(\varphi =[\![\omega ,\eta ]\!]\) in this case. For any subgroup \(G \le {\text {Diff}}(M)\), we will denote by \({\mathcal {V}}^{G}(M)\) the space of G-invariant valuations; i.e. \(\mu \in {\mathcal {V}}(M)\) such that \(\mu (gA) = \mu (A)\) for all \(A \in {\mathcal {R}}(M)\) and \(g \in G\).
The kernel of the map \((\omega ,\eta )\mapsto [\![\omega ,\eta ]\!]\) was determined by Bernig and Bröcker in [8] as follows. Given \(\omega \in \Omega ^{n-1}(SM)\), there exists \(\xi \in \Omega ^{n-2}(SM)\) such that
is a multiple of \(\alpha \), the canonical contact form on SM. The unique n-form \(D\omega \) satisfying this condition is called the Rumin differential of \(\omega \) (see [24]). Then \([\![\omega ,\eta ]\!]= 0\) if and only if
One of the most striking aspects of Alesker’s theory of valuations on manifolds is the existence of a natural product on \({\mathcal {V}}(M)\), which turns this space into an algebra with \(\chi \) as the unit element. The realization by Fu that this product is closely tied to kinematic formulas opened the door to the recent development of integral geometry in several spaces, including the complex space forms [1, 10, 11].
Another important algebraic structure is the convolution of valuations found by Bernig and Fu in linear spaces (cf. [9], but also [5]). This is a product on the dense subspace \({\text {Val}}^\infty (V):={\text {Val}}(V)\cap {\mathcal {V}}(V)\) characterized as follows. Given \(A \in {\mathcal {K}}(V)\), with smooth and positively curved boundary, we have \(\mu _A(\cdot ):= {\text {vol}}(\cdot + A)\in {\text {Val}}^\infty (V)\). The convolution is determined by
where \(+\) refers to the Minkowski sum. In particular, \({\text {vol}}\) is the unit element of this operation.
2.2 Real space forms
The fundamental examples of valuations in Euclidean space \({\mathbb {R}}^m\) are the intrinsic volumes \(\mu _k\). These are implicitly defined by the Steiner formula
where \({\mathbb {B}}^m\) is the unit ball and \(\omega _i\) is the volume of the i-dimensional unit ball. In particular \(\mu _0 = \chi \), \(\mu _{m-1} = \frac{1}{2}\,\textrm{perimeter}\), and \(\mu _n = \textrm{vol}_{m}\) are intrinsic volumes.
We will denote by \({\mathbb {S}}^{m}_{\lambda }\) the m-dimensional complete and simply connected Riemannian manifold of constant curvature \(\lambda \). That is, the sphere \(S^m(\sqrt{\lambda })\) for \(\lambda > 0\), Euclidean space \({\mathbb {R}}^n\) for \(\lambda = 0\), and hyperbolic space \(H^m(\sqrt{-\lambda })\) for \(\lambda < 0\). Let \(G_{\lambda ,{\mathbb {R}}}\) be the group of orientation preserving isometries of \({\mathbb {S}}^{m}_{\lambda }\); i.e. \(G_{\lambda ,{\mathbb {R}}}\cong SO(m+1)\) for \(\lambda >0\), and \(G_{\lambda ,{\mathbb {R}}} \cong SO(m) \ltimes {\mathbb {R}}^m\) for \(\lambda = 0\), while \(G_{\lambda ,{\mathbb {R}}}\cong PSO(m,1)\) for \(\lambda < 0\). We will denote by \({\mathcal {V}}_{\lambda ,{\mathbb {R}}}^m\) the space of \(G_{\lambda ,{\mathbb {R}}}\)-invariant valuations of \({\mathbb {S}}_\lambda ^m\).
Let \(\kappa _0,\dots ,\kappa _{m-1}\in \Omega ^{m-1}(S{\mathbb {S}}^{m}_{\lambda })^{{{G}}_{\lambda ,{\mathbb {R}}}}\) be the differential forms defined in [13, §0.4.4]. In the same paper it was shown that the \({\mathbb {R}}\)-algebra of \({G}_{\lambda ,{\mathbb {R}}}\)-invariant differential forms is generated by \(\kappa _0,\dots ,\kappa _{m-1},\alpha ,d\alpha \). It follows by [11, Prop. 2.6] that the following valuations constitute a basis of \({\mathcal {V}}_{\lambda ,{\mathbb {R}}}^{m}\)
In euclidean space \({\mathbb {R}}^m\) these valuations are proportional to the intrinsic volumes:
For general \(\lambda \), the \(\sigma _i^\lambda \) are proportional to the valuations \(\tau _i^\lambda \) appearing in [7, 17]
As we will see, the normalization taken for the \(\sigma _i^\lambda \) makes the tube formulas in \({\mathcal {V}}_{\lambda ,{\mathbb {R}}}^m\) specially simple. A stronger reason in favor of this normalization is Theorem 4.12.
2.3 Complex space forms
We denote by \({\mathbb {C}}P^{n}_{\lambda }\) the complete, simply connected n-dimensional Kähler manifold of constant holomorphic curvature \(4\lambda \); i.e. the complex projective space (with the suitably normalized Fubini-Study metric) for \(\lambda > 0\), the complex euclidean space \({\mathbb {C}}^n\) for \(\lambda = 0\), and the complex hyperbolic space for \(\lambda < 0\). For \(\lambda \ne 0\) we let \(G_{\lambda ,{\mathbb {C}}}\) be the full isometry group of \({\mathbb {C}}P^{n}_{\lambda }\). For \(\lambda =0\) we put \(G_{\lambda ,{\mathbb {C}}} = U(n) \ltimes {\mathbb {C}}^n\). We denote by \({\mathcal {V}}_{\lambda ,{\mathbb {C}}}^n\) the space of \(G_{\lambda ,{\mathbb {C}}}\)-invariant valuations on \({\mathbb {C}}P^{n}_{\lambda }\).
Let \(\{\beta _{k,q},\gamma _{k,q}\} \subset \Omega ^{2n-1}(S{\mathbb {C}}P^{n}_{\lambda })^{G_{\lambda ,{\mathbb {C}}}}\) be the differential forms introduced in [10] for \(\lambda =0\), and extended to the curved case \(\lambda \ne 0\) in [11]. Let also
where \(d\textrm{vol}\) is the Riemannian volume element. It was shown in [10, 11] that these valuations \(\mu _{k,q}^\lambda \) with \(\max \{0, k-n\} \le q \le \frac{k}{2} \le n\) constitute a basis of \({\mathcal {V}}^{n}_{\lambda ,{\mathbb {C}}}\). It is convenient to emphasize that the \(\mu _{k,q}^\lambda \) do not coincide with the hermitian intrinsic volumes \(\mu _{k,q}^M\) for \(M={\mathbb {C}}P^{n}_{\lambda }\) introduced in [12].
For \(\lambda =0\) we simply write \(\mu _{k,q}\) instead of \(\mu _{k,q}^0\). We will also use the so-called Tasaki valuations
It will be useful to consider the following linear isomorphisms:
More generally, whenever we have a valuation \(\nu \) in \({\text {Val}}^{U(n)}\) we will denote \(\nu ^\lambda := {\mathcal {F}}_{\lambda ,{\mathbb {C}}}(\nu )\). For instance \(\tau _{k,q}^\lambda =\mathcal {F_{\lambda ,{\mathbb {C}}}}(\tau _{k,q})\).
3 Tube formulas in linear spaces
Let V be an m-dimensional euclidean vector space. Given \(t\ge 0\), let \({\textbf{T}}_t:{\text {Val}}(V)\rightarrow {\text {Val}}(V)\) be given by
where \({\mathbb {B}}^m\) is the unit ball. We will call \({\textbf{T}}_t\) the tubular operator. Let also \(\partial :{\text {Val}}(V) \rightarrow {\text {Val}}(V)\) be the operator given by
This operator has sometimes been denoted by \(\Lambda \) in the literature, but following [10] we reserve the symbol \(\Lambda \) for a certain normalization of \(\partial \) (see (18)).
The properties of the Minkowski sum ensure that \({\textbf{T}}_{t+s} = {\textbf{T}}_t \circ {\textbf{T}}_s={\textbf{T}}_s \circ {\textbf{T}}_t\). Differentiating with respect to s at zero yields
It follows that
For each \(\mu \in {\text {Val}}(V)\), the map \(t \mapsto {\textbf{T}}_t\mu \) is a polynomial in t of degree m by (12) and the Steiner formula (7) (or by [23]). Hence
Note also that, by (15) and (16), the derivative operator \(\partial \) is \((m+1)\)-nilpotent; i.e. \(\partial ^{m+1} = 0\).
Let us compute the tube formula for the intrinsic volume \(\mu _i\) for each \(0 \le i \le m\) using (17). For that purpose we first compute \(\partial \). Since \({\textbf{T}}_{t+s} = {\textbf{T}}_s\circ {\textbf{T}}_t\) we have
On the other hand
Differentiating at \(s=0\) and comparing coefficients yields
Finally, using (17), we get
which is (2).
In order to compute the tube formulas for invariant valuations in \({\mathbb {C}}^n\) (i.e. to determine \({\textbf{T}}_t\) on \({\text {Val}}^{U(n)}\)), it will be useful to recall the \(\mathfrak {sl}_2\)-module structure of \({\text {Val}}^{U(n)}\) found in [10]. Consider the linear maps \(\Lambda , L, H:{\text {Val}}^\infty (V)\rightarrow {\text {Val}}^\infty (V)\), defined as follows
where \(\cdot \) refers to the Alesker product.
Proposition 3.1
On \({\text {Val}}^{O(m)}\) the operators \(\Lambda ,L\) are given by
while on \({\text {Val}}^{U(n)}\) one has
which implies
Proof
The first two equalities are [7, eqs. (2.3.12) and (2.3.13)]. The rest is [10, Lemma 5.2]. \(\square \)
Proposition 3.2
([7, Prop. 2.3.10 (3)]) The operators \(\Lambda , L,H\) define an \(\mathfrak {sl}_2\)-module structure on both \({\text {Val}}^{O(m)}\) and \({\text {Val}}^{U(n)}\); i.e. \([L,\Lambda ]=H\), \([H,L]=2L, [H,\Lambda ]=-2\Lambda \).
The decomposition into irreducible components is as follows
where \(V^{(m)}\) is the \((m+1)-\)dimensional irreducible \(\mathfrak {sl}_2\)-representation. In particular, for \(0 \le 2r \le n\), there exists a unique, up to a multiplicative constant, primitive element (i.e. anihilated by \(\Lambda \)) in each irreducible component of \({\text {Val}}^{U(n)}\). By the so-called Lefschetz decomposition, the L-orbits of these primitive elements consitute a basis of \({\text {Val}}^{U(n)}\). This basis was explicitly computed in [10] as follows.
Proposition 3.3
([10, eq. (76)]) The following valuations
are \(\Lambda \)-primitive; i.e. \(\Lambda \pi _{2r,r}=0\). The family
forms a basis of \({\text {Val}}^{U(n)}\).
In particular the irreducible components of \({\text {Val}}^{U(n)}\) are the following subspaces
We are now able to compute the tube formulas in the complex case using (17).
Theorem 3.4
Proof
By [10, Lemma 5.6],
and then
Using (17), we obtain the tube formula
\(\square \)
These tube formulas can also be given in terms of the valuations \(\tau _{k,q}\). To this end, we next compute their Lefschetz decomposition.
Proposition 3.5
The Lefschetz decomposition of \(\tau _{k,r}\) is given by
Proof
Consider the linear map \(\psi : {\text {Val}}^{U(n)} \rightarrow {\text {Val}}^{U(n)}\) mapping \(\tau _{k,r}\) to the left hand side of (33). We need to show that \(\psi ={\text {id}}\). Let us check that this endomorphism commutes with both \(\Lambda \) and L. To check commutation with \(\Lambda \), we only need to verify the following
Comparing term by term, the previous identities boil down to
which is trivial.
Commutation with L is straightforward using \(L\pi _{k,i} = \pi _{k+1,i}\).
Given that \(\psi \) commutes with the operators \(\Lambda \) and L and \({\text {Val}}^{U(n)}\) is multiplicity-free, Schur’s lemma implies that for each \(0 \le 2r \le n\), there exists a constant \(c_r\) such that \(\left. \psi \right| _{{\mathcal {I}}_{0}^{n,r}} = c_r {\text {id}}\).
Let \(a_{2r,j}\) and \(b_{2r,i}\) be the coefficients of \(\pi _{2r,j}\) and \(\tau _{2r,i}\) in (33) and (26) respectively, so that \(\psi (\tau _{2r,i}) = \sum _{j= 0}^i a_{2r,j} \pi _{2r,j}\) and \(\pi _{2r,r} = \sum _{i = 0}^r b_{2r,i} \tau _{2r,i}\). Then
Comparing the coefficient of \(\pi _{2r,r}\) on both sides we get \(c_r=b_{2r,r}a_{2r,r}=1\) for each \(0 \le 2r \le n\). Hence \(\psi = {\text {id}}\), which proves (33). \(\square \)
By plugging (28) and (33) in (30) one gets the tube formulas \({\textbf{T}}_t\tau _{k,p}\) in terms of the \(\tau _{i,j}\).
4 Tube formulas in Riemannian manifolds
4.1 Tubular and derivative operators
Next we extend to any complete Riemannian manifold M the tubular operator \({\textbf{T}}_t\) introduced in the previous section on linear spaces. Let T be the Reeb vector field on SM, which is characterized by \(i_T \alpha = 1\) and \({\mathcal {L}}_T\alpha = 0\), where \({\mathcal {L}}\) is the Lie derivative. The Reeb flow \(\phi : SM \times {\mathbb {R}}\rightarrow SM\), defined as the flow of T, is a family of contactomorphisms and coincides with the geodesic flow on SM (see e.g. [18, Theorem 1.5.2]).
Definition 4.1
(Tubular and derivative operators) Given \(t\ge 0\), we define the tubular operator \({\textbf{T}}_t\) by
where \(p_t :SM \times [0,t] \rightarrow SM\) is the projection on the first factor, and \(\phi _t=\phi (\cdot ,t)\). We define the derivative operator \(\partial =\partial _M\) by
To show that these definitions are consistent, suppose \(\mu = [\![\omega ,\eta ]\!]= 0\), and let us check that \({\textbf{T}}_t\mu = 0\) for all \(t \ge 0\), i.e.
By (5) we have \(\pi ^*\eta = -D\omega =-d(\omega +\xi \wedge \alpha )\). Hence
as \(\alpha \) vanishes on N(A). Since \([\![\omega ,\eta ]\!]= 0\), we have \(\int _{N(A)}\omega = -\int _A\eta \). Therefore \({\textbf{T}}_t \mu = 0\).
Let us next establish some basic properties of these operators.
Lemma 4.1
Proof
Given a compact smooth submanifold \(N\subset SM\),
Since \(i_T\) and \(\phi _t^*\) commute, the result follows. \(\square \)
Proposition 4.2
For \(\mu = [\![\omega ,\eta ]\!]\),
-
(i)
\(\partial \mu = [\![i_T \left( d\omega + \pi ^*\eta \right) ,0]\!]\)
-
(ii)
\({\textbf{T}}_{t+s}\mu =({\textbf{T}}_{t}\circ {\textbf{T}}_{s})\mu \)
Proof
Modulo exact forms we have
Together with Lemma 4.1, this yields
Evaluating at \(t=0\), this gives (i).
In order to prove (ii), it is enough to check that both sides have the same derivative with respect to s, as they clearly agree for \(s=0\). By (35), we have
Since \(\phi _t^*\) and \(i_T\) commute, it follows from (35) that \(\frac{d}{ds}{\textbf{T}}_{t+s}=\frac{d}{ds}{\textbf{T}}_{t}\circ {\textbf{T}}_{s}\). \(\square \)
Fix \(\mu \in {\mathcal {V}}(M)\). It follows from Proposition 4.2 (ii) that
If \(\mu \in {\mathcal {V}}(M)^G\) for a group G acting on M by isometries, then also \({\textbf{T}}_t\mu \in {\mathcal {V}}(M)^G\). Hence, in case \({\mathcal {V}}^G(M)\) is finite-dimensional, computing \({\textbf{T}}_t\mu \) boils down to solving the Cauchy problem (36) with initial condition \({\textbf{T}}_0\mu =\mu \); i.e.
This is the approach we will follow to obtain the tube formulas for invariant valuations in complex space forms. Note that (37) coincides with (16) except that \(\partial \) does not need to be nilpotent for general M.
4.2 Tubes in Riemannian manifolds
Let M be a complete Riemannian manifold and let \(d:M \times M \rightarrow [0,\infty )\) be the Riemannian distance on M. For \(t\ge 0\), the tube of radius t around a subset \(A\subset M\) is defined as
where
Next we review some basic facts about tubes around sets of positive reach (introduced by Federer in euclidean spaces and by Kleinjohann in Riemannian manifolds). For such sets A we will prove that \({\textbf{T}}_t \mu (A)=\mu (A_t)\) for any \(\mu \in {\mathcal {V}}(M)\) and sufficiently small t.
Definition 4.2
(Sets of positive reach) A set of positive reach in M is a closed subset \(A\subset M\) for which there exists an open neighborhood \(U_A\supset A\) such that for every \( p\in U_A{\setminus } A\) there exists a unique point \(f_A(p)\in A\) such that \(d(p,f_A(p))=d_A(p)\), and a unique minimizing geodesic joining p with \(f_A(p)\). We denote by \({\mathcal {R}}(M)\) the class of compact sets of positive reach in M.
By the previous definition, there is a well-defined map
where \(\gamma \) is the unique minimizing geodesic such that \(\gamma (0) = f_A(p)\) and \(\gamma (d_A(p)) = p\).
It was shown by Kleinjohann [22, Satz 3.3] that \(N(A):= F_A(U_A{\setminus } A)\) is a naturally oriented compact Lipschitz submanifold of SM. The corresponding current, also denoted by N(A), is called the normal cycle of A. It follows from Proposition 4.6 below that N(A) is legendrian (i.e. it vanishes on multiples of \(\alpha \)).
Proposition 4.3
([22, Satz 3.3, Korollar 2.7]) Given a set of positive reach A in M there exists \(r=r_A>0\) such that \(A_r\subset U_A\) and
-
(i)
for \(0<t<r\) the restriction \(\left. F_A\right| _{\partial A_t}\) gives a bilipschitz homeomorphism between \(\partial A_t\) and N(A), preserving the natural orientations,
-
(ii)
the distance function \(d_A\) is of class \(C^1\) in \(A_{r}{\setminus } A\) and
$$\begin{aligned} \phi _{d_A(p)}(F_A(p))=(p,\nabla d_A(p)),\qquad \partial A_t=d_A^{-1}(\{t\}) \end{aligned}$$for \(0<t<r\). In particular, each level set \(\partial A_t\) with \(0<t<r\) is a \(C^1\)-regular hypersurface with unit normal vector field \(\nabla d_A\).
The following propositions are certainly well-known.
Proposition 4.4
For \(0<s<r=r_A\) the set \(A_s\) has positive reach and on \(A_r{\setminus } A_s\) we have
In particular \((A_s)_t=A_{t+s}\) for \(t+s<r\).
Proof
Let \(p \in A_r{\setminus } A_s\), and put \(d=d_A(p)\). Let \(\gamma :[0,d]\rightarrow A_r\) be the unique minimizing geodesic with \(\gamma (0)=f_A(p)\) and \(\gamma (d)=p\). In particular \(|\gamma '|=1\) and thus \(\gamma (s)\in A_s\).
Assume that \(\left. \gamma \right| _{[s,d]}\) does not minimize the distance between p and \(A_s\), i.e., there exists a smooth curve \(\alpha : [0,1] \rightarrow M\) with \(q:=\alpha (0)\in A_s\), \(\alpha (1)=p\) and length \(\ell (\alpha ) < d-s\). It follows that
a contradiction. We conclude that \(\gamma |_{[s,d]}\) realizes the distance \(d_{A_s}(p)\). Hence \(d_{A_s}(p)=d_A(p)-s\) and
\(\square \)
Proposition 4.5
For \(0<s<r_A\), the restriction \(\left. \phi _s\right| _{N(A)}\) is a bilipschitz homeomorphism between N(A) and \(N(A_s)\).
Proof
Take t with \(s<t<\min (r_A,s+r_{A_s})\). By Proposition 4.3, both \(\left. F_A\right| _{\partial A_t}:\partial A_t \rightarrow N(A)\) and \(\left. F_{A_s}\right| _{\partial A_t} :\partial A_t \rightarrow N(A_s)\) are bilipschitz homeomorphisms. By (39) we have
The statement follows. \(\square \)
Proposition 4.6
For \(0<t<r_A\) the composition \(\pi \circ \phi \) gives a bijective Lipschitz map between \(N(A) \times (0,t]\) and \(A_t {\setminus } A\).
Proof
Since \(\pi ,\phi \) are smooth, the restriction of \(\pi \circ \phi \) to the Lipschitz manifold \(N(A)\times (0,t]\) is clearly Lipschitz.
Given \((\xi ,s)\in N(A)\times (0,t]\), we know by the previous proposition that \(\phi (\xi ,s)\in N(A_s)\) and thus \(\pi \circ \phi (x,s)\in \partial A_s\subset A_t{\setminus } A\).
To check surjectivity, given \(p\in A_t{\setminus } A\) take \(\xi =F_A(p), s=d_A(p)\) and note that \(\pi \circ \phi (\xi ,s)=p\).
As for injectivity, suppose \(\pi \circ \phi (\xi _1,t_1)=\pi \circ \phi (\xi _2,t_2)=:p\) for some \((\xi _1,t_1),(\xi _2,t_2)\in N(A)\times (0,t]\). By the previous proposition p belongs to both \(\partial A_{t_1},\partial A_{t_2}\), so \(t_1=t_2\). For \(s\in [0,t_1]\), the geodesics \(\gamma _1(s)=\pi \circ \phi (\xi _1,s),\gamma _2(s)=\pi \circ \phi (\xi _2,s)\) realize the distance between p and A. Since \(A_s\subset A_{r_A}\subset U_A\), we have \(\gamma _1=\gamma _2\) and thus \(\xi _1=\xi _2\). \(\square \)
Corollary 4.7
For every \(A\in {\mathcal {R}}(M)\) and \(\mu \in {\mathcal {V}}(M)\) we have \(\mu (A_t) = {\textbf{T}}_t\mu (A)\) for \(0\le t\le r_A\).
Proof
Let \(\mu = [\![\omega ,\eta ]\!]\). By Propositions 4.5 and 4.6 and the coarea formula,
\(\square \)
Remark
In the subclass \({\mathcal {P}}(M)\subset {\mathcal {R}}(M)\) of compact submanifolds with corners, the normal cycle is more naturally defined as follows. For \(A \in {\mathcal {P}}(M)\) and \(p \in A\), let
Let us check that indeed \(N'(A)\) equals \(N(A)=F_A(U_A)\). Covering A by local charts (locally modelled on \({\mathbb {R}}^k\times [0,\infty )^l\subset {\mathbb {R}}^m\)), and considering the copy of \(N'(A)\) in the cosphere bundle of M, one sees that \(N'(A)\) is a compact topological manifold.
It is also easy to show that \(N(A)\subset N'(A)\). It then follows by the invariance of domain theorem that N(A) is an open subset of \(N'(A)\). Since \(N'(A)\) is a Hausdorff space and N(A) is compact, we also have that N(A) is a closed subset of \(N'(A)\). Since the number of connected components of both \(N(A),N'(A)\) clearly equals the number of connected components of A, we necessarily have \(N(A)=N'(A)\).
4.3 Derivative operators in \({\mathbb {S}}^{m}_{\lambda }\) and \({\mathbb {C}}P^{n}_{\lambda }\)
Given \(\lambda \in {\mathbb {R}}\) let \(\partial _{\lambda ,{\mathbb {R}}}:{\mathcal {V}}_{\lambda ,{\mathbb {R}}}^m\rightarrow {\mathcal {V}}_{\lambda ,{\mathbb {R}}}^m\) be the restriction of \(\partial _{{\mathbb {S}}^{m}_{\lambda }}\) to \({\mathcal {V}}_{\lambda ,{\mathbb {R}}}^m\), and let \(\partial _{\lambda ,{\mathbb {C}}}\) be the restriction of \(\partial _{{\mathbb {C}}P^{m}_{\lambda }}\) to \({\mathcal {V}}_{\lambda ,{\mathbb {C}}}^n\).
Proposition 4.8
where it is understood that \(\sigma _{-1}^\lambda =0\).
Let us emphasize that (40) would make formal sense but does not hold for \(i=m-1\).
Proof
By [15, Lemma 3.1], putting \(\kappa _m=0\), we have
Contracting with T yields
By Proposition 4.2 the result follows. \(\square \)
Lemma 4.9
The following equalities hold modulo \(\alpha ,d\alpha \)
-
(i)
For \(k > 2q\)
$$\begin{aligned} \begin{aligned} \dfrac{\omega _{2n-k}}{\omega _{2n-k+1}}i_{T} d\beta _{k,q}&\equiv 2 (n-k+q+1)\gamma _{k-1,q}\\ {}&\quad +{(k-2q+1)}\beta _{k-1,q-1} \\&\quad -\frac{\lambda }{2\pi } {(k-2q+1)(2n-k+1)}\beta _{k+1,q}. \end{aligned} \end{aligned}$$(43) -
(ii)
For \(n > k-q\)
$$\begin{aligned} \begin{aligned} \dfrac{\omega _{2n-2q}}{\omega _{2n-2q+1}}i_T d\gamma _{2q,q}&\equiv \beta _{2q-1,q-1} \\&\quad -\frac{\lambda }{2\pi } \frac{(q+2)(2n-2q+1)}{n-q}\beta _{2q+1,q} \\&\quad -\frac{\lambda }{2\pi } \frac{(n-q-1)(2n-2q+1)}{n-q}\gamma _{2q+1,q}. \end{aligned} \end{aligned}$$(44)
Proof
This is a straightforward computation using [1, Lemma 3.3, Lemma 3.6]. \(\square \)
Proposition 4.10
For \(k > 2q\)
and
Proof
Equality (45) follows from Proposition 4.2, using (43) and the following (see [1, Proposition 2.7])
Let us now prove (46). Note first that from (44) and (47) we get
Then, by Proposition 4.2 and observing that \(a_{n} = 2\)
A straightforward computation using \(k\omega _k=2\pi \omega _{k-2}\) shows
and the result follows. \(\square \)
Note that by (18) the linear map \(\Phi _0:{\text {Val}}^{U(n)}\rightarrow {\text {Val}}^{U(n)}\) given by \(\left. \Phi _0\right| _{{\text {Val}}_k^{U(n)}}=\omega _{2n-k}{\text {id}}\) satisfies
Remarkably, a similar identity holds for all \(\lambda \), which will be crucial for our determination of tube formulas in \({\mathbb {C}}P^{n}_{\lambda }\).
Theorem 4.11
The linear isomorphism
fulfills
Proof
By combining Proposition 4.10, Proposition 3.1 and the fact \(\frac{\omega _n}{\omega _{n-2}} = \frac{2\pi }{n}\), this is straightforward to check:
\(\square \)
A similar phenomenon holds in real space forms, but restricted to a hyperplane of \({\mathcal {V}}_{\lambda ,{\mathbb {R}}}^m\).
Theorem 4.12
The linear monomorphism
fulfills
Proof
By Proposition 4.8 and Theorem 3.4
\(\square \)
Note the difference of dimensions between the source and the target of \(\Psi _\lambda \). We will show that there is no isomorphism between \({\text {Val}}^{O(m)}\) and \({\mathcal {V}}_{\lambda ,{\mathbb {R}}}^m\) intertwining \(\partial \) and \(\Lambda -\lambda L\). This is essentially due to the fact that (41) and (42) differ from (40).
5 A model space for tube formulas
We next perform some abstract computations that will easily lead to the tube formulas in both complex and real space forms via (62) and (64). The same approach will allow us to determine the kernel, the image, and the spectrum of the derivative operator \(\partial \) on these spaces.
5.1 A system of differential equations
It is well-known that the operators \(X = x\frac{\partial }{\partial y}\), \(Y = y \frac{\partial }{\partial x}\) and \(H=[X,Y]\) induce an \(\mathfrak {sl}_2\)-structure on \({\mathbb {C}}[x,y]\). The decomposition into irreducible components is \({\mathbb {C}}[x,y]=\bigoplus _{m\ge 0} V^{(m)}\) where \(V^{(m)}\) is the subspace of m-homogeneous polynomials:
One has \(H(x^ky^{m-k})=(m-2k) x^k y^{m-k}\).
Motivated by Theorem 4.11, we consider \(Y_\lambda = Y - \lambda X\), which is a derivation on \({\mathbb {C}}[x,y]\). It will be sometimes convenient to consider the monomials \({m\atopwithdelims ()k}x^ky^{m-k}\). In these terms
Our goal here is to solve the following Cauchy problem: find \(p_k:{\mathbb {R}}\rightarrow V^{(m)}\) such that
i.e. to compute
We will use the standard notation
which is an analytic function in both \(\lambda \) and t, and \(\cos _\lambda (t):= \frac{d}{dt}\sin _\lambda (t)\).
Proposition 5.1
For any \(\lambda ,t \in {\mathbb {R}}\), we have
Proof
Since clearly
we have
In the same way we can compute \(\exp (tY_\lambda )y\). \(\square \)
The following standard and elementary fact will be useful.
Lemma 5.2
Let \({\textbf{A}}\) be a finite-dimensional algebra. A vector field on \({\textbf{A}}\) is a derivation iff its flow \(\phi _t\) satisfies
In other words, each \(\phi _t\) is an \({\textbf{A}}\)-morphism.
Theorem 5.3
The solution of the Cauchy problem (50) is
where
Proof
Since \(Y_\lambda \) is a derivation, \(\exp (tY_\lambda )\) is a \({\mathbb {C}}[x,y]\)-morphism by the previous lemma. Hence
Comparing with (51) yields (54).
It remains to prove (55). Putting \(s=\sin _\lambda (t),c=\cos _\lambda (t)\) we have
where we changed \(a=k-h,b=j-h\). Using
yields (55). \(\square \)
5.2 Eigenvalues and eigenvectors of \(Y_\lambda \)
Given \(f:V \rightarrow V\) an endomorphism of \({\mathbb {C}}\)-vector spaces, we denote by \({\text {spec}}(f)\) the set of eigenvalues of f and by \(E_\alpha (f)\) the eigenspace associated to each \(\alpha \in {\text {spec}}(f)\).
Lemma 5.4
The endomorphism \(\left. Y_\lambda \right| _{V^{(m)}}\) is diagonalizable with simple multiplicities and
where \(e_1:= \sqrt{-\lambda }x + y\) and \(e_2:= -\sqrt{-\lambda }x + y\).
Proof
The result is trivial to check for \(m=1\) as
Since \(Y_\lambda \) is a derivation
Hence
as stated. \(\square \)
Remark
It is interesting to notice that the spectra of \(Y_\lambda \) and \(\sqrt{-\lambda } H\), when restricted to each \(V^{(m)}\), are identical. These two operators are thus intertwined e.g. by the linear isomorphism \(x^ky^{m-k}\mapsto e_1^{k}e_2^{m-k}\).
5.3 Image of \(Y_\lambda \)
Using Lemma 5.4, we can conclude that \(\left. Y_\lambda \right| _{V^{(m)}}\) is bijective if and only if m is odd. If m is even, then the kernel is one-dimensional. An explicit description is the following.
Proposition 5.5
If m is even, then
where
Proof
By the binomial formula
if k is even, and \(Z_{m,\lambda }(x^k y^{m-k}) = 0\) if k is odd. Therefore
This shows that \(\text {im}(Y_\lambda )\) is a subspace of \(\ker Z_{m,\lambda }\). Given that \(Z_{m,\lambda }\) is not zero, we have \(\text {dim} \ker Z_{m,\lambda } = m\), and by Lemma 5.4, we know that the image of \(\left. Y_\lambda \right| _{V^{(m)}}\) has the same dimension. This yields (58). \(\square \)
Next we compute, for even m and given \(\varphi \) in the image of \(\left. Y_\lambda \right| _{V^{(m)}}\), the preimage \(Y_\lambda ^{-1}(\{\varphi \})\).
Consider
A simple computation using (49) shows
where \(c_{m,k}=0\) if \(m-k\) is even, and otherwise
With these ingredients at hand, for even m, we can now compute a preimage by \(Y_\lambda \) of any element in \({\text {im}} Y_\lambda \) as follows.
Proposition 5.6
Let \(\Pi :V^{(m)} \rightarrow V^{(m)}\) be given by \(\left( {\begin{array}{c}m\\ k\end{array}}\right) x^k y^{m-k} \mapsto P_{m,k+1}\). If m is even then
Proof
Let \(0< k < m\). Since \((m-k+1)c_{m,k} - \lambda (k+1)c_{m,k+2} = 0\), using (49) we get
For \(k = 0\) and \(k = m\),
Since \(c_{m,m} = 0\), and \(c_{m,2} = 0\) if m is even, the result follows. \(\square \)
6 Tube formulas in \({\mathbb {S}}^{m}_{\lambda }\) and \({\mathbb {C}}P^{n}_{\lambda }\)
Here we will obtain our main result: the tube formulas for invariant valuations of \({\mathbb {C}}P^{n}_{\lambda }\) (i.e. the tubular operator \({\textbf{T}}_t\) on \({\mathcal {V}}_{\lambda ,{\mathbb {C}}}^n\)). We will also recover Santaló’s tube formulas for \({\mathcal {V}}_{\lambda ,{\mathbb {R}}}^m\) (cf. [25]) in a way that explains the similarities between the real and the complex space forms.
6.1 Tube formulas in complex space forms
Recalling (25) and Proposition 3.3, we get an isomorphism \(I:W_n\rightarrow {\text {Val}}^{U(n)}\) of \(\mathfrak {sl}_2\)-modules from
to \({\text {Val}}^{U(n)}\) by putting \(I(y^{2n-4r})=\pi _{2r,r}\) (i.e. mapping Y-primitive elements to \(\Lambda \)-primitive elements) and
By Theorem 4.11, the map \(J_{\lambda ,{\mathbb {C}}}:=\Phi _\lambda \circ I:W_{n}\rightarrow \mathcal V_{\lambda ,{\mathbb {C}}}^n\) fulfills
We define
and arrive at our main theorem.
Theorem 6.1
The tubular operator \({\textbf{T}}_t\) in \({\mathcal {V}}_{\lambda ,{\mathbb {C}}}^n\) is given by
where
Proof
By (37), using (62) and (63), and putting \(m=2n-4r\), we get
Using (55) the result follows. \(\square \)
The tube formulas in terms of the \(\tau _{k,i}^\lambda \) can be obtained from Theorem 6.1 using (28) and (33) which hold verbatim replacing \(\pi ^\lambda _{k,r},\tau _{k,r}^\lambda \) for \(\pi _{k,r},\tau _{k,r}\).
Remark
The tube formula for the volume \(\sigma _{2n,0}^{\lambda } = {\text {vol}}_{{\mathbb {C}}P^{n}_{\lambda }}\) is given by the following simple expression
which is Theorem 4.3 of [11], since \(\sigma _{j,0}^\lambda = \omega _{2n-j}\tau _{j,0}^{\lambda } = \Phi _\lambda (\mu _j)\). The tube formulas \({\textbf{T}}_t \sigma _{2n-2r,r}\) are equally simple.
Remark
An interesting feature of the previous tube formulas is the following self-similarity property, which is explained by (62). Let
Then one has \({\textbf{T}}_t\circ {\textbf{G}}^{n,j}={\textbf{G}}^{n,j}\circ {\textbf{T}}_t\).
Remark
It is also worth noting that \({\mathcal {V}}_{\lambda ,{\mathbb {C}}}^n=\bigoplus _{0 \le 2r \le n}{\mathcal {I}}_{\lambda }^{n,r}\) where
and that these subspaces are \(\partial _{\lambda ,{\mathbb {C}}}\)-invariant. In particular, given \(\varphi \in {\mathcal {I}}_{\lambda }^{n,r}\) one has \({\textbf{T}}_t(\varphi )\in {\mathcal {I}}_{\lambda }^{n,r}\).
6.2 Tube formulas in real space forms
Let \(I:V^{(m)}\rightarrow {\text {Val}}^{O(m)}\) be the isomorphism of irreducible \(\mathfrak {sl}_2\)-representations determined by \(I(y^{m})=\chi \); i.e.
where we used (19). By Theorem 4.12, the map \(J_{\lambda ,{\mathbb {R}}}=\Psi _\lambda \circ I\) satisfies
The map \(J_{\lambda ,{\mathbb {R}}}\) is explicitly given by
The image of \(J_{\lambda ,{\mathbb {R}}}\) is the hyperplane \(\mathcal H_\lambda ^{m+1}:={\text {im}} J_{\lambda ,{\mathbb {R}}}= {\text {span}}\left\{ \sigma _{0}^\lambda ,\dots ,\sigma _{m}^\lambda \right\} \).
Theorem 6.2
The tubular operator on \({\mathcal {V}}_{\lambda ,{\mathbb {R}}}^{m+1}\) is given as follows. For \( i = 0,\dots ,m\),
In particular
and thus
These formulas where first obtained by Santaló [25].
Proof
By (37), (64) and (65), we have for \(0 \le i \le m\),
This proves (66) of which (67) is a particular case. Integrating with respect to t yields (68). \(\square \)
Remark
It is worth pointing out the similarity between tube formulas in real and complex space forms. More precisely, note that the isomorphism
between the subspaces \(\mathcal H_{\lambda }^{2n-4r+1}\subset {\mathcal {V}}^{2n-4r+1}_{\lambda ,{\mathbb {R}}} \) and \({\mathcal {I}}_{\lambda }^{n,r}\subset {\mathcal {V}}_{\lambda ,{\mathbb {C}}}^n\) commutes with the tubular operator \({\textbf{T}}_t\). This is explained by (62) and (64).
Remark
Recently, Hofstätter and Wannerer [21] have found a map \(\mathcal {V}^{2n+1}_{\lambda ,{\mathbb {R}}}\rightarrow {\mathcal {V}}_{\lambda ,{\mathbb {C}}}^n\) which also commutes with \(\textbf{T}_t\). Next we describe their results and how they relate to \(\textbf{F}_{n,0}^\lambda \). For \(\lambda >0\) let \(\pi _\lambda : {\mathbb {S}}^{2n+1}_{\lambda } \rightarrow {\mathbb {C}}P^{n}_{\lambda }\) be the Hopf fibration. For a proper submersion \(p:M\rightarrow N\), Alesker proved the existence of a push-forward of valuations \(p_*:\mathcal V(M)\rightarrow \mathcal V(N)\) characterized by
Hofstätter and Wannerer have computed the push-forward of invariant valuations through the Hopf fibration. More presicely they have shown that \((\pi _\lambda )_*\) commutes with \(\textbf{T}_t\) and deduced from this fact that
It follows from (40) that
6.3 Spectral analysis of the derivative map
Here we compute the eigenvalues and eigenvectors of \(\partial _{\lambda ,{\mathbb {R}}}\) and \(\partial _{\lambda ,{\mathbb {C}}}\). Note that the tube formulas for such valuations are extremely simple: if \(\partial _{} \mu =a\mu \) with \(a\in {\mathbb {C}}\), then \({\textbf{T}}_t\mu =e^{at}\mu \).
Proposition 6.3
For \(0 \le 2r \le n\), the restriction of \(\partial _{\lambda ,{\mathbb {C}}}\) to \({{\mathcal {I}}_{\lambda }^{n,r}}\) has the following (simple) eigenvalues and eigenspaces:
Hence \(\partial _{\lambda ,{\mathbb {C}}}\) diagonalizes on \(\mathcal V_{\lambda ,{\mathbb {C}}}^n\) with the following eigenspaces:
for \(-n\le j\le n\).
Proof
Everything follows from Lemma 5.4 and (62). \(\square \)
Proposition 6.4
-
(i)
In \({\mathbb {S}}^{2n}_{\lambda }\) the derivative operator is diagonalizable with
$$\begin{aligned} {\text {spec}}(\partial _{\lambda ,{\mathbb {R}}})&= \left\{ 0,\pm \sqrt{-\lambda },{\pm 3\sqrt{-\lambda }},\dots , \pm (2n-1)\sqrt{-\lambda } \right\} , \end{aligned}$$(72)$$\begin{aligned} E_0(\partial _{\lambda ,{\mathbb {R}}})&= {\text {span}}_{\mathbb {C}}\{\chi \} \end{aligned}$$(73)$$\begin{aligned} E_{(2k-2n+1)\sqrt{-\lambda }}(\partial _{\lambda ,{\mathbb {R}}})&= {\text {span}}_{\mathbb {C}}\{J_{\lambda ,{\mathbb {R}}}(e_1^{k}e_2^{2n-k-1})\}, \quad 0 \le k \le 2n-1 \end{aligned}$$(74) -
(ii)
In \({\mathbb {S}}^{2n+1}_{\lambda }\) the derivative operator is not diagonalizable since
$$\begin{aligned} {\text {spec}}(\partial _{\lambda ,{\mathbb {R}}})&= \left\{ 0,0,\pm 2\sqrt{-\lambda },{\pm 4\sqrt{-\lambda }},\dots ,\pm 2n\sqrt{-\lambda } \right\} , \end{aligned}$$(75)$$\begin{aligned} E_0(\partial _{\lambda ,{\mathbb {R}}})&= {\text {span}}_{\mathbb {C}}\{\chi \} \end{aligned}$$(76)$$\begin{aligned} E_{(2k-2n)\sqrt{-\lambda }}(\partial _{\lambda ,{\mathbb {R}}})&= {\text {span}}_{\mathbb {C}}\{J_{\lambda ,{\mathbb {R}}}(e_1^{k}e_2^{2n-k})\}, \quad 0 \le k \le 2n. \end{aligned}$$(77)
Proof
- (i):
-
By Lemma 5.4 and (64) we have that \((2k-2n+1)\sqrt{-\lambda }\), \(0 \le k \le 2n-1\), is an eigenvalue of \({\partial _{\lambda ,{\mathbb {R}}}}\) with eigenspace given by (74). The Euler characteristic is clearly an eigenvector with zero eigenvalue. We thus have at least \(2n+1\) eigenvalues. Since this is precisely the dimension of \({\mathcal {V}}_{\lambda ,{\mathbb {R}}}^{2n}\), the statement follows.
- (ii):
-
In light of Lemma 5.4 and (64), we ascertain that \((2k-2n)\sqrt{-\lambda }\), \(0 \le k \le 2n\), is an eigenvalue of \({\partial _{\lambda ,{\mathbb {R}}}}\) and the corresponding eigenspace is described by (77).
Our next objective is to prove that while the algebraic multiplicity of the zero eigenvalue is two, its geometric multiplicity is only one. This will entail finding a valuation \(\mu \) that satisfies \(\partial _{\lambda ,{\mathbb {R}}}^2 \mu = 0\), while also ensuring that \(\partial _{\lambda ,{\mathbb {R}}} \mu \ne 0\). Consider \(\sigma _{2n}^\lambda =J_{\lambda ,{\mathbb {R}}}(x^{2n}) \in {\mathcal {V}}^{2n+1}_{\lambda ,{\mathbb {R}}}\). In the notation of Lemma 5.4,
$$\begin{aligned} x = \frac{1}{2\sqrt{-\lambda }}(e_1 - e_2), \qquad x^{2n} = (-4\lambda )^{-n}\sum _{i = 0}^{2n}(-1)^i \left( {\begin{array}{c}2n\\ i\end{array}}\right) e_1^{i}e_{2}^{2n-i}. \end{aligned}$$Hence
$$\begin{aligned} \partial _{\lambda ,{\mathbb {R}}}\sigma _{2n+1}^\lambda = \sigma _{2n}^\lambda = (-4\lambda )^{-n}\sum _{i = 0}^{2n}(-1)^i \left( {\begin{array}{c}2n\\ i\end{array}}\right) J_{\lambda ,{\mathbb {R}}}(e_1^{i}e_{2}^{2n-i}). \end{aligned}$$Consider
$$\begin{aligned} \nu := (-4\lambda )^{-n} \sum ^{2n}_{\begin{array}{c} i = 0\\ i\ne n \end{array}}\left( {\begin{array}{c}2n\\ i\end{array}}\right) \dfrac{(-1)^i}{(2i-2n)\sqrt{-\lambda }} J_{\lambda ,{\mathbb {R}}}(e_1^{i}e_{2}^{2n-i}), \end{aligned}$$and note that, by Lemma 5.4,
$$\begin{aligned} \partial _{\lambda ,{\mathbb {R}}}\nu := (-4\lambda )^{-n} \sum ^{2n}_{\begin{array}{c} i = 0\\ i\ne n \end{array}}\left( {\begin{array}{c}2n\\ i\end{array}}\right) (-1)^{i} J_{\lambda ,{\mathbb {R}}}(e_1^{i}e_{2}^{2n-i}), \end{aligned}$$since \(e_1^{n}e_2^n \in \ker Y_\lambda \). Finally, we define \(\mu = \sigma _{2n+1}^\lambda -\nu \). Then
$$\begin{aligned} \partial _{\lambda ,{\mathbb {R}}} \mu = (-4\lambda )^{-n}\left( {\begin{array}{c}2n\\ n\end{array}}\right) (-1)^n J_{\lambda ,{\mathbb {R}}}(e_1^{n}e_2^{n})\ne 0, \end{aligned}$$while
$$\begin{aligned} \partial _{\lambda ,{\mathbb {R}}}^2 \mu&= (-4\lambda )^{-n}\left( {\begin{array}{c}2n\\ n\end{array}}\right) (-1)^n \partial _{\lambda ,{\mathbb {R}}} J_{\lambda ,{\mathbb {R}}}(e_1^{n}e_2^{n}) \\&= (-4\lambda )^{-n}\left( {\begin{array}{c}2n\\ n\end{array}}\right) (-1)^n J_{\lambda ,{\mathbb {R}}}(Y_{\lambda }(e_1^{n}e_2^{n})) = 0. \end{aligned}$$It follows that \(\dim \ker \partial _{\lambda ,{\mathbb {R}}}<\dim \ker \partial _{\lambda ,{\mathbb {R}}}^2\). Noting that \(\chi \in \ker \partial _{\lambda ,{\mathbb {R}}}\) this implies the statement.
\(\square \)
Remark
We conclude from Prosposition 6.4 and Lemma 5.4 that there is no isomorphism between \({\text {Val}}^{O(m)}\) and \({\mathcal {V}}_{\lambda ,{\mathbb {R}}}^m\) intertwining \(\Lambda - \lambda L\) and \(\partial _{\lambda ,{\mathbb {R}}}\). Indeed, these two operators have different spectra no matter the parity of m.
6.4 Stable valuations in complex space forms
We say that a valuation \(\varphi \in {\mathcal {V}}(M)\) on a Riemannian manifold M is stable if \(\partial \mu =0\), or equivalently, if \({\textbf{T}}_t\mu =\mu \) for all t. By Propositions 6.4 and 6.3, up to multiplicative constants, the Euler characteristic is the unique isometry-invariant stable valuation in \({\mathbb {S}}^{m}_{\lambda }\). The complex case is more interesting.
Proposition 6.5
The unique (up to multiplicative constants) stable valuation on \({\mathcal {I}}_\lambda ^{n,r}\) is given by
Proof
By Lemma 5.4 the kernel of \(Y_\lambda \) on the space \(V^{(m)}\) of homogeneous polynomials of degree \(m=2n-4r\) is spanned by
Therefore the kernel of \(\partial _{\lambda ,{\mathbb {C}}}\) in \({\mathcal {I}}_{\lambda }^{n,r}\) is spanned by \(\psi _{2r}=J_\lambda (e_1^{n-2r}e_2^{n-2r})\), for each \(0 \le 2r \le n\). \(\square \)
Next we express the Euler characteristic as a combination of the stable valuations \(\psi _{2r}\). Note in particular that \(\chi \) is not confined to any \(\partial \)-invariant subspace \({\mathcal {I}}_{\lambda }^{n,r}\).
Proposition 6.6
Proof
Since \(\chi \) is stable, it can be expressed as \(\chi =\sum _j a_j \psi _{2j}\). By [11, Theorem 3.11]
The coefficient of \(\tau _{2r,r}^{\lambda }\) in this expansion is
By Proposition 3.3, we have
whence
Hence
and the result follows. \(\square \)
6.5 Image of \(\partial _{\lambda ,{\mathbb {C}}}\) and \(\partial _{\lambda ,{\mathbb {R}}}\)
Next we describe the image of the operators \(\partial _{\lambda ,{\mathbb {C}}}\) and \(\partial _{\lambda ,{\mathbb {R}}}\), and we compute the preimage of any element belonging to them.
Proposition 6.7
Given any \(\varphi = \sum _{k,r} a_{k,r}\sigma _{k,r}^\lambda \in {\mathcal {V}}_{\lambda ,{\mathbb {C}}}^n\), we have \( \varphi \in {\text {im}}\partial _{\lambda ,{\mathbb {C}}}\) if and only if
Proof
Note that \(\varphi =\sum _{r}\varphi _r\) with \(\varphi _r=\sum _{k} a_{k,r}\sigma _{k,r}^\lambda \) is the decomposition of \(\varphi \) corresponding to \({\mathcal {V}}_{\lambda ,{\mathbb {C}}}^n=\bigoplus _{r=0}^{\left\lfloor n/2\right\rfloor } {\mathcal {I}}_{\lambda }^{n,r}\). By (62) and Proposition 5.5 we have \( \varphi \in {\text {im}}\partial _{\lambda ,{\mathbb {C}}}\) if and only if for every r
where we used (59). \(\square \)
Proposition 6.8
Given \(\varphi = \sum _{k,r} a_{k,r}\sigma _{k,r}^\lambda \in {\mathcal {V}}_{\lambda ,{\mathbb {C}}}^n\) satisfying (78) we have
where \(P_{m,l}\) is given by (60).
Proof
This follows at once from Proposition 5.6 after decomposing \(\varphi =\sum _r\varphi _r\) as in the previous proof. \(\square \)
Proposition 6.9
The image of \(\partial _{\lambda ,{\mathbb {R}}}\) in \({\mathcal {V}}_{\lambda ,{\mathbb {R}}}^m\) is the hyperplane \({\mathcal {H}}_\lambda ^m\) generated by \(\sigma _0^{\lambda },\dots ,\sigma ^{\lambda }_{m-1}\). Moreover
where \(\phi ^k=\sum _{j \ge 0}\left( \frac{\lambda }{4}\right) ^j \tau _{k+2j}^\lambda \). In particular
Recall from [7, eq. (118)] that \(\phi ^k= \int _{G_{\lambda ,{\mathbb {C}}}}\chi (\cdot \cap g {\mathbb {S}}^{m-k}_{\lambda })dg\) where dg is a properly normalized Haar measure on \(G_{\lambda ,{\mathbb {C}}}\), and \({\mathbb {S}}^{m-k}_{\lambda }\) is an \((m-k)\)-dimensional totally geodesic submanifold in \({\mathbb {S}}^{m}_{\lambda }\).
Proof
where the term between brackets appears only if \(m-k\) is even. Using Proposition 4.8, this yields (82). The rest of the statement follows. \(\square \)
Remark
References
Abardia, J., Gallego, E., Solanes, G.: The Gauss–Bonnet theorem and Crofton-type formulas in complex space forms. Isr. J. Math. 187, 287–315 (2012)
Alesker, S.: Theory of valuations on manifolds. I. Linear spaces. Isr. J. Math. 156, 311–339 (2006)
Alesker, S.: Theory of valuations on manifolds. II. Adv. Math. 207(1), 420–454 (2006)
Alesker, S.: Theory of valuations on manifolds: a survey. Geom. Funct. Anal. 17(4), 1321–1341 (2007)
Alesker, S., Bernig, A.: Convolution of valuations on manifolds. J. Differ. Geom. 107(2), 203–240 (2017)
Alesker, S., Fu, J.H.G.: Theory of valuations on manifolds. III. Multiplicative structure in the general case. Trans. Am. Math. Soc. 360(4), 1951–1981 (2008)
Alesker, S., Fu, J.H.G.: Integral geometry and valuations. Advanced Courses in Mathematics. In: Gallego, E., Solanes, G. (eds.) CRM Barcelona. Birkhäuser/Springer, Basel, 2014. Lectures from the Advanced Course on Integral Geometry and Valuation Theory held at the Centre de Recerca Matemàtica (CRM), Barcelona, September 6–10 (2010)
Bernig, A., Bröcker, L.: Valuations on manifolds and Rumin cohomology. J. Differ. Geom. 75(3), 433–457 (2007)
Bernig, A., Fu, J.H.G.: Convolution of convex valuations. Geom. Dedicata 123, 153–169 (2006)
Bernig, A., Fu, J.H.G.: Hermitian integral geometry. Ann. Math. (2) 173(2), 907–945 (2011)
Bernig, A., Fu, J.H.G., Solanes, G.: Integral geometry of complex space forms. Geom. Funct. Anal. 24(2), 403–492 (2014)
Bernig, A., Fu, J.H.G., Solanes, G., Wannerer, T.: The Weyl tube theorem for Kähler manifolds (2022)
Fu, J.H.G.: Kinematic formulas in integral geometry. Indiana Univ. Math. J. 39(4), 1115–1154 (1990)
Fu, J.H.G.: Curvature measures of subanalytic sets. Am. J. Math. 116(4), 819–880 (1994)
Fu, J.H.G.: Some remarks on Legendrian rectifiable currents. Manuscr. Math. 97(4), 175–187 (1998)
Fu, J.H.G., Pokorný, D., Rataj, J.: Kinematic formulas for sets defined by differences of convex functions. Adv. Math. 311, 796–832 (2017)
Fu, J.H.G., Wannerer, T.: Riemannian curvature measures. Geom. Funct. Anal. 29(2), 343–381 (2019)
Geiges, H.: An introduction to contact topology. Cambridge Studies in Advanced Mathematics, vol. 109. Cambridge University Press, Cambridge (2008)
Gray, A., Vanhecke, L.: The volumes of tubes in a Riemannian manifold. Rend. Sem. Mat. Univ. Politec. Torino 39(3), 1–50 (1983)
Gray, A.: Tubes. Addison-Wesley Publishing Company, Advanced Book Program, Redwood City (1990)
Hofstätter, G., Wannerer, T.: Pushforwards of Intrinsic Volumes, in preparation
Kleinjohann, N.: Nächste Punkte in der Riemannschen Geometrie. Math. Z. 176(3), 327–344 (1981)
McMullen, P.: Valuations and Euler-type relations on certain classes of convex polytopes. Proc. Lond. Math. Soc. (3) 35(1), 113–135 (1977)
Rumin, M.: Un complexe de formes différentielles sur les variétés de contact. C. R. Acad. Sci. Paris Sér. I Math. 310(6), 401–404 (1990)
Santaló, L.A.: On parallel hypersurfaces in the elliptic and hyperbolic \(n\)-dimensional space. Proc. Am. Math. Soc. 1, 325–330 (1950)
Santaló, L.A.: Integral geometry and geometric probability. Cambridge Mathematical Library, 2nd edn. Cambridge University Press, Cambridge (2004). With a foreword by Mark Kac
Solanes, G.: Integral geometry and the Gauss–Bonnet theorem in constant curvature spaces. Trans. Am. Math. Soc. 358(3), 1105–1115 (2006)
Weyl, H.: On the volume of tubes. Am. J. Math. 61(2), 461–472 (1939)
Funding
Open Access Funding provided by Universitat Autonoma de Barcelona.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Work partially supported by the FEDER/MICIU/AEI grants PGC2018-095998-B-I00, PID2021-125625NB-I00 and the AGAUR grant 2021-SGR-01015. The first author is supported by the Serra Hunter Programme and the MICIU/AEI María de Maeztu grant CEX2020-001084-M. The second author is supported by the FEDER/MICIU/AEI grant PRE2019-091402.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Solanes, G., Trillo, J.A. Tube formulas for valuations in complex space forms. Math. Ann. (2024). https://doi.org/10.1007/s00208-024-02929-2
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00208-024-02929-2