Curvature bounded conjugate symmetric statistical structures with complete metric

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.


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 → R n+1 be an elliptic affine sphere whose Blaschke metric is complete.Then the induced structure on M is trivial.Consequently the affine sphere is an ellipsoid.Theorem 1.2.(E.Calabi) Let f : M → R n+1 be a hyperbolic or parabolic affine sphere whose Blaschke metric is complete.Then the Ricci tensor of the metric is negative semidefinite.
The aim of this paper is to generalize these theorems.In particular, we prove Theorem 1.3.Let (g, ∇) be a trace-free conjugate symmetric statistical structure on a manifold M .Assume that g is complete on M .If the sectional ∇-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 ∇-curvature is non-negative everywhere then the statistical structure is trivial, that is, ∇ = ∇.If the sectional ∇-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 ∇-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 ∇-curvature is relatively new, see [6], [1], the theorems proved in this paper show that the notion is meaningful.
In the proofs of Theorems 3.1 and Theorem 4.1 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 a 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.

Preliminaries
2.1.Definitions of statistical structures.We shall shortly recall basic notions of statistical geometry.For details we refer to [6].Let g be a positive definite Riemannian tensor field on a manifold M .Denote by ∇ the Levi-Civita connection for g.A statistical structure is a pair (g, ∇) where ∇ is a torsion-free connection such that the following Codazzi condition is satisfied: 1) is called a statistical connection for g.
For any connection ∇ one defines its conjugate (dual) connection ∇ relative to g by the formula It is known that the pairs (g, ∇) and (g, ∇) are simultaneously statistical structures.
From now on we assume that ∇ is a statistical connection for g.We have where R and R are the curvature tensors for ∇ and ∇, respectively.Denote by Ric and Ric the corresponding Ricci tensors.Note that, in general, these Ricci tensors are not necessarily symmetric.The curvature and the Ricci tensor of ∇ will be denoted by R and Ric, respectively.The function is the scalar curvature of (g, ∇).Similarly, one can define the scalar curvature ρ for (g, ∇) but, by (3), ρ = ρ.We also have the usual scalar curvature ρ for g.Denote by K the difference tensor between ∇ and ∇, that is, It is now clear that the symmetry of ∇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 (∇ We define the statistical cubic form A by (7) It is clear that a statistical structure can be defined equivalently as a pair (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 tr g K(•, •) = 0 (equivalently, tr g A(X, •, •) = 0 for every X; equivalently, tr K X = 0 for every X).The trace-freeness is also equivalent to the condition that ∇ν g = 0, where ν 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.It is known that Writing the same equality for ∇ and adding both equalities we get which can be shortly written as Using (9) and assuming that a given statistical structure is trace-free one gets, see [6], In particular, if (g, ∇) is trace-free then (13) 2 Ric(X, X) ≥ Ric(X, X) + Ric(X, X).
Lemma 2.1.Let (g, ∇) be a statistical structure.The following conditions are equivalent: The family of statistical structures satisfying one of the above conditions is as important in the theory of statistical structures as the family of affine spheres in affine differential geometry.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 = R implies the symmetry of Ric.
Taking now the trace relative to g on both sides of (12) and taking into account that ρ = ρ, we get for a trace-free statistical structure.

2.3.
Sectional ∇-curvature.The notion of a sectional ∇-curvature was first introduced in [6].Namely, the tensor field is a Riemannian curvature tensor.In particular, it satisfies the condition In general, this condition is not satisfied by the curvature tensor 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 ∇-curvature by ( 17) for a vector plane π ∈ T x M , x ∈ M and e 1 , e 2 any orthonormal basis of π.It is a well-defined notion.
In general, Schur's lemma does not hold for the sectional ∇-curvature.But, if a statistical structure is conjugate symmetric (in this case R = R) then the second Bianchi identity holds and, consequently, the Schur lemma holds, see [6].Thus, in Theorems 3.4 and 4.1 from [4] (which were the inspiration for our investigation), the functions R are, in fact, constant if n > 2.
2.4.Statistical structures on affine hypersurfaces.The theory of affine hypersurfaces in 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.
Let f : M → R n+1 be a locally strongly convex hypersurface.For simplicity assume that M is connected and orientable.Let ξ be a transversal vector field on M .We have the induced volume form ν ξ on M defined as follows We also have the induced connection ∇ and the second fundamental form g defined by the Gauss formula where D is the standard flat connection on R n+1 .Since the hypersurface is locally strongly convex, the second fundamental form g is definite.By multiplying ξ by −1 if necessary, we can assume that g is positive definite.A transversal vector field is called equiaffine if ∇ν ξ = 0.This condition is equivalent to the fact that ∇g is symmetric, i.e. (g, ∇) 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 ∇ is automatically symmetric.A hypersurface equipped with an equiaffine transversal vector field and the induced structure is called an equiaffine hypersurface.
Recall now the notion of the shape operator.Having a fixed equiaffine transversal vector field ξ and differentiating it we get the Weingarten formula The tensor field S is called the shape operator for ξ.If R is the curvature tensor for the induced connection ∇ then This is the Gauss equation for R. The Gauss equation for the dual structure is the following It follows that the dual connection is projectively flat if n > 2. The dual connection is also projectively flat for 2-dimensional surfaces equipped with an equiaffine transversal vector field, that is, ∇Ric is symmetric.The form g(SX, Y ) is symmetric for any equiaffine transversal vector field.
We have the volume form ν g determined by g on M .In general, this volume form is not covariant constant relative to ∇.The starting point of the classical affine differential geometry is the theorem saying that there is a unique equiaffine transversal vector field ξ such that ν ξ = ν 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 = 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 = κ id for some constant κ.
If we choose a positive definite Blaschke metric on a connected locally strongly convex affine sphere then we call the sphere elliptic if κ > 0, parabolic if κ = 0 and hyperbolic if κ < 0.
2.5.Conjugate symmetric statistical structures non-realizable on affine spheres.As we have already mentioned, if ∇ is a connection on a hypersurface induced by an equiaffine transversal vector field then the conjugate connection ∇ is projectively flat.Therefore the projective flatness of the conjugate connection is a necessary condition for (g, ∇) 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 Riccisymmetric statistical structure, but we will not need it in this paper.Note also that, if (g, ∇) is a conjugate symmetric statistical structure then ∇ and ∇ 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 ∇ is projectively flat then so is ∇.Since R = R, ∇ is Ricci-symmetric.By the fundamental theorem mentioned above (g, ∇) can be locally realized on an equiaffine surface in 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 ∇ is projectively flat.It follows that if (g, ∇) is conjugate symmetric then it is locally realizable on an equiaffine hypersurface if only if ∇ or ∇ 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, ∇) is the induced statistical structure on an affine sphere, the metric g is not of constant sectional curvature and α = 1, −1 is a real number, then ∇ α := ∇ + αK is not projectively flat and therefore it cannot be realized (even locally) on any affine sphere.Of course, (g, ∇ α ) 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, ∇ α ) 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.
Let M = R n , where n ≥ 4, be equipped with the standard flat metric tensor field g.Let x 1 , ..., x n be the canonical coordinate system and e 1 , ..., 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 (20) A ijk = 0 if at least two of indices i, j, k are equal, A ijk ∈ R + if the indices i, j, k are mutually distinct.
Then ∇K = 0 and, consequently, R = R. Observe now that the connection ∇ = ∇ − K, where g(K(X, Y ), Z) = A(X, Y, Z), is not projectively flat and therefore, (g, ∇) cannot be realized on any Blaschke hypersurface, even locally.Indeed, suppose that ∇ is projectively flat.Then we must have g(R(e i , e j )e j , e l ) = 0 for i = j and l = i, j.On the other hand, by (8), we have g(R(e i , e j )e j , e l ) = g([K ei , K ej ]e j , e l ) = −g(K ej K ei e j , e l ) = −g(K(e i , e j ), K(e j , e l )) = − n s=1 A ijs A jls .
By (20) it is clear that the function − n s=1 A ijs A jls is negative if n ≥ 4. Another version of this example (with ∇K = 0) is given by the symmetric A ijk , where (21) A ijk = 0 if at least two of indices i, j, k are equal, (A ijk ) x = x 1 + ... + (x i ) + ... + (x j ) + ... + (x k ) + ... + x n f or i < j < k, where (x l ) means that the coordinate x l was removed from the sum.One can easily check that ∇A is symmetric.Indeed, we want to check that In cases a) and b) both sides of the required equality vanish.In the last case, where all indices are mutually distinct, on both sides of the required equality we get 1.
In the same manner as in the previous example one sees that ∇ is not projectively flat on (R + ) n if n ≥ 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.

Curvature bounded conjugate symmetric trace-free statistical structures
Let n = dim M and (g, ∇) be a statistical structure on M .From now on we assume that the structure is trace-free and conjugate symmetric.Assume moreover that for every vector plane π ⊂ T x M and x ∈ M .Denote by ε the difference The quantities H 1 , H 2 and ε can be functions on 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 ( 23) Theorem 3.1.Let (g, ∇) be a trace-free conjugate symmetric statistical structure on an n-dimensional manifold M .Assume that (M, g) is complete and the sectional ∇-curvature k satisfies the inequalities (24) on M , where H 3 is a non-positive number and ε is a non-negative function on M .Then the Ricci tensor Ric of g satisfies the inequalities The scalar curvature ρ of g satisfies the inequalities Proof.In what follows the scalar multiplication g will be also denoted by , .The following lemma is crucial in the following proof Lemma 3.2.Let V be any unit vector of T p M .Denote by T V the (0, 4)-tensor given by Assume that for some H 3 ∈ R, ε ∈ R + and for all vector planes π ⊂ T p M .Then Proof of Lemma 3.2.Let e 1 , ..., e n be an eigenbasis of K V and K V e i = λ i e i for i = 1, ..., n.Then ψ V = λ 2 1 + ... + λ 2 n .We have where k ij = k(e i ∧ e j ).Since k ij = k ji and k ii = 0, we obtain (33) In the last term we now replace λ n by −λ 1 − ... − λ n−1 .We get Thus, using the assumption (22) and the condition λ n = −λ 1 − ... − λ n−1 , we get Therefore, it is sufficient to prove The left hand side of this inequality can be written and then estimated as follows (35) Namely, since We now can make father estimations in (35) as follows We shall use the following inequality proved, in fact, on p. 84 in [5].
Proposition 3.3.For a trace-free statistical structure we have where We shall now estimate the function g(A ′ , A) from below.We have (44) g(A ′ , A) = i,j,k,l (R(e i , e k )A)(e i , e j , e l ), A(e k , e j , e l ) = − i,j,k,l [A(R(e i , e k )e i , e j , e l )A(e k , e j , e l ) +A(e i , R(e i , e k )e j , e l )A(e k , e j , e l )] − i,j,k,l A(e i , e j , R(e i , e k )e l )A(e k , e j , e l ).
In the last term we interchange the indices j and l.Since A is symmetric, we get (45) g(A ′ , A) = − i,j,k,l [A(R(e i , e k )e i , e j , e l )A(e k , e j , e l ) +2A(e i , R(e i , e k )e j , e l )A(e k , e j , e l )].
For a fixed index l we have [A(R(e i , e k )e i , e j , e l )A(e k , e j , e l ) + 2A(e i , R(e i , e k )e j , e l )A(e k , e j , e l )] We shall now cite a theorem on weak solutions of differential inequalities for the Laplacian of non-negative functions.The following version of this theorem, proved in [3], is sufficient for our purposes Theorem 3.4.Let (M, g) be a complete Riemannian manifold with Ricci tensor bounded from below.Suppose that ψ is a non-negative continuous function and a weak solution of the differential inequality where k > 1 is an integer and b 0 > 0, b 1 ≥ 0,..., b k ≥ 0. Let N be the largest root of the polynomial equation We have, see ( 14), that is, Ric is bounded from below.Since H 3 ≤ 0, by Theorem 3.4 and (47) we have Let X be a unit vector.Using (12) we now obtain Combining this with (51) one gets the following estimation of the Ricci tensor Ric In order to estimate the scalar curvature ρ we use (15) and (52).We get The proof of Theorem 3.1 is completed.✷ Theorem 3.1 can be obviously formulated as follows Theorem 3.5.Let (g, ∇) be a trace-free conjugate symmetric statistical structure on an n-dimensional manifold M .Assume that (M, g) is complete and the sectional ∇-curvature k satisfies the inequality (22) on M , where is a non-positive number and ε is a non-negative function on M .Then the Ricci tensor Ric of g satisfies the inequalities The scalar curvature ρ of g satisfies the inequalities Remark 3.6.The estimation of the Ricci tensor Ric from below in the above theorems is easy and it follows from (13).The estimation of the Ricci tensor Ric from 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 Ric ≤ −(n − 1) 2 H 3 (it should be Ric ≤ 0).The estimation of the scalar curvature in Theorems 1.3, 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 ∇-curvature We shall prove Theorem 4.1.Let (M, g) be a complete Riemannian manifold with a conjugate symmetric trace-free statistical structure (g, ∇).If the sectional ∇-curvature is non-negative on M then the statistical structure is trivial, i.e. ∇ = ∇.
This theorem can be deduced from the considerations of the previous section, but it can be proved in an easier way, as it is shown below.Namely, consider the non-negative function ϕ on M given by ( 57) where U x is the unit hypersphere in T x M , x ∈ M .The function ϕ is continuous on M .Let p ∈ M be a fixed point and V ∈ U p be a vector for which A(U, U, U ) attains its maximum on 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 , ..., e n is an orthonormal eigenbasis of K V with corresponding eigenvalues λ 1 , ..., λ n then = ( ∇X ( ∇Y A))(V, V, V ) + 3( ∇Y A)( ∇X V, V, V ) + 3( ∇X A)( ∇Y V, V, V ) +3A( ∇X ∇Y V, V, V ) + 6A( ∇Y V, ∇X V, V ) − dΦ( ∇X Y ).

Proof of Theorem 1.3
We shall now prove Theorem 1.3.Assume that the statistical sectional curvature is bounded from below and above, that is, the inequalities

H 2 ≤
k(π) ≤ H 1 are satisfied, where H 1 , H 2 are real numbers.If H 2 < 0 then H 3 = H 2 − n−22 ε < 0 and we can use Theorem 3.1 to get the first assertion of Theorem 1.3.If H 2 ≥ 0 then we can use Theorem 4.1.The fact that the Ricci tensor of g is bounded trivially follows from the fact that the ordinary sectional curvature of g is equal to the sectional ∇-curvature.If H 2 > 0 then Ric ≥ (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.
R(e i , e k )e i , e j )A e l (e k , e j ) + 2A e l (e i , R(e i , e k )e j )A e l (e k , e j )]