Curvatures of Statistical Structures

28/10/2015
Auteurs : Barbara Opozda
Publication GSI2015
OAI : oai:www.see.asso.fr:11784:14279

Résumé

Curvature properties for statistical structures are studied. The study deals with the curvature tensor of statistical connections and their duals as well as the Ricci tensor of the connections, Laplacians and the curvature operator. Two concepts of sectional curvature are introduced. The meaning of the notions is illustrated by presenting few exemplary theorems.

Curvatures of Statistical Structures

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application/pdf Curvatures of Statistical Structures Barbara Opozda
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Curvature properties for statistical structures are studied. The study deals with the curvature tensor of statistical connections and their duals as well as the Ricci tensor of the connections, Laplacians and the curvature operator. Two concepts of sectional curvature are introduced. The meaning of the notions is illustrated by presenting few exemplary theorems.
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Curvature properties for statistical structures are studied. The study deals with the curvature tensor of statistical connections and their duals as well as the Ricci tensor of the connections, Laplacians and the curvature operator. Two concepts of sectional curvature are introduced. The meaning of the notions is illustrated by presenting few exemplary theorems.

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Curvatures of statistical structures Barbara Opozda Paris, October 2015 Barbara Opozda () Curvatures of statistical structures Paris, October 2015 1 / 29 Statistical structures - statistical setting M - open subset of Rn Λ - probability space with a fixed σ-algebra p : M × Λ (x, λ) → p(x, λ) ∈ R - smooth relative to x such that px (λ) := p(x, λ) is a probability measure on Λ — probability distribution (x, λ) := log(p(x, λ)) gij (x) := Ex [(∂i )(∂j )], where Ex is the expectation relative to the probability px ∀x ∈ M, ∂1, ..., ∂n - the canonical frame on M g – Fisher information metric tensor field on M Cijk(x) = Ex [(∂i )(∂j )(∂k )] - cubic form (g, C) – statistical structure on M Barbara Opozda () Curvatures of statistical structures Paris, October 2015 2 / 29 Statistical structures (Codazzi structures)– geometric setting; three equivalent definitions M – manifold, dim M = n I) (g, C), C - totally symmetric (0, 3)-tensor field on M, that is, C(X, Y , Z) = C(Y , X, Z) = C(Y , Z, X) ∀X, Y , Z ∈ Tx M, x ∈ M C – cubic form II) (g, K), K – symmetric (1, 2)-tensor field (i.e., K(X, Y ) = K(Y , X)) and symmetric relative to g, that is, g(X, K(Y , Z)) = g(Y , K(X, Z)) is symmetric for all arguments. C(X, Y , Z) = g(X, K(Y , Z)) Barbara Opozda () Curvatures of statistical structures Paris, October 2015 3 / 29 III) (g, ), - torsion-free connection such that ( X g)(Y , Z) = ( Y g)(X, Z) (1) — statistical connection T – any tensor field of type (p, q) on M, T – of type (p, q + 1) T(X, Y1, ..., Yq) = ( X T)(Y1, ..., Yq) In particular, g(X, Y , Z) = ( X g)(Y , Z) (1) ⇔ g is a symmetric cubic form ˆ - Levi-Civita connection for g K(X, Y ) := X Y − ˆ X Y K – difference tensor g(X, Y , Z) = −2g(X, K(Y , Z)) = −2C(X, Y , Z) Barbara Opozda () Curvatures of statistical structures Paris, October 2015 4 / 29 A statistical structure is trivial if and only if K = 0 or equivalently C = 0 or equivalently = ˆ . KX Y := K(X, Y ) E := tr g K = K(e1, e1) + ... + K(en, en) = (tr Ke1 )e1 + ... + (tr Ken )en E – mean difference vector field E = 0 ⇔ tr KX = 0 ∀X ∈ TM ⇔ tr g C(X, ·, ·) = 0 ∀X ∈ TM E = 0 ⇒ trace-free statistical structure Fact. (g, ) – trace-free if and only if νg = 0, where νg – volume form determined by g Barbara Opozda () Curvatures of statistical structures Paris, October 2015 5 / 29 Examples Riemannian geometry of the second fundamental form M – locally strongly hypersurface in Rn+1 – the second fundamental form h satisfies the Codazzi equation h(X, Y , Z) = h(Y , X, Z), where is the induced connection (the Levi-Civita connection of the first fundamental form) (h, ) - statistical structure Similarly one gets statistical structures on hypersurfaces in space forms. Barbara Opozda () Curvatures of statistical structures Paris, October 2015 6 / 29 Equiaffine geometry of hypersurfaces in the standard affine space Rn+1 M – locally strongly convex hypersurface in Rn+1 ξ – a transversal vector field D – standard flat connection on Rn+1, X, Y ∈ X(M), ξ - transversal vector field DX Y = X Y + h(X, Y )ξ − Gauss formula – induced connection, h – second fundamental form (metric tensor field) DX ξ = −SX + τ(X)ξ − Weingarten formula If τ = 0, ξ is called equiaffine. In this case the Codazzi equation is satisfied h(X, Y , Z) = h(Y , X, Z) (h, ) – statistical structure Barbara Opozda () Curvatures of statistical structures Paris, October 2015 7 / 29 Barbara Opozda () Curvatures of statistical structures Paris, October 2015 8 / 29 Barbara Opozda () Curvatures of statistical structures Paris, October 2015 8 / 29 Barbara Opozda () Curvatures of statistical structures Paris, October 2015 8 / 29 Barbara Opozda () Curvatures of statistical structures Paris, October 2015 8 / 29 Barbara Opozda () Curvatures of statistical structures Paris, October 2015 9 / 29 Barbara Opozda () Curvatures of statistical structures Paris, October 2015 9 / 29 Barbara Opozda () Curvatures of statistical structures Paris, October 2015 9 / 29 Barbara Opozda () Curvatures of statistical structures Paris, October 2015 10 / 29 Barbara Opozda () Curvatures of statistical structures Paris, October 2015 10 / 29 Barbara Opozda () Curvatures of statistical structures Paris, October 2015 10 / 29 Geometry of Lagrangian submanifolds in Kaehler manifolds N – Kaehler manifold of real dimension 2n and with complex structure J M – Lagrangian submanifold of N - n-dimensional submanifold such that JTM orthogonal to TM, i.e. JTM is the normal bundle (in the metric sense) for M ⊂ N D – the Kaehler connection on N DX Y = X Y + JK(X, Y ) g – induced metric tensor field on M (g, K) – statistical structure It is trace-free ⇔ M is minimal in N. Barbara Opozda () Curvatures of statistical structures Paris, October 2015 11 / 29 Most of statistical structures are outside the three classes of examples. For instance, in order that a statistical structure is locally realizable on an equiaffine hypersurface it is necessary that is projectively flat. Barbara Opozda () Curvatures of statistical structures Paris, October 2015 12 / 29 Dual connections, curvature tensors g – metric tensor field on M, – any connection Xg(Y , Z) = g( X Y , Z) + g(Y , X Z) (2) – dual connection (g, ) – statistical structure if and only if (g, ) – statistical structure R(X, Y )Z – (1, 3) - curvature tensor for If R = 0 the structure is called Hessian R(X, Y )Z – curvature tensor for g(R(X, Y )Z, W ) = −g(R(X, Y )W , Z) (3) In particular, R = 0 ⇔ R = 0. Barbara Opozda () Curvatures of statistical structures Paris, October 2015 13 / 29 ˆ – Levi-Civita connection for g, = ˆ + K, = ˆ − K ˆR – curvature tensor for ˆ R(X, Y ) = ˆR(X, Y ) + ( ˆ X K)Y − ( ˆ Y K)X + [KX , KY ] (4) , where [KX , KY ] = KX KY − KY KX R(X, Y ) = ˆR(X, Y ) − ( ˆ X K)Y + ( ˆ Y K)X + [KX , KY ] (5) R(X, Y ) + R(X, Y ) = 2ˆR(X, Y ) + 2[KX , KY ] (6) Barbara Opozda () Curvatures of statistical structures Paris, October 2015 14 / 29 Sectional curvatures R does not have to be skew-symmetric relative to g, i.e. g(R(X, Y )Z, W ) = −g(R(X, Y )W , Z), in general. Lemma * The following conditions are equivalent: 1) g(R(X, Y )Z, W ) = −g(R(X, Y )W , Z) ∀X, Y , Z, W 2) R = R 3) ˆ K is symmetric, that is, ( ˆ K)(X, Y , Z) = ( ˆ X K)(Y , Z) = ( ˆ Y K)(X, Z) = ( ˆ K)(Y , X, Z) ∀X, Y , Z. For hypersurfaces in Rn+1 each of the above conditions describes an affine sphere Barbara Opozda () Curvatures of statistical structures Paris, October 2015 15 / 29 R := R+R 2 [K, K](X, Y )Z := [KX , KY ]Z R(X, Y )Z and [K, K](X, Y )Z are Riemann-curvature-like tensors – they are skew-symmetric in X, Y , satisfy the first Bianchi identity, R(X, Y ), [K, K](X, Y ) are skew-symmetric relative to g ∀X, Y π – vector plane in Tx M, X, Y – orthonormal basis of π sectional curvature for g – ˆk(π) := g(ˆR(X, Y )Y , X) sectional K-curvature – k(π) := g([K, K](X, Y )Y , X) sectional -curvature – k (π) := g(R(X, Y )Y , X) Barbara Opozda () Curvatures of statistical structures Paris, October 2015 16 / 29 In general, Schur’s lemma does not hold for k and k. We have, however, Lemma Assume that M is connected, dim M > 2 and the sectional - curvature (the sectional K-curvature) is point-wise constant. If one of the equivalent conditions in Lemma * holds then the sectional -curvature (the sectional K-curvature) is constant on M. sectional K-curvature The easiest situation which should be taken into account is when the sectional K-curvature is constant for all vector planes in Tx M. In this respect we have Barbara Opozda () Curvatures of statistical structures Paris, October 2015 17 / 29 Theorem If the sectional K-curvature is constant and equal to A for all vector planes in Tx M then there is an orthonormal basis e1, ..., en of Tx M and numbers λ1, ..., λn, µ1, ..., µn−1 such that Ke1 =       λ1 µ1 ... µ1       Kei =              µ1 ... µi−1 µ1 · · · µi−1 λi µi ... µi              Ken =       µ1 ... µn−1 µ1 · · · µn−1 λn       Barbara Opozda () Curvatures of statistical structures Paris, October 2015 18 / 29 continuation of the theorem Moreover µi = λi − λ2 i − 4Ai−1 2 , Ai = Ai−1 − µ2 i , for i = 1, ..., n − 1 where A0 = A. The above representation of K is not unique, in general. If additionally tr g K = 0 then A 0, λn = 0 and λi , µi for i = 1, ..., n − 1 are expressed as follows λi = (n − i) −Ai−1 n − i + 1 , µi = − −Ai−1 n − i + 1 . In particular, in the last case the numbers λi , µi depend only on A and the dimension of M. Barbara Opozda () Curvatures of statistical structures Paris, October 2015 19 / 29 Example 1. Ke1 =       λ λ/2 ... λ/2       Kei =              λ/2 ... 0 λ/2 · · · 0 0 0 ... 0              Ken =       λ/2 ... 0 λ/2 · · · 0 0       The sectional K-curvature is constant = λ2/4 Barbara Opozda () Curvatures of statistical structures Paris, October 2015 20 / 29 Example 2. K-curvature vanishes, i.e. [K, K] = 0. There is an orthonormal frame e1, ..., e1 such that Ke1 =       λ1 0 ... 0       Kei =              0 ... 0 0 · · · 0 λi 0 ... 0              Ken =       0 ... 0 0 · · · 0 λn       Barbara Opozda () Curvatures of statistical structures Paris, October 2015 21 / 29 Some theorems on the sectional K-curvature (g, K) – trace-free if E = tr g K = 0 Theorem Let (g, K) be a trace-free statistical structure on M with symmetric ˆ K. If the sectional K-curvature is constant then either K = 0 (the statistical structure is trivial) or ˆR = 0 and ˆ K = 0. Theorem Let ˆ K = 0. Each of the following conditions implies that ˆR = 0: 1) the sectional K-curvature is negative, 2) [K,K]=0 and K is non-degenerate, i.e. X → KX is a monomorphism. Barbara Opozda () Curvatures of statistical structures Paris, October 2015 22 / 29 Theorem K is as in Example 1. at each point of M, ˆ K is symmetric, div E is constant on M (E = tr g K). Then the sectional curvature for g by any plane containing E is non-positive. Moreover, if M is connected it is constant. If ˆ E = 0 then ˆ K = 0 and the sectional curvature (of g) by any plane containing E vanishes. Theorem If the sectional K-curvature is non-positive on M and [K, K] · K = 0 then the sectional K-curvature vanishes on M. Corollary If (g, K) is a Hessian structure on M with non-negative sectional curvature of g and such that ˆR · K = 0 then ˆR = 0. Barbara Opozda () Curvatures of statistical structures Paris, October 2015 23 / 29 Theorem The sectional K-curvature is negative on M, ˆR · K = 0. Then ˆR = 0. Theorem Let M be a Lagrangian submanifold of N, where N is a Kaehler manifold of constant holomorphic curvature 4c, the sectional curvature of the first fundamental form g on M is smaller than c on M and ˆR · K = 0, where K is the second fundamental tensor of M ⊂ N. Then ˆR = 0. Barbara Opozda () Curvatures of statistical structures Paris, October 2015 24 / 29 -sectional curvature All affine spheres are statistical manifolds of constant sectional -curvature A Riemann curvature-like tensor defines the curvature operator. For instance, for the curvature tensor R = (R + R)/2 we have the curvature operator R : Λ2TM → Λ2TM given by g(R(X ∧ Y ), Z ∧ W ) = g(R(Z, W )Y , X) A curvature operator is symmetric relative to the canonical extension of g to the bundle Λ2TM. Hence it is diagonalizable. In particular, it can be positive definite, negative definite etc. The assumption that R is positive definite is stronger than the assumption that the sectional -curvature is positive. Barbara Opozda () Curvatures of statistical structures Paris, October 2015 25 / 29 Theorem Let M be a connected compact oriented manifold and (g, ) be a trace-free statistical structure on M. If R = R and the curvature operator determined by the curvature tensor ˆR is positive definite on M then the sectional -curvature is constant. Theorem Let M be a connected compact oriented manifold and (g, ) be a trace-free statistical structure on M. If the curvature operator for R = R+R 2 is positive on M then the Betti numbers b1(M) = ... = bn−1(M) = 0. Barbara Opozda () Curvatures of statistical structures Paris, October 2015 26 / 29 sectional curvature for g ˆk(π) = g(ˆR(X, Y )Y , X), X, Y – an orthonormal basis for π Theorem Let M be a compact manifold equipped with a trace-free statistical structure (g, ) such that R = R. If the sectional curvature ˆk for g is positive on M then the structure is trivial, that is = ˆ . In the 2-dimensional case we have Theorem Let M be a compact surface equipped with a trace-free statistical structure (g, ). If M is of genus 0 and R = R then the structure is trivial. Barbara Opozda () Curvatures of statistical structures Paris, October 2015 27 / 29 B. Opozda, Bochner’s technique for statistical manifolds, Annals of Global Analysis and Geometry, DOI 10.1007/s10455-015-9475-z B. Opozda, A sectional curvature for statistical structures, arXiv:1504.01279[math.DG] Barbara Opozda () Curvatures of statistical structures Paris, October 2015 28 / 29 Hessian structures (g, ) – Hessian if R = 0. Then R = 0 and ˆR = −[K, K]. (g, ) is Hessian if and only if ˆ K is symmetric and ˆR = −[K, K]. All Hessian structure are locally realizable on affine hypersurfaces in Rn+1 equipped with Calabi’s structure. If they are trace-free they are locally realizable on improper affine spheres. If the difference tensor is as in Example 1. and the structure is Hessian then K = 0. Barbara Opozda () Curvatures of statistical structures Paris, October 2015 29 / 29