# Curvature bounded conjugate symmetric statistical structures with complete metric

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## Abstract

In the paper two important theorems about complete affine spheres are generalized to the case of statistical structures on abstract manifolds. The assumption about constant sectional curvature is replaced by the assumption that the curvature satisfies some inequalities.

## Keywords

Affine sphere Conjugate symmetric statistical structure Sectional \(\nabla \)-curvature Differential inequality## Mathematics Subject Classification

Primary: 53C05 53C20 53A15 53C21## 1 Introduction

In the paper we refer to the following well-known theorems of affine differential geometry

### Theorem 1.1

(W. Blaschke, A. Deicke, E. Calabi) Let \(f:M\rightarrow \mathbf {R}^{n+1}\) be an elliptic affine sphere whose Blaschke metric is complete. Then the induced structure on *M* is trivial, that is, the induced affine connection is the Levi–Civita connection of the Blaschke metric. Consequently, the affine sphere is an ellipsoid.

### Theorem 1.2

(E. Calabi) Let \(f:M\rightarrow \mathbf {R}^{n+1}\) be a hyperbolic or parabolic affine sphere whose Blaschke metric is complete. Then the Ricci tensor of the metric is negative semi-definite.

The above theorems deal with affine spheres which constitute one of the most important categories studied in the classical affine differential geometry. The mystery of affine spheres is that although they are defined so naturally and analogously to the Riemannian case (i.e. affine lines determined by the affine normal vector field meet at one point or are parallel), they are, as the whole class, unknown. On the other hand, they have exceptionally nice properties. Many particular examples of affine spheres are known; the whole class is divided into subclasses (for instance, elliptic, hyperbolic and parabolic), but it is not seen what a satisfactory description of the whole class might look like. Therefore, it is a way of studying the class to impose additional geometric conditions on a sphere and to prove that it lies in a better-known class of manifolds equipped with some geometric structure. The underlying Riemannian geometry is the first candidate here. In the above theorems the additional condition is the completeness of the Blaschke metric.

The aim of this paper is to generalize Theorems 1.1 and 1.2 to the case of statistical manifolds and to the case where a curvature, which is constant on affine spheres, is only bounded. Statistical manifolds are generalizations of affine hypersurfaces in the sense that the structure on the so-called equiaffine hypersurfaces is a statistical structure but statistical structures are not, in general, realized on affine hypersurfaces, even locally. Conjugate symmetric statistical structures are as important in the geometry of statistical structures as affine spheres in the theory of affine hypersurfaces. Within the two geometries they can be characterized by the same condition. The condition is that the curvature tensor of the affine connection of these structures has the same symmetries as the Riemannian curvature tensor, see Sect. 2 or [6].

We shall prove, in particular, the following result.

### Theorem 1.3

Let \((g,\nabla ) \) be a trace-free conjugate symmetric statistical structure on a manifold *M*. Assume that *g* is complete on *M*. If the sectional \(\nabla \)-curvature is bounded from below and above on *M* then the Ricci tensor of *g* is bounded from below and above on *M*. If the sectional \(\nabla \)-curvature is non-negative everywhere then the statistical structure is trivial, that is, \(\nabla ={\hat{\nabla }}\). If the sectional \(\nabla \)-curvature is bounded from 0 by a positive constant then, additionally, *M* is compact and its first fundamental group is finite.

More precise and more general formulations of this theorem give Theorems 3.1 and 4.1. The meaning of the generalization can be explained as follows. The induced structure on an affine sphere is a conjugate symmetric trace-free statistical structure. But the statistical connection on an affine sphere is projectively flat and its sectional \(\nabla \)-curvature is constant. In the theorems we propose the projective flatness is not needed, which means that the statistical structure can be non-realizable on any Blaschke hypersurface even locally. Moreover, the assumption about the constant curvature is replaced by the assumption that the curvature satisfies some inequalities. Since the notion of the sectional \(\nabla \)-curvature is relatively new, see [1, 6], the theorems proved in this paper show that the notion is meaningful.

In the proof of the first part of Theorem 1.3 we use the same main tool as in Calabi’s theorems, that is, a theorem on weak solutions of differential inequalities for the Laplacian of non-negative functions on complete Riemannian manifolds. In fact, we shall use only a particular version of this theorem. Note that the crucial step in the proof of Theorem 3.1 is an estimation obtained in Lemma 3.2. In the case of affine spheres (Theorem 1.2) the corresponding part of the proof is trivial.

An inspiration for the study of the problems this paper deals with was [4] where the first attempt to a generalization of Theorem 1.1 was made. Let us quote one of Noguchi’s results which in the language of this paper can be displayed as follows.

### Theorem 1.4

([4, Theorem 4.1]) Let \((g,\nabla )\) be a trace-free conjugate symmetric statistical structure on a manifold *M*. Assume that the sectional \(\nabla \)-curvature is point-wise constant and non-negative on *M* and *g* is complete. Then the structure is trivial, that is, \(\nabla =\hat{\nabla }\).

We now know that if a statistical structure is conjugate symmetric, then Schur’s lemma holds for the sectional \(\nabla \)-curvature, see Sect. 2.2 or [6]. It implies that in the above theorem the statistical structure can be locally realized on an affine sphere if \(n\ge 3\). But the theorems we discuss here are of global nature and it means that Theorem 1.4 is more general than Theorem 1.1.

## 2 Preliminaries

### 2.1 Definitions of statistical structures

*g*be a positive definite Riemannian tensor field on a manifold

*M*. Denote by \(\hat{\nabla }\) the Levi–Civita connection for

*g*. A statistical structure is a pair \((g,\nabla ) \), where \(\nabla \) is a torsion-free connection such that the following Codazzi condition is satisfied

*g*.

*g*by the formula

*g*. We have

*R*and \({\overline{R}}\) are the curvature tensors for \(\nabla \) and \({\overline{\nabla }}\), respectively. Denote by \(\mathrm{{Ric}}\) and \(\overline{\mathrm{{Ric}}}\) the corresponding Ricci tensors. Note that in general, these Ricci tensors are not necessarily symmetric. The curvature and the Ricci tensor of \(\hat{\nabla }\) will be denoted by \(\hat{R}\) and \(\widehat{\mathrm{{Ric}}}\), respectively. The function

*g*.

*K*the difference tensor between \(\nabla \) and \(\hat{\nabla }\), that is,

*K*(

*X*,

*Y*) will stand for \(K_XY\). Since \(\nabla \) and \(\hat{\nabla }\) are without torsion,

*K*as a (1, 2)-tensor is symmetric. We have \( (\nabla _Xg)(Y,Z)=(K _Xg)(Y,Z)=-g(K_XY,Z)-g(Y,K_XZ)\). It is now clear that the symmetry of \(\nabla g\) and

*K*implies the symmetry of \(K_X\) relative to

*g*for each

*X*. The converse also holds. Namely, if \(K_X\) is symmetric relative to

*g*then we have \(( \nabla _Xg)(Y,Z) = -2g(K_XY,Z)\).

*A*by

*g*,

*K*), where

*K*is a symmetric tensor field of type (1, 2) which is also symmetric relative to

*g*, or as a pair (

*g*,

*A*), where

*A*is a symmetric cubic form.

A statistical structure is trace-free if \(\mathrm{tr}\, _gK(\cdot ,\cdot )=0\) (equivalently, \(\mathrm{tr}\, _gA(X,\cdot ,\cdot )=0\) for every *X*; equivalently, \(\mathrm{tr}\, K_X=0\) for every *X*). The trace-freeness is also equivalent to the condition that \(\nabla \nu _g=0\), where \(\nu _g\) is the volume form determined by *g*. In affine differential geometry the trace-freeness is called the apolarity. The assumption about the trace-freeness of a statistical structure is essential in all the theorems mentioned in the Introduction.

### 2.2 Relations between curvature tensors of statistical structures

### Lemma 2.1

- (1)
\(R={\overline{R}}\),

- (2)
\(\hat{\nabla }K\) is symmetric (equiv. \({\hat{\nabla }} A\) is symmetric),

- (3)
*g*(*R*(*X*,*Y*)*Z*,*W*) is skew symmetric relative to*Z*,*W*.

Note that \(\hat{\nabla }K\) in (2) stands for the (1, 3)-tensor field defined by the formula \(\hat{\nabla }K(X,Y,Z)=(\hat{\nabla }_XK)(Y,Z)\). Of course the same deals with \(\hat{\nabla }A\). A statistical structure satisfying (2) in the above lemma was called in [2] conjugate symmetric. We shall adopt this definition.

Note that the condition \(R={\overline{R}}\) implies the symmetry of \(\mathrm{{Ric}}\).

*g*on both sides of (12) and taking into account that \(\rho ={\overline{\rho }}\), we get

### 2.3 Sectional \(\nabla \)-curvature

*R*. In the case where a given statistical structure is conjugate symmetric the curvature tensor

*R*satisfies this condition. In [6] we defined the sectional \(\nabla \)-curvature by

In general, Schur’s lemma does not hold for the sectional \(\nabla \)-curvature. But, if a statistical structure is conjugate symmetric (in this case \({\mathcal {R}}=R\)), then some type of the second Bianchi identity holds and, consequently, Schur’s lemma holds, see [6].

### 2.4 Statistical structures on affine hypersurfaces

The theory of affine hypersurfaces in \(\mathbf {R}^{n+1}\) is a natural source of statistical structures. For the theory we refer to [3] or [5]. We recall here only some selected facts.

*M*is connected and orientable. Let \(\xi \) be a transversal vector field on

*M*. We have the induced volume form \(\nu _\xi \) on

*M*defined as follows

*g*defined by the Gauss formula

*D*is the standard flat connection on \(\mathbf {R}^{n+1}\). Since the hypersurface is locally strongly convex, the second fundamental form

*g*is definite. By multiplying \(\xi \) by \(-1\) if necessary, we can assume that

*g*is positive definite. A transversal vector field is called equiaffine if \(\nabla \nu _\xi =0\). This condition is equivalent to the fact that \(\nabla g\) is symmetric, i.e. \((g,\nabla )\) is a statistical structure. It means, in particular, that for a statistical structure obtained on a hypersurface by a choice of an equiaffine transversal vector field, the Ricci tensor of \(\nabla \) is automatically symmetric. A hypersurface equipped with an equiaffine transversal vector field, and the induced structure is called an equiaffine hypersurface.

*R*is the curvature tensor for the induced connection \(\nabla \), then

*R*. The Gauss equation for the dual structure is the following

We have the volume form \(\nu _g\) determined by *g* on *M*. In general, this volume form is not covariant constant relative to \(\nabla \). The starting point of the classical affine differential geometry is the theorem saying that there is a unique equiaffine transversal vector field \(\xi \) such that \(\nu _\xi =\nu _g\). This unique transversal vector field is called the affine normal vector field or the Blaschke affine normal. The second fundamental form for the affine normal is called the Blaschke metric. A non-degenerate hypersurface endowed with the affine Blaschke normal is called a Blaschke hypersurface. The induced statistical structure is trace-free on a Blaschke hypersurface. If the affine lines determined by the affine normal vector field meet at one point or are parallel, then the hypersurface is called an affine sphere. In the first case the sphere is called proper in the second one improper. The class of affine spheres is very large. There exist many conditions characterizing affine spheres. For instance, a Blaschke hypersurface is an affine sphere if and only if \(R={\overline{R}}\). Therefore, conjugate symmetric statistical manifolds can be regarded as generalizations of affine spheres. For connected affine spheres the shape operator *S* is a constant multiple of the identity, i.e. \(S=\kappa \, \mathrm{id}\,\) for some constant \(\kappa \).

If we choose a positive definite Blaschke metric on a connected locally strongly convex affine sphere, then we call the sphere elliptic if \(\kappa >0\), parabolic if \(\kappa =0\) and hyperbolic if \(\kappa <0\).

### 2.5 Conjugate symmetric statistical structures non-realizable on affine spheres

As we have already mentioned, if \(\nabla \) is a connection on a hypersurface induced by an equiaffine transversal vector field then the conjugate connection \(\overline{\nabla }\) is projectively flat. Therefore, the projective flatness of the conjugate connection is a necessary condition for \((g,\nabla )\) to be realizable as the induced structure on a hypersurface equipped with an equiaffine transversal vector field. In fact, one of the fundamental theorems in affine differential geometry (see, e.g. [5]) says, roughly speaking, that it is also a sufficient condition for the local realizability of a Ricci symmetric statistical structure, but we will not need it in this paper. Note also that if \((g,\nabla )\) is a conjugate symmetric statistical structure then \(\nabla \) and \({\overline{\nabla }}\) are simultaneously projectively flat. Indeed, it is obvious for \(n>2\). If \(n=2\) we can argue as follows. It suffices to prove that if \({\overline{\nabla }}\) is projectively flat then so is \(\nabla \). Since \(R={\overline{R}}\), \(\nabla \) is Ricci symmetric. By the fundamental theorem mentioned above, \((g,\nabla )\) can be locally realized on an equiaffine surface in \(\mathbf {R}^3\). By Lemma 12.5 from [6] the surface is an equiaffine sphere, that is, the shape operator is locally a constant multiple of the identity, and hence, \(\nabla \) is projectively flat. It follows that if \((g,\nabla )\) is conjugate symmetric then it is locally realizable on an equiaffine hypersurface if only if \(\nabla \) or \({\overline{\nabla }}\) is projectively flat.

We shall now consider trace-free conjugate symmetric statistical structures. The following fact was observed in [6], see Proposition 4.1 there. If \((g,\nabla )\) is the induced statistical structure on an affine sphere, the metric *g* is not of constant sectional curvature and \(\alpha \ne 1, -1\) is a real number, then \(\nabla ^\alpha :=\hat{\nabla }+\alpha K\) is not projectively flat and therefore it cannot be realized (even locally) on any affine sphere. Of course, \((g,\nabla ^\alpha )\) is again a statistical conjugate symmetric structure (by 2) of Lemma 2.1) and since the initial structure was trace-free (because an affine sphere is endowed with the Blaschke structure), \((g,\nabla ^\alpha )\) is trace-free as well. Note also that there are very few affine spheres whose Blaschke metric has constant sectional curvature, see [3], which means that the assumption that *g* is not of constant sectional curvature is not restrictive.

The following example shows another easy way of producing conjugate symmetric trace-free statistical structures which are non-realizable (even locally) on affine spheres.

*g*. Let \(x^1,\ldots ,x^n\) be the canonical coordinate system and \(e_1,\ldots ,e_n\) be the canonical orthonormal frame. Define the cubic form \(A=(A_{ijk})\) on

*M*, where \(A_{ijk}=A(e_i,e_j,e_k)\), by

In the same manner as in the previous example, one sees that \(\overline{\nabla }\) is not projectively flat on \((\mathbf {R}^+)^n\) if \(n\ge 4\).

The considerations of this subsection show that the class of conjugate symmetric trace-free statistical manifolds is much larger than the class of affine spheres, even in the local setting.

## 3 Curvature bounded conjugate symmetric trace-free statistical structures

*M*. From now on we assume that the structure is trace-free and conjugate symmetric. Assume moreover that

*M*(not satisfying any smoothness assumptions), but in the main theorem of this section, that is, in Theorem 3.1, \(H_3\) must be a real number. The condition (22) can be written as

### Theorem 3.1

*n*-dimensional manifold

*M*. Assume that (

*M*,

*g*) is complete and the sectional \(\nabla \)-curvature

*k*satisfies the inequalities (24) on

*M*, where \(H_3\) is a non-positive number and \(\varepsilon \) is a non-negative function on

*M*. Then the Ricci tensor \({\widehat{\mathrm{{Ric}}}}\) of

*g*satisfies the inequalities

*g*satisfies the inequalities

### Proof

In what follows the scalar multiplication *g* will be also denoted by \(\langle \ , \ \rangle \). The following lemma is crucial in the following proof. \(\square \)

### Lemma 3.2

*V*be any unit vector of \(T_pM\). Denote by \(T_V\) the (0, 4)-tensor given by

### Proof of Lemma 3.2

*s*the following formula holds

*s*is a tensor field of type (0,

*k*), then

*p*and denote the obtained vector fields by the same letters

*X*,

*Y*,

*X*, \(e_1,\ldots ,e_n\), respectively. Of course, \({\hat{\nabla }} X={\hat{\nabla }} Y={\hat{\nabla }} Z=0\), \({\hat{\nabla }} e_1=0\),..., \({\hat{\nabla }} e_n=0\) at

*p*. The frame field \(e_1,\ldots , e_n\) is orthonormal. Since \(\hat{\nabla }A\) is symmetric, one gets at

*p*

### Proposition 3.3

*j*and

*l*. Since

*A*is symmetric, we get

*l*we have

*V*in Lemma 3.2 and we get

### Theorem 3.4

*M*,

*g*) be a complete Riemannian manifold with Ricci tensor bounded from below. Suppose that \(\psi \) is a non-negative continuous function and a weak solution of the differential inequality

*N*be the largest root of the polynomial equation

*X*be a unit vector. Using (12) we now obtain

Theorem 3.1 can be obviously formulated as follows

### Theorem 3.5

*n*-dimensional manifold

*M*. Assume that (

*M*,

*g*) is complete and the sectional \(\nabla \)-curvature

*k*satisfies the inequality (22) on

*M*, where \(H_1= H_3+\frac{n}{2}\varepsilon \), \(H_2=H_1-\varepsilon \), \(H_3\) is a non-positive number and \(\varepsilon \) is a non-negative function on

*M*. Then the Ricci tensor \({\widehat{\mathrm{{Ric}}}}\) of

*g*satisfies the inequalities

*g*satisfies the inequalities

### Remark 3.6

The estimation of the Ricci tensor \({\widehat{\mathrm{{Ric}}}}\) from below in the above theorems is easy, and it follows from (13). The estimation of the Ricci tensor \({\widehat{\mathrm{{Ric}}}}\) from the above is not optimal in Theorems 3.1, 3.5. Namely, in the case of a hyperbolic sphere, that is, in the case where \(H_1=H_2=H_3<0\), Theorem 3.1 gives the estimation \({\widehat{\mathrm{{Ric}}}}\le -(n-1)^2H_3\). (It should be \({\widehat{\mathrm{{Ric}}}}\le 0\).) The estimation of the scalar curvature in Theorems 3.1, 3.5 is optimal and, in the above proof, it is not deduced from the estimation of the Ricci tensor.

## 4 Conjugate symmetric trace-free statistical structures with non-negative sectional \(\nabla \)-curvature

We shall prove

### Theorem 4.1

Let (*M*, *g*) be a complete Riemannian manifold with a conjugate symmetric trace-free statistical structure \((g,\nabla )\). If the sectional \(\nabla \)-curvature is non-negative on *M*, then the statistical structure is trivial, i.e. \(\nabla ={\hat{\nabla }}\).

*M*given by

*M*. Let \(p\in M\) be a fixed point and \(V\in {\mathcal {U}}_p\) be a vector for which

*A*(

*U*,

*U*,

*U*) attains its maximum on \({\mathcal {U}}_p\). One observes (see, e.g. [6] the proof of Theorem 5.6) that

*V*is an eigenvector of \(K_V\) and if \(e_1=V, e_2,\ldots , e_n\) is an orthonormal eigenbasis of \(K_V\) with corresponding eigenvalues \(\lambda _1,\ldots ,\lambda _n\) then

*p*. We obtain a smooth orthonormal frame field. Denote the vector fields again by \(V=e_1\)\(e_2,\ldots ,e_n\). Then we have at

*p*

*A*(

*V*,

*V*,

*V*). Of course, \(\varPhi _p=\varphi _p\) and \(\varPhi \le \varphi \) everywhere. We have at

*p*

*p*. We now have at

*p*

*p*

*N*. Using the equality \({\hat{R}}=R-[K,K]\) and the relations \(\lambda _1-2\lambda _i\ge 0\), \(\varPhi =\lambda _1\ge 0\), \(\varPhi =\varphi \) at

*p*, we get at

*p*

We also proved

### Proposition 4.2

*M*,

*g*) be a complete Riemannian manifold and \((g,\nabla )\) a trace-free conjugate symmetric statistical structure on

*M*. If the sectional \(\nabla \)-curvature is bounded from below by a non-positive number

*N*, then for any unit tangent vector \(U\in TM\) we have

## 5 Proof of Theorem 1.3

*g*is bounded trivially follows from the fact that the ordinary sectional curvature of

*g*is equal to the sectional \(\nabla \)-curvature. If \(H_2>0\) then \({\widehat{\mathrm{{Ric}}}}\ge (n-1)H_2>0\). By Myers’ theorem,

*M*is compact and its first fundamental group is finite. This completes the proof of Theorem 1.3.

## Notes

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