Sasakian Statistical Manifolds II

Abstract

This article is a digest of [2, 3] with additional remarks on invariant submanifolds of Sasakian statistical manifolds.

1 Introduction

This article is a digest of [2, 3] with additional remarks on invariant submanifolds of Sasakian statistical manifolds.

We set $\varOmega =\{1, \ldots , n+1 \}$ as a sample space, and denote by $\mathcal {P}^{+}(\varOmega )$ the set of positive probability densities, that is, $\mathcal {P}^{+}(\varOmega )=\{ p: \varOmega \rightarrow \mathbb {R}_+ \ | \ \sum _{x \in \varOmega } p(x)=1\ \}$, where $\mathbb {R}_+$ is the set of positive real numbers. Let M be a smooth manifold as a parameter space, and $s : M \ni u \mapsto p(\cdot , u) \in \mathcal {P}^{+}(\varOmega )$ an injection with the property that $p(x, \cdot ):M \rightarrow \mathbb {R}_+$ is smooth for each $x \in \varOmega$. Consider a family of positive probability densities on $\varOmega$ parametrized by M in this manner. We define a (0, 2)-tensor field on M by

$$\begin{aligned} g_u(X,Y)=\sum _{x \in \varOmega } \{X \log p(x, \cdot )\} \{Y \log p(x, \cdot )\} p(x,u) \end{aligned}$$

for tangent vectors $X,Y \in T_u M$. We say that an injection $s : M \rightarrow \mathcal {P}^{+}(\varOmega )$ is a statistical model if $g_u$ is nondegenerate for each $u\in M$, namely, if g is a Riemannian metric on M, which is called the Fisher information metric for s. Define $\varphi : M \rightarrow \mathbb {R}^{n+1}$ for a statistical model s by $\varphi (u)={}^t[ 2\sqrt{p(1, u)}, \ldots , 2\sqrt{p(n+1, u)}]$. It is known that the metric on M induced by $\varphi$ from the Euclidean metric on $\mathbb {R}^{n+1}$ coincides with the Fisher information metric g. Since the image $\varphi (M)$ lies on the n-dimensional hypersphere $S^{n}(2)$ of radius 2, the Fisher information metric is considered as the Riemannian metric induced from the standard metric of the hypersphere. For example, we set

$$\begin{aligned}&M=\{ u={}^t[u^1, \ldots , u^n] \in \mathbb {R}^n \ | \ u^j>0, \ \sum _{l=1}^n u^l<1 \ \}, \\&s: M \ni u \mapsto p(x,u)= \left\{ \begin{array}{lcl} u^k, &{} &{}x=k \in \{1, \ldots , n\}, \\ 1-\sum _{l=1}^n u^l, &{}\ &{} x=n+1. \end{array} \right. \end{aligned}$$

Then $\varphi (M)=S^n(2) \cap (\mathbb {R}_+)^{n+1}$ and the Fisher information metric is the restriction of the standard metric of $S^n(2)$. It shows that a hypersphere with the standard metric plays an important role in information geometry. It is an interesting question whether a whole hypersphere plays another part there.

In this article, we give a certain statistical structure on an odd-dimensional hypersphere, and explain its background.

2 Sasakian Statistical Structures

Throughout this paper, M denotes a smooth manifold, and $\varGamma (E)$ denotes the set of sections of a vector bundle $E \rightarrow M$. All the objects are assumed to be smooth. For example, $\varGamma (TM^{(p,q)})$ means the set of all the $C^\infty$ tensor fields on M of type (p, q).

At first, we will review the basic notion of Sasakian manifolds, which is a classical topic in differential geometry (See [5] for example). Let $g \in \varGamma (TM^{(0,2)})$ be a Riemannian metric, and denote by $\nabla ^g$ the Levi-Civita connection of g. Take $\phi \in \varGamma (TM^{(1,1)})$ and $\xi \in \varGamma (TM)$.

A triple $(g, \phi , \xi )$ is called an almost contact metric structure on M if the following equations hold for any $X,Y \in \varGamma (TM)$:

$$\begin{aligned}&\phi \ \xi =0, \quad g(\xi , \xi )=1, \\&\phi ^2 X=-X+g(X, \xi )\xi , \\&g(\phi X, Y)+g(X, \phi Y)=0. \end{aligned}$$

An almost contact metric structure on M is called a Sasakian structure if

$$\begin{aligned} ({\nabla }^g_X \phi )Y=g(Y,\xi )X-g(Y,X)\xi \end{aligned}$$

(1)

holds for any $X,Y \in \varGamma (TM)$. We call a manifold equipped with a Sasakian structure a Sasakian manifold.

It is known that on a Sasakian manifold the formula

$$\begin{aligned} {\nabla }^g_X \xi =\phi X \end{aligned}$$

(2)

holds for $X \in \varGamma (TM)$. A typical example of a Sasakian manifold is a hypersphere of odd dimension as mentioned below.

We now review the basic notion of statistical manifolds to fix the notation (See [1] and references therein). Let $\nabla$ be an affine connection of M, and $g \in \varGamma (TM^{(0,2)})$ a Riemannian metric. The pair $(\nabla , g)$ is called a statistical structure on M if (i) $\nabla _XY-\nabla _YX-[X, Y]=0$ and (ii) $(\nabla _X g)(Y,Z)=(\nabla _Y g)(X,Z)$ hold for any $X, Y, Z \in \varGamma (TM)$. By definition, $(\nabla ^g, g)$ is a statistical structure on M.

We denote by $R^\nabla$ the curvature tensor field of $\nabla$, and by $\nabla ^*$ the dual connection of $\nabla$ with respect to g, and set $S=S^{(\nabla , g)} \in \varGamma (TM^{(1,3)})$ as the mean of the curvature tensor fields of $\nabla$ and of $\nabla ^*$, that is, for $X, Y, Z \in \varGamma (TM)$,

$$\begin{aligned}&R^\nabla (X,Y)Z=\nabla _X \nabla _Y Z -\nabla _Y \nabla _X Z-\nabla _{[X,Y]} Z, \nonumber \\&Xg(Y,Z)=g(\nabla _X Y, Z)+g(Y, \nabla ^*_X Z), \nonumber \\&S(X,Y)Z= \dfrac{1}{2}\{ R^\nabla (X,Y)Z+R^{\nabla ^*}(X,Y)Z\}. \end{aligned}$$

(3)

A statistical manifold $(M, \nabla , g)$ is called a Hessian manifold if $R^\nabla =0$. If so, we have $R^{\nabla ^*}=S=0$ automatically.

For a statistical structure $(\nabla , g)$ on M, we set $K=\nabla -\nabla ^g$. Then the following hold:

$$\begin{aligned} \begin{array}{c} K \in \varGamma (TM^{(1,2)}), \\ K_XY=K_YX, \quad g(K_XY,Z)=g(Y, K_XZ) \end{array} \end{aligned}$$

(4)

for any $X, Y, Z \in \varGamma (TM)$. Conversely, if K satisfies (4), the pair $(\nabla = \nabla ^g+K, g)$ is a statistical structure on M.

The formula

$$\begin{aligned} S(X,Y)Z=R^g(X,Y)Z+[K_X, K_Y] Z \end{aligned}$$

(5)

holds, where $R^g=R^{\nabla ^g}$ is the curvature tensor field of the Levi-Civita connection of g.

For a statistical structure $(\nabla , g)$, we often use the expression like $(\nabla =\nabla ^g+K, g)$, and write $K_XY$ by K(X, Y).

Definition 1

A quadruplet $(\nabla =\nabla ^g+K, g,\phi ,\xi )$ is called a Sasakian statistical structure on M if (i) $(g, \phi , \xi )$ is a Sasakian structure and (ii) $(\nabla , g)$ is a statistical structure on M, and (iii) $K\in \varGamma (TM^{(1,2)})$ for $(\nabla , g)$ satisfies

$$\begin{aligned} K(X,\phi Y)+\phi K(X,Y)=0 \quad \text { for }\ X, Y\in \varGamma (TM). \end{aligned}$$

(6)

These three conditions are paraphrased in the following three conditions ([3, Theorem 2.17]: (i’) $(g, \phi , \xi )$ is an almost contact metric structure and (ii) $(\nabla , g)$ is a statistical structure on M, and (iii’) they satisfy

$$\begin{aligned}&\nabla _X(\phi Y)-\phi \nabla ^*_X Y=g(\xi , Y)X-g(X,Y)\xi ,\end{aligned}$$

(7)

$$\begin{aligned}&\nabla _X\xi =\phi X+g(\nabla _X \xi , \xi )\xi . \end{aligned}$$

(8)

We get the following formulas for a Sasakian statistical manifold:

$$\begin{aligned} K(X,\xi )= \lambda g(X, \xi ) \xi , \quad g(K(X,Y), \xi )=\lambda g(X,\xi ) g(Y, \xi ), \end{aligned}$$

(9)

where

$$\begin{aligned} \lambda =g(K(\xi ,\xi ),\xi ). \end{aligned}$$

(10)

Proposition 2

For a Sasakian statistical manifold $(M, \nabla , g, \phi , \xi )$,

$$\begin{aligned} S(X,Y)\xi =g(Y,\xi )X-g(X,\xi )Y \end{aligned}$$

(11)

holds for $X,Y \in \varGamma (TM)$.

Proof

By (9), we have $[K_X, K_Y]\xi =0$, from which (5) implies $S=R^g$. It is known that $R^g$ is written as the right hand side of (11) (See [5]).    $\square$

A quadruplet $(\widetilde{M}, \widetilde{\nabla } =\nabla ^{\widetilde{g}}+\widetilde{K}, \widetilde{g}, \widetilde{J})$ is called a holomorphic statistical manifold if $(\widetilde{g}, \widetilde{J})$ is a Kähler structure, $(\widetilde{\nabla }, \widetilde{g})$ is a statistical structure on $\widetilde{M}$, and

$$\begin{aligned} \widetilde{K}(X, \widetilde{J}Y)+\widetilde{J}\widetilde{K}(X, Y) =0 \end{aligned}$$

(12)

holds for $X, Y \in \varGamma (T\widetilde{M})$. The notion of Sasakian statistical manifold can be also expressed in the following: The cone over M defined below is a holomorphic statistical manifold. Let $(M, \nabla =\nabla ^g+K, g, \phi , \xi )$ be a statistical manifold with an almost contact metric structure. Set $\widetilde{M}$ as $M \times \mathbb {R}_+$, and define a Riemannian metric $\widetilde{g}=r^2 g+(dr)^2$ on $\widetilde{M}$. Take a vector field $\varPsi =r \dfrac{\partial }{\partial r} \in \varGamma (T\widetilde{M})$, and define $\widetilde{J} \in \varGamma (T\widetilde{M}^{(1,1)})$ by $\widetilde{J}\varPsi =\xi$ and $\widetilde{J}X=\phi X-g(X,\xi )\varPsi$ for any $X\in \varGamma (TM)$. Then, $(\widetilde{g}, \widetilde{J})$ is an almost Hermitian structure on $\widetilde{M}$, and furthermore, $(g, \phi , \xi )$ is a Sasakian structure on M if and only if $(\widetilde{g}, \widetilde{J})$ is a Kähler structure on $\widetilde{M}$. We construct connection $\widetilde{\nabla }$ on $\widetilde{M}$ by

$$\begin{aligned} \left\{ \begin{array}{l} \widetilde{\nabla }_\varPsi \varPsi =-\lambda \xi + \varPsi , \\ \widetilde{\nabla }_X \varPsi =\widetilde{\nabla }_\varPsi X=X-\lambda g(X,\xi )\varPsi , \\ \widetilde{\nabla }_XY=\nabla _XY-g(X,Y)\varPsi , \end{array} \right. \end{aligned}$$

that is,

$$\begin{aligned} \widetilde{K}(\varPsi , \varPsi )=-\lambda \xi , \quad \widetilde{K}(X, \varPsi )=-\lambda g(X,\xi )\varPsi , \quad \widetilde{K}(X,Y)=K(X,Y) \end{aligned}$$

for $X,Y \in \varGamma (TM)$, where $\lambda$ is in (10). We then have that $(M, \nabla , g, \phi , \xi )$ is a Sasakian statistical manifold if and only if $(\widetilde{M}, \widetilde{\nabla }, \widetilde{g}, \widetilde{J})$ is a holomorphic statistical manifold (A general statement is given as [2, Proposition 4.8 and Theorem 4.10]). It is derived from the fact that the formula (12) holds if and only if both (6) and (9) hold.

Example 3

Let $S^{2n-1}$ be a unit hypersphere in the Euclidean space $\mathbb {R}^{2n}$. Let J be a standard almost complex structure on $\mathbb {R}^{2n}$ considered as $\mathbb {C}^{n}$, and set $\xi =-JN$, where N is a unit normal vector field of $S^{2n-1}$. Define $\phi \in \varGamma (T(S^{2n-1})^{(1,1)})$ by $\phi (X)=JX-\langle JX, N \rangle N$. Denote by g the standard metric of the hypersphere. Then such a $(g, \phi , \xi )$ is known as a standard Sasakian structure on $S^{2n-1}$. We set

$$\begin{aligned} K(X,Y)=g(X,\xi )g(Y,\xi )\xi \end{aligned}$$

(13)

for any $X,Y \in \varGamma (TS^{2n-1})$. Since K satisfies (4) and (6), we have a Sasakian statistical structure $(\nabla =\nabla ^g+ K, g, \phi , \xi )$ on $S^{2n-1}$.

Proposition 4

Let $(M, g, \phi , \xi )$ be a Sasakian manifold. Set $\nabla$ as $\nabla ^g+f K$ for $f \in C^\infty (M)$, where K is given in (13). Then $(\nabla , g, \phi , \xi )$ is a Sasakian statistical structure on M. Conversely, we define $\nabla _XY=\nabla ^g_XY+L(X,Y)V$ for some unit vector field V and $L \in \varGamma (TM^{(0,2)})$. If $(\nabla , g, \phi , \xi )$ is a Sasakian statistical structure, then $L \otimes V$ is written as $L(X,Y)V= f g(X,\xi )g(Y,\xi )\xi$ for some $f\in C^\infty (M)$, as above.

Proof

The first half is obtained by direct calculation. To get the second half, we have by (4),

$$\begin{aligned} 0= & {} L(X,Y)V-L(Y,X)V=\{L(X,Y)-L(Y,X)\}V, \nonumber \\ 0= & {} g(L(X,Y)V,Z)-g(Y, L(X,Z)V)=g(L(X,Y)Z-L(X,Z)Y, V). \end{aligned}$$

(14)

Substituting V for Z in (14), we have

$$\begin{aligned} L(X,Y)=L(V,V)g(X,V)g(Y,V). \end{aligned}$$

Accordingly, we get by (6),

$$\begin{aligned} 0=L(X, \phi Y)V+\phi \{L(X,Y)V\} =L(V,V) g(X,V) \{ -g(Y, \phi V) V +g(Y,V) \phi V\}, \end{aligned}$$

which implies that $\phi V=0$ if $L(V,V) \ne 0$, and hence $V= \pm \xi$.    $\square$

3 Invariant Submanifolds

Let $(\widetilde{M}, \widetilde{g}, \widetilde{\phi }, \widetilde{\xi })$ be a Sasakian manifold, and M a submanifold of $\widetilde{M}$. We say that M is an invariant submanifold of $\widetilde{M}$ if (i) $\widetilde{\xi }_u \in T_u M$, (ii) $\widetilde{\phi } X \in T_u M$ for any $X \in T_u M$ and $u \in M$. Let $g \in \varGamma (TM^{(0,2)})$, $\phi \in \varGamma (TM^{(1,1)})$ and $\xi \in \varGamma (TM)$ be the restriction of $\widetilde{g}$, $\widetilde{\phi }$ and $\widetilde{\xi }$, respectively. Then it is shown that $(g, \phi , \xi )$ is a Sasakian structure on M.

A typical example of an invariant submanifold of a Sasakian manifold $S^{2n-1}$ in Example 3 is an odd dimensional unit sphere. Furthermore, we have the following example. Let $\iota : Q \rightarrow \mathbb {C}P^{n-1}$ be a complex hyperquadric in the complex projective space, and $\widetilde{Q}$ the principal fiber bundle over Q induced by $\iota$ from the Hopf fibration $\pi : S^{2n-1} \rightarrow \mathbb {C}P^{n-1}$. We denote the induced homomorphism by $\widetilde{\iota } : \widetilde{Q} \rightarrow S^{2n-1}$. Then it is known that $\widetilde{\iota }(\widetilde{Q})$ is an invariant submanifold (See [4, 5]).

We briefly review the statistical submanifold theory to study invariant submanifolds of a Sasakian statistical manifold. Let $(\widetilde{M}, \widetilde{\nabla }, \widetilde{g})$ be a statistical manifold, and M a submanifold of $\widetilde{M}$. Let g be the metric on M induced from $\widetilde{g}$, and consider the orthogonal decomposition with respect to $\widetilde{g}$: $T_u\widetilde{M}=T_u M \oplus T_u M^{\perp }$. According to this decomposition, we define an affine connection $\nabla$ on M, $B \in \varGamma (TM^\perp \otimes TM^{(0,2)})$, $A \in \varGamma ((TM^\perp )^{(0,1)} \otimes TM^{(1,1)})$, and a connection $\nabla ^\perp$ of the vector bundle $TM^\perp$ by

$$\begin{aligned} \widetilde{\nabla }_XY=\nabla _XY+B(X,Y), \quad \widetilde{\nabla }_XN=-A_NX+\nabla ^\perp _XN \end{aligned}$$

(15)

for $X, Y \in \varGamma (TM)$ and $N \in \varGamma (TM^\perp )$. Then $(\nabla , g)$ is a statistical structure on M. In the same fashion, we define an affine connection $\nabla ^*$ on M, $B^* \in \varGamma (TM^\perp \otimes TM^{(0,2)})$, $A^* \in \varGamma ((TM^\perp )^{(0,1)} \otimes TM^{(1,1)})$, and a connection $(\nabla ^\perp )^*$ of $TM^\perp$ by using th dual connection $\widetilde{\nabla }^*$ instead of $\widetilde{\nabla }$ in (15).

We remark that $\widetilde{g}(B(X,Y), N)=g(A^*_NX,Y)$ for $X,Y \in \varGamma (TM)$ and $N \in \varGamma (TM^\bot )$, and remark that $\nabla ^*$ coincides with the dual connection of $\nabla$ with respect to g. See [1] for example.

Theorem 5

Let $(\widetilde{M}, \widetilde{\nabla }, \widetilde{g}, \widetilde{\phi }, \widetilde{\xi })$ be a Sasakian statistical manifold, and M an invariant submanifold of $\widetilde{M}$ with $g, \phi , \xi , \nabla , B, A, \nabla ^\perp , \nabla ^*, B^*, A^*, (\nabla ^\perp )^*$ defined as above. Then the following hold:

(i) A quintuplet $(M, \nabla , g, \phi , \xi )$ is a Sasakian statistical manifold.

(ii) $B(X, \xi )=B^*(X, \xi )=0$ for any $X \in \varGamma (TM)$.

(iii) $B(X, \phi Y)=B(\phi X, Y)=\widetilde{\phi } B^*(X,Y)$ for any $X, Y \in \varGamma (TM)$. In particular, $\mathrm {tr}_gB=\mathrm {tr}_gB^*=0$.

(iv) If B is parallel with respect to the Van der Weaden-Bortolotti connection $\widetilde{\nabla }'$ for $\widetilde{\nabla }$, then B and $B^*$ vanish. Namely, if $(\widetilde{\nabla }'_XB)(Y,Z)= {\nabla }^\perp _X B(Y,Z)-B(\nabla _XY,Z)-B(Y, \nabla _XZ)=0$ for $Z \in \varGamma (TM)$, then $B^*(X,Y)=0$.

(v) $\widetilde{g}(\widetilde{S}(X, \widetilde{\phi }X)\widetilde{\phi }X -S(X,\phi X) \phi X, X) =2 \widetilde{g}(B^*(X,X), B(X,X))$ for $X \in \varGamma (TM)$, where $S=S^{(\nabla , g)}$ and $\widetilde{S}=S^{(\widetilde{\nabla }, \widetilde{g})}$ as in (3).

Corollary 6

Let $(\widetilde{M}, \widetilde{\nabla }, \widetilde{g}, \widetilde{\phi }, \widetilde{\xi })$ be a Sasakian statistical manifold of constant $\widetilde{\phi }$-sectional curvature c, and M an invariant submanifold of $\widetilde{M}$. The induced Sasakian statistical structure on M has constant $\phi$-sectional curvature c if and only if $\widetilde{g}(B^*(X,X), B(X,X))=0$ for any $X \in \varGamma (TM)$ orthogonal to $\xi$.

If we take the Levi-Civita connection as $\widetilde{\nabla }$, the properties above reduce to the ones for an invariant submanifold of a Sasakian manifold. It is known that an invariant submanifold of a Sasakian manifold of constant $\widetilde{\phi }$-sectional curvature c is of constant $\phi$-sectional curvature c if and only if it is totally geodesic. It is obtained by setting $B=B^*$ in Corollary 6. It is an interesting question whether there is an interesting invariant submanifold having nonvanishing B with the above property.

Outline of Proof of Theorem 5. The proof of (i) can be omitted.

By (i) and (8), we calculate that $\nabla _X \xi +B(X,\xi ) = \widetilde{\nabla }_X \xi = \widetilde{\phi }X +\widetilde{g}(\widetilde{\nabla }_X \xi , \widetilde{\xi }) \widetilde{\xi } = \phi X+g(\nabla _X\xi , \xi )\xi$. Comparing the normal components, we have (ii).

By (7), we have $\widetilde{g}(Y, \widetilde{\xi })X- \widetilde{g}(Y, X)\widetilde{\xi } =\widetilde{\nabla }_X(\widetilde{\phi }Y) -\widetilde{\phi }\widetilde{\nabla }^*_X Y = {\nabla }_X({\phi }Y)+B(X, \phi Y) -\widetilde{\phi }({\nabla }^*_X Y+B^*(X,Y)) = {g}(Y, {\xi })X-{g}(Y, X){\xi } +B(X, \phi Y)-\widetilde{\phi }B^*(X,Y)$. Comparing the normal components, we have (iii).

By (i) and (ii), we get that $0 = {\nabla }^\perp _X B(Y,\xi )-B(\nabla _X Y, \xi )-B(Y, \nabla _X \xi ) = -B(Y, \phi X)=-\widetilde{\phi }B^*(X,Y)$, which implies (iv).

To get (v), we use the Gauss equation in the submanifold theory. The tangential component of $R^{\widetilde{\nabla }}(X,Y)Z$ is given as

$$\begin{aligned} R^\nabla (X,Y)Z-A_{B(Y,Z)}X+A_{B(X,Z)}Y, \end{aligned}$$

for $X,Y,Z \in \varGamma (TM)$, which implies that

$$\begin{aligned}&2\widetilde{g}( \widetilde{S}(X,Y)Z,W) =2g(S(X,Y)Z,W) \\&\qquad - \,\widetilde{g}(B^*(X,W), B(Y,Z)) +\widetilde{g}(B^*(Y,W), B(X,Z)) \\&\quad \qquad - \, \widetilde{g}(B(X,W), B^*(Y,Z)) +\widetilde{g}(B(Y,W), B^*(X,Z)). \end{aligned}$$

Therefore, we prove (v) from (iii).    $\square$

To get Corollary 6, we have only to review the definition. A Sasakian statistical structure $(\nabla , g, \phi , \xi )$ is said to be of constant $\phi$-sectional curvature c if the sectional curvature defined by using S equals c for each $\phi$-section, the plane spanned by X and $\phi X$ for a unit vector X orthogonal to $\xi$: $g(S(X,\phi X)\phi X, X)=c g(X,X)^2$ for $X \in \varGamma (TM)$ such that $g(X,\xi )=0$.