Foliations on Affinely Flat Manifolds Information Geometry

28/08/2013
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Foliations on Affinely Flat Manifolds Information Geometry

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' & $ % Foliations on Affinely Flat Manifolds Information Geometry Robert Wolak Jagiellonian University, Krakow (Poland) joint work with Michel Nguiffo Boyom UMR CNRS 5149. 3M Département de Mathématiques et de Modélisation Université Montpellier2, Montpellier GSI2013 - Geometric Science of Information MINES ParisTech Paris 28-08-2013 - 30-08-2013 GSI2013 Foliations on Affinely Flat Manifolds. 1/17 ' & $ % Contents 1. Algebraic preliminaries Koszul-Vinberg algebra (KV-algebra) Algebroid of Koszul-Vinberg Twisted KV-cochain complex Chevalley-Eilenberg complex of the twisted module 2. Foliations on locally flat manifolds 3. Fisher information metric 4. Dual pairs of connections 5. Foliations on locally flat manifolds cont. Ehresmann connections Topological properties GSI2013 Foliations on Affinely Flat Manifolds. 2/17 ' & $ % An algebra A is an R-vector space endowed with a bilinear map µ : A × A → A. This map µ is the multiplication map of A. For a, b ∈ A, ab will stand for µ(a, b). Given an algebra A, the Koszul-Vinberg anomaly (KV-anomaly) of A is the three-linear map KV : A3 → A defined by KV (a, b, c) = (ab)c − a(bc) − (ba)c + b(ac) Definition An algebra A is called a Koszul-Vinberg algebra (KV-algebra) if its KV anomaly vanishes identically. GSI2013 Foliations on Affinely Flat Manifolds. 3/17 ' & $ % Definition An algebroid of Koszul-Vinberg is a couple (V, a) where V is a vector bundle over the base manifold M and whose module of sections Γ(V ) has the structure of a Koszul-Vinberg algebra over R and a is a homomorphism of the vector bundle V into TM satisfying the following properties: (i) (fs).s′ = f(ss′ ) ∀s, s′ ∈ Γ(V ), ∀f ∈ C∞ (M, R) ; (ii) s.(fs′ ) = (a(s)f)s′ + f(ss′ ). Remark (i) If we equip Γ(V ) with the bracket [s, s′ ] = ss′ − s′ s then the Koszul-Vinberg algebroid (V, a) becomes a Lie algebroid. (ii) The condition [s, fs′ ] = (a(s)f)s′ + f[s, s′ ] ensures that a is a homomorphism of Lie algebras, i.e. a([s, s′ ]) = [a(s), a(s′ )] The vector space spanned by the commutators [a, b] = ab − ba of a KV-algebra A is a Lie algebra denoted by AL. GSI2013 Foliations on Affinely Flat Manifolds. 4/17 ' & $ % Let A be a Koszul-Vinberg algebra; a A -module is a vector space W equipped with a right action and a left action of A related by the following equalities: for any a, b ∈ A, and any w ∈ W we have a(bw) − (ab)w = b(aw) − (ba)w and a(wb) − (aw)b = w(ab) − (wa)b. Let A = (X(M), ·) be an algebra with the multiplication given by X · Y = DXY , then A is a Koszul-Vinberg algebra and the space T(M) of tensors on M is a two sided A-module. T(M) is a bigraded by subspaces Tp,q (M) of tensors of type (p, q), GSI2013 Foliations on Affinely Flat Manifolds. 5/17 ' & $ % Twisted KV-cochain complex Let A be a KV-algebra and let W be a two-sided KV-module over A. We equip the vector space W with the left module structure A × W −→ W defined by a ∗ w = aw − wa, ∀a ∈ A, w ∈ W. (1) One has KV (a, b, w) = (a, b, w) − (b, a, w) = 0, where (a, b, w) = (ab) ∗ w − a ∗ (b ∗ w). Definition The left KV-module structure defined by (1) is called the twisted KV-module structure derived from the two-sided KV-module W. The vector space W endowed with the twisted module structure is denoted by Wτ . GSI2013 Foliations on Affinely Flat Manifolds. 6/17 ' & $ % The map (a, w) −→ a ∗ w defines on Wτ a left module structure over the Lie algebra AL. The complex CCE (AL, Wτ ) is called the Chevalley-Eilenberg complex of the twisted module. Let A be a KV-algebra and let W be a two-sided KV-module over A. We consider the graded vector space CKV (A, Wτ ) = M q∈Z Cq KV (A, Wτ ) where Cq KV (A, Wτ ) = {0} if q < 0, C0 KV (A, Wτ ) = Wτ for q ≥ 1, Cq KV (A, Wτ ) = HomR(⊗q A, Wτ ). If no risk of confusion, C(A, Wτ ) will stand for CKV (A, Wτ ). GSI2013 Foliations on Affinely Flat Manifolds. 7/17 ' & $ % Let us define the linear mapping d : Cq (A, Wτ ) −→ Cq+1 (A, Wτ ): ∀w ∈ Wτ , f ∈ Cq (A, Wτ ), a ∈ A and ζ = a1 ⊗ ... ⊗ aq+1 ∈ ⊗q+1 A, (dw)(a) = −aw + wa, (df)(ζ) = q+1 X i=1 (−1)i {ai ∗ (f(∂iζ)) − f(ai.∂iζ)} (2) the action ai.∂iζ is defined by the standard tensor product extension. Theorem (i) The pair (C(A, Wτ ), d) is a cochain complex whose qth cohomology space is denoted by Hq KV (A, Wτ ). (ii) The graded space CN (A, Wτ ) = W ⊕ P q>0 Hom(∧q A, Wτ ) is a subcomplex of (C(A, Wτ ), d) whose cohomology coincides with the cohomology of the Chevalley-Eilenberg complex CCE (AL, Wτ ). GSI2013 Foliations on Affinely Flat Manifolds. 8/17 ' & $ % (M, ∇) - a locally flat manifold. A∇ = (X(M), ∇) - the KV-algebra associated to (M, ∇) Wτ = C∞ (M) - the left KV-module over A∇ under the covariant derivative C0(A∇, Wτ ) the vector subspace of C∞ (A∇, Wτ ) formed by cochains of order 0, thus C0(A∇, Wτ ) consists of C∞ (M)-multilinear mappings. Theorem The second cohomology space H2 0 (A∇, Wτ ) can be decomposed as it follows: H2 0 (A∇, Wτ ) = H2 dR(M) ⊕ H0 (A∇, Hom(S2 A∇, Wτ )) (3) where H2 dR(M) is the 2nd de Rham cohomology space of M. GSI2013 Foliations on Affinely Flat Manifolds. 9/17 ' & $ % H(A∇, Wτ ) = L q≥0 Hq (A∇, Wτ ) - a geometric invariant of (M, ∇), bq(∇) = dim Hq 0 (A∇, C∞ (M)) - qth Betti number of (M, ∇) bq(M) = dim Hq dR(M, R). - the classical qth Betti number of M. bq(M) ≤ bq(∇). M. Nguiffo Boyom, F. Ngakeu, P. M. Byande, R. Wolak, KV-cohomology and differential geometry of affinely flat manifolds. information geometry, African Diaspora Journal of Mathematics, Special Volume in Honor of Prof. Augustin Banyaga Vol. 14, 2, pp. 197–226 (2012) GSI2013 Foliations on Affinely Flat Manifolds. 10/17 ' & $ % Definition Let (M, ∇) be a locally flat manifold. (i) A totally geodesic foliation F of (M, ∇) is called affine foliation. (ii) A totally geodesic foliation F of M is tranversally euclidean if its normal bundle TM/TF is endowed with a ∇-parallel (pseudo) euclidean scalar product. Q(M) = HomC∞(M) (S2 A, Wτ ), the vector space of tensorial quadratic forms on (sections of) TM. For σ ∈ H0 KV (A, Q(M)), let σ̄ be the quadratic form on TM/ ker σ deduced from σ and let sign(σ) be the Morse index of σ̄. We define the following numerical invariants: Definition We set: ρ∇(M) = min{ρ∇(σ) = dim ker σ, σ ∈ H0 (A, Q(M))} and S∇(M) = min{S∇(σ) = dim ker σ + sign(σ), σ ∈ H0 (A, Q(M))}. GSI2013 Foliations on Affinely Flat Manifolds. 11/17 ' & $ % (Ξ, Ω) a measurable set. Θ ⊂ Rn be a connected subset. Definition A connected open subset Θ ⊂ Rn is an n-dimensional statistical model for a measurable set (Ξ, Ω) if there exists a real valued positive function p : Θ × Ξ → R subject to the following requirements. (i) For every fixed ξ ∈ Ξ the function θ → p(θ, ξ) is smooth. (ii) For every fixed θ ∈ Θ the function ξ → p(θ, ξ) is a probability density in (Ξ, Ω) viz Z Ξ p(θ, ξ)dξ = 1. (iii) For every fixed ξ ∈ Ξ there exists a couple (θ, θ′ ) such that p(θ, ξ) 6= p(θ′ , ξ) ∇ a torsion free linear connection in the manifold Θ and let one set ln(θ, ξ) = log(p(θ, ξ)). GSI2013 Foliations on Affinely Flat Manifolds. 12/17 ' & $ % At each point θ ∈ Θ we define the the family {q(θ,ξ))} of bilinear forms. Let (X, Y ) be a couple of smooth vector fields in Θ. We put q(θ,ξ)(X, Y ) = −(∇dln)(X, Y )(θ, ξ). Since ∇ is torsion free q(θ,ξ)(X, Y ) is symmetric w.r.t. the couple (X, Y ). Definition The Fisher information g of the local model (Θ, p) is the mathematical expectation of the bilinear form q(θ,ξ), g(X, Y )(θ) = Z Ξ p(θ, ξ)q(θ,ξ)(X, Y )dξ. The Fisher information g does not depend on the choice of the symmetric connection ∇. The Fisher information g is a semi definite positive. When g is definite it is called Fisher metric of the model (Θ, p). GSI2013 Foliations on Affinely Flat Manifolds. 13/17 ' & $ % The dualistic relation between linear connections. Definition In a Riemannian manifold (M, g) a couple (∇, ∇∗ ) of linear connections are dual of each other if the identity Xg(Y, Z) = g(∇X Y, Z) + g(Y, ∇∗ X Z) holds for all vector fields X, Y, Z on the manifold M. A dual pair (∇, ∇∗ ) in a Riemannian manifold (M, g). Assume that both (M, ∇) and (M, ∇∗ ) are locally flat structures. They define the pair ([ρ∇], [ρ∇∗ ]) of conjugation class of canonical representations. Therefore we have the following two properties. Theorem The pair ([ρ∇], [ρ∇∗ ]) does not depend on the choice of the riemannian structure g. GSI2013 Foliations on Affinely Flat Manifolds. 14/17 ' & $ % Theorem Every locally flat manifold (M, ∇) whose 2-dimensional twisted cohomology H2 0 (A∇, Wτ ) differs from the de Rham cohomology space H2 dR(M) is either a flat (pseudo)-Riemannian manifold or is foliated by a pair (F, F∗) of g-orthogonal foliations for every Riemannian metric g. Moreover, these foliations are totally geodesic w.r.t. the g-dual pair (D, D∗ ) (respectively). GSI2013 Foliations on Affinely Flat Manifolds. 15/17 ' & $ % TM = TF ⊕ TF∗ Define a torsion free linear connection by setting D̃(X1,X2)(Y1, Y2) = (DX1 Y1 + [X2, Y1], D∗ X2 Y2 + [X1, Y2]) for all (X1, X2), (Y1, Y2) ∈ Γ(TF) × Γ(TF∗). D̃ is the unique torsion free linear connection which preserves (F, F∗). GSI2013 Foliations on Affinely Flat Manifolds. 16/17 ' & $ % Assume that one of the connections (D, D∗ ) is geodesically complete. The foliations are Ehresmann connections for the other. – the universal coverings of leaves of the foliation F, respectively, F∗, are D-affinely isomorphic, respectively, D∗-affinely isomorphic. – the universal covering M̃ of the manifold M is the product K × L where K is the universal covering of leaves of the foliation F and L is the universal covering of leaves of the foliation F∗. Assume that the connection D is complete. Then the restriction of D to leaves of F is complete and each leaf of F is a geodesically complete locally flat manifold, so its universal covering is diffeomorphic Rp where p is the dimension of leaves of F. The same is true if the connection D∗ is complete. GSI2013 Foliations on Affinely Flat Manifolds. 17/17 ' & $ % Merci Thank you