Transformations and Coupling Relations for Affine Connections

28/10/2015
Auteurs : James Tao, Jun Zhang
Publication GSI2015
OAI : oai:www.see.asso.fr:11784:14290

Résumé

The statistical structure on a manifold M is predicated upon a special kind of coupling between the Riemannian metric g and a torsion-free affine connection ∇ on the TM, such that ∇ g is totally symmetric, forming, by definition, a “Codazzi pair” { ∇ , g}. In this paper, we first investigate various transformations of affine connections, including additive translation (by an arbitrary (1,2)-tensor K), multiplicative perturbation (through an arbitrary invertible operator L on TM), and conjugation (through a non-degenerate two-form h). We then study the Codazzi coupling of ∇ with h and its coupling with L, and the link between these two couplings. We introduce, as special cases of K-translations, various transformations that generalize traditional projective and dual-projective transformations, and study their commutativity with L-perturbation and h-conjugation transformations. Our derivations allow affine connections to carry torsion, and we investigate conditions under which torsions are preserved by the various transformations mentioned above. Our systematic approach establishes a general setting for the study of Information Geometry based on transformations and coupling relations of affine connections – in particular, we provide a generalization of conformal-projective transformation.

Transformations and Coupling Relations for Affine Connections

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application/pdf Transformations and Coupling Relations for Affine Connections James Tao, Jun Zhang
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The statistical structure on a manifold M is predicated upon a special kind of coupling between the Riemannian metric g and a torsion-free affine connection ∇ on the TM, such that ∇ g is totally symmetric, forming, by definition, a “Codazzi pair” { ∇ , g}. In this paper, we first investigate various transformations of affine connections, including additive translation (by an arbitrary (1,2)-tensor K), multiplicative perturbation (through an arbitrary invertible operator L on TM), and conjugation (through a non-degenerate two-form h). We then study the Codazzi coupling of ∇ with h and its coupling with L, and the link between these two couplings. We introduce, as special cases of K-translations, various transformations that generalize traditional projective and dual-projective transformations, and study their commutativity with L-perturbation and h-conjugation transformations. Our derivations allow affine connections to carry torsion, and we investigate conditions under which torsions are preserved by the various transformations mentioned above. Our systematic approach establishes a general setting for the study of Information Geometry based on transformations and coupling relations of affine connections – in particular, we provide a generalization of conformal-projective transformation.
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The statistical structure on a manifold M is predicated upon a special kind of coupling between the Riemannian metric g and a torsion-free affine connection ∇ on the TM, such that ∇ g is totally symmetric, forming, by definition, a “Codazzi pair” { ∇ , g}. In this paper, we first investigate various transformations of affine connections, including additive translation (by an arbitrary (1,2)-tensor K), multiplicative perturbation (through an arbitrary invertible operator L on TM), and conjugation (through a non-degenerate two-form h). We then study the Codazzi coupling of ∇ with h and its coupling with L, and the link between these two couplings. We introduce, as special cases of K-translations, various transformations that generalize traditional projective and dual-projective transformations, and study their commutativity with L-perturbation and h-conjugation transformations. Our derivations allow affine connections to carry torsion, and we investigate conditions under which torsions are preserved by the various transformations mentioned above. Our systematic approach establishes a general setting for the study of Information Geometry based on transformations and coupling relations of affine connections – in particular, we provide a generalization of conformal-projective transformation.

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Transformations and Coupling Relations for Affine Connections James Tao (Harvard University, Cambridge MA) Jun Zhang (University of Michigan, Ann Arbor MI) Oct 29, 2015 James Tao (Harvard University, Cambridge MA) Jun Zhang (University of Michigan, Ann Arbor MI)Transformations and Coupling Relations for Affine Connections Outline 1 Transformation of affine connections (with torsion) h-conjugation: by a two-form h gauge transform: by an operator L additive translation: by a (1,2)-tensor K 2 Commutative relations and “commutativity prisms” keeping track of “torsion” as going through the transformations 3 Transformations that preserve Codazzi coupling pg, q more general than “conformal-projective transformation”? James Tao (Harvard University, Cambridge MA) Jun Zhang (University of Michigan, Ann Arbor MI)Transformations and Coupling Relations for Affine Connections Outline 1 Transformation of affine connections (with torsion) h-conjugation: by a two-form h gauge transform: by an operator L additive translation: by a (1,2)-tensor K 2 Commutative relations and “commutativity prisms” keeping track of “torsion” as going through the transformations 3 Transformations that preserve Codazzi coupling pg, q more general than “conformal-projective transformation”? James Tao (Harvard University, Cambridge MA) Jun Zhang (University of Michigan, Ann Arbor MI)Transformations and Coupling Relations for Affine Connections Outline 1 Transformation of affine connections (with torsion) h-conjugation: by a two-form h gauge transform: by an operator L additive translation: by a (1,2)-tensor K 2 Commutative relations and “commutativity prisms” keeping track of “torsion” as going through the transformations 3 Transformations that preserve Codazzi coupling pg, q more general than “conformal-projective transformation”? James Tao (Harvard University, Cambridge MA) Jun Zhang (University of Michigan, Ann Arbor MI)Transformations and Coupling Relations for Affine Connections Statistical manifold and Codazzi coupling On a differentiable manifold M, one independently prescribes: 1 a pseudo-Riemannian metric g; 2 an affine connection . Codazzi coupling of g and The pair pg, q is said to be Codazzi-coupled if p Z gqpX, Y q “ p X gqpZ, Y q. This notion is a generalization of Levi-Civita coupling (i.e., parallelism of g with respect to ). It can be shown that p , gq is Codazzi-coupled ÐÑ and ˚ have same torsion. Statistical manifold: definition A manifold pM, g, q where (i) is torsion-free and (ii) pg, q is Codazzi-coupled. James Tao (Harvard University, Cambridge MA) Jun Zhang (University of Michigan, Ann Arbor MI)Transformations and Coupling Relations for Affine Connections Statistical manifold and Codazzi coupling On a differentiable manifold M, one independently prescribes: 1 a pseudo-Riemannian metric g; 2 an affine connection . Codazzi coupling of g and The pair pg, q is said to be Codazzi-coupled if p Z gqpX, Y q “ p X gqpZ, Y q. This notion is a generalization of Levi-Civita coupling (i.e., parallelism of g with respect to ). It can be shown that p , gq is Codazzi-coupled ÐÑ and ˚ have same torsion. Statistical manifold: definition A manifold pM, g, q where (i) is torsion-free and (ii) pg, q is Codazzi-coupled. James Tao (Harvard University, Cambridge MA) Jun Zhang (University of Michigan, Ann Arbor MI)Transformations and Coupling Relations for Affine Connections Statistical manifold and Codazzi coupling On a differentiable manifold M, one independently prescribes: 1 a pseudo-Riemannian metric g; 2 an affine connection . Codazzi coupling of g and The pair pg, q is said to be Codazzi-coupled if p Z gqpX, Y q “ p X gqpZ, Y q. This notion is a generalization of Levi-Civita coupling (i.e., parallelism of g with respect to ). It can be shown that p , gq is Codazzi-coupled ÐÑ and ˚ have same torsion. Statistical manifold: definition A manifold pM, g, q where (i) is torsion-free and (ii) pg, q is Codazzi-coupled. James Tao (Harvard University, Cambridge MA) Jun Zhang (University of Michigan, Ann Arbor MI)Transformations and Coupling Relations for Affine Connections Statistical manifold and Codazzi coupling On a differentiable manifold M, one independently prescribes: 1 a pseudo-Riemannian metric g; 2 an affine connection . Codazzi coupling of g and The pair pg, q is said to be Codazzi-coupled if p Z gqpX, Y q “ p X gqpZ, Y q. This notion is a generalization of Levi-Civita coupling (i.e., parallelism of g with respect to ). It can be shown that p , gq is Codazzi-coupled ÐÑ and ˚ have same torsion. Statistical manifold: definition A manifold pM, g, q where (i) is torsion-free and (ii) pg, q is Codazzi-coupled. James Tao (Harvard University, Cambridge MA) Jun Zhang (University of Michigan, Ann Arbor MI)Transformations and Coupling Relations for Affine Connections Conjugate connection g-conjugation of a connection Given any pg, q, conjugate connection ˚ can be defined: ZgpX, Y q “ gp Z X, Y q ` gpX, ˚ Z Y q. It can be verified that (i) ˚ is indeed a connection and (ii) the ˚ action on is involutive: p ˚ q˚ “ . Defining a connection by conjugacy with a non-degenerate two-form h: can be done unambiguously only when h is symmetric or skew-symmetric; otherwise “left conjugate” and “right conjugate”, in reference to the slot hp¨, ¨q, will not be the same. James Tao (Harvard University, Cambridge MA) Jun Zhang (University of Michigan, Ann Arbor MI)Transformations and Coupling Relations for Affine Connections Conjugate connection g-conjugation of a connection Given any pg, q, conjugate connection ˚ can be defined: ZgpX, Y q “ gp Z X, Y q ` gpX, ˚ Z Y q. It can be verified that (i) ˚ is indeed a connection and (ii) the ˚ action on is involutive: p ˚ q˚ “ . Defining a connection by conjugacy with a non-degenerate two-form h: can be done unambiguously only when h is symmetric or skew-symmetric; otherwise “left conjugate” and “right conjugate”, in reference to the slot hp¨, ¨q, will not be the same. James Tao (Harvard University, Cambridge MA) Jun Zhang (University of Michigan, Ann Arbor MI)Transformations and Coupling Relations for Affine Connections Gauge transformation of connection Let L denote TM isomorphism. The gauge transformation of by L, denoted Lp q, is defined as (for vector fields X, Y ): pLp qqX Y “ L´1 p X pLY qq. pL, q is said to be Codazzi-coupled if p X LqY “ p Y LqX, where p X LqY ” X pLY q ´ Lp X Y q. Proposition (Schwenk-Schellschmidt and Simon, 2009) Let be an affine connection, and L be a tangent bundle isomorphism. Then the following are equivalent: 1 p , Lq is Codazzi-coupled. 2 and Lp q have equal torsions. 3 pLp q, L´1 q is Codazzi-coupled. James Tao (Harvard University, Cambridge MA) Jun Zhang (University of Michigan, Ann Arbor MI)Transformations and Coupling Relations for Affine Connections Gauge transformation of connection Let L denote TM isomorphism. The gauge transformation of by L, denoted Lp q, is defined as (for vector fields X, Y ): pLp qqX Y “ L´1 p X pLY qq. pL, q is said to be Codazzi-coupled if p X LqY “ p Y LqX, where p X LqY ” X pLY q ´ Lp X Y q. Proposition (Schwenk-Schellschmidt and Simon, 2009) Let be an affine connection, and L be a tangent bundle isomorphism. Then the following are equivalent: 1 p , Lq is Codazzi-coupled. 2 and Lp q have equal torsions. 3 pLp q, L´1 q is Codazzi-coupled. James Tao (Harvard University, Cambridge MA) Jun Zhang (University of Michigan, Ann Arbor MI)Transformations and Coupling Relations for Affine Connections Gauge transformation of connection Let L denote TM isomorphism. The gauge transformation of by L, denoted Lp q, is defined as (for vector fields X, Y ): pLp qqX Y “ L´1 p X pLY qq. pL, q is said to be Codazzi-coupled if p X LqY “ p Y LqX, where p X LqY ” X pLY q ´ Lp X Y q. Proposition (Schwenk-Schellschmidt and Simon, 2009) Let be an affine connection, and L be a tangent bundle isomorphism. Then the following are equivalent: 1 p , Lq is Codazzi-coupled. 2 and Lp q have equal torsions. 3 pLp q, L´1 q is Codazzi-coupled. James Tao (Harvard University, Cambridge MA) Jun Zhang (University of Michigan, Ann Arbor MI)Transformations and Coupling Relations for Affine Connections Linking g-conjugation with L-gauge transform We proved the following characterization theorem for g-conjugation of a connection in terms of any L: Characterization Theorem Let be a connection and ˚ its conjugate connection w.r.t. a metric g. Denote ωpX, Y q “ gpLX, Y q for arbitrary TM isomorphism L. Then ω “ 0 if and only if Lp ˚ q “ . Explicitly written: ˚ Z X “ Z X ` Lp Z L´1 qX. Proof used the identify (for any invertible operator L): ChpX, Y , Zq “ Cg pLpXq, Y , Zq ` gpp Z LqX, Y q, where CpX, Y , Zq ” p Z gqpX, Y q, hpX, Y q ” gpLpXq, Y q. James Tao (Harvard University, Cambridge MA) Jun Zhang (University of Michigan, Ann Arbor MI)Transformations and Coupling Relations for Affine Connections Translation of a connection by K-tensor Translation by a (1,2)-tensor: X Y Ñ X Y ` KpX, Y q. It is torsion-preserving iff K is symmetric: KpX, Y q “ KpY , Xq. Examples of K-translations (i) P_ pτq : X Y ÞÑ X Y ` τpXqY , P_ -transformation; (ii) Ppτq : X Y ÞÑ X Y ` τpY qX, P-transformation; (iii) Projpτq : X Y ÞÑ X Y ` τpY qX ` τpXqY , called projective transformation, always torsion-preserving; (iv) Dph, V q : X Y ÞÑ X Y ´ hpY , XqV , called “dual-projective transformation”, torsion-preserving when h symmetric. Here τ is an arbitrary one-form, h is a non-degenerate two-form, X, Y , V are all vector fields. James Tao (Harvard University, Cambridge MA) Jun Zhang (University of Michigan, Ann Arbor MI)Transformations and Coupling Relations for Affine Connections Translation of a connection by K-tensor Translation by a (1,2)-tensor: X Y Ñ X Y ` KpX, Y q. It is torsion-preserving iff K is symmetric: KpX, Y q “ KpY , Xq. Examples of K-translations (i) P_ pτq : X Y ÞÑ X Y ` τpXqY , P_ -transformation; (ii) Ppτq : X Y ÞÑ X Y ` τpY qX, P-transformation; (iii) Projpτq : X Y ÞÑ X Y ` τpY qX ` τpXqY , called projective transformation, always torsion-preserving; (iv) Dph, V q : X Y ÞÑ X Y ´ hpY , XqV , called “dual-projective transformation”, torsion-preserving when h symmetric. Here τ is an arbitrary one-form, h is a non-degenerate two-form, X, Y , V are all vector fields. James Tao (Harvard University, Cambridge MA) Jun Zhang (University of Michigan, Ann Arbor MI)Transformations and Coupling Relations for Affine Connections Interactions of h-conjugation, L-gauge, K-translation Let g, L, τ be as above. Let gL denote gpL¨, ¨q, ΓL denote L-gauge transformation, Cpgq denote conjugation w.r.t. g, and ¯τ be the vector field such that gpX, ¯τq “ τpXq. ‚ ‚ ‚ ‚ ‚ ‚ Ppτq Cpgq ΓL CpgLq Dpg,¯τq ΓL Cpgq DpgL,L´1p¯τqq CpgLq James Tao (Harvard University, Cambridge MA) Jun Zhang (University of Michigan, Ann Arbor MI)Transformations and Coupling Relations for Affine Connections Interactions of h-conjugation, L-gauge, K-translation ‚ ‚ ‚ ‚ ‚ ‚ P_pτq Cpgq ΓL CpgLq P_p´τq ΓL Cpgq P_p´τq CpgLq James Tao (Harvard University, Cambridge MA) Jun Zhang (University of Michigan, Ann Arbor MI)Transformations and Coupling Relations for Affine Connections Conformal-projective transformation (CPT) Conformal-projective transformation (CPT) is defined (Kurose, 2002) as, for any smooth functions ψ and φ, gpX, Y q ÞÑ eψ`φ gpX, Y q X Y ÞÑ X Y ´ gpX, Y q gradg ψ ` XpφqY ` Y pφqX CPT include, as special cases, projective transformation of conformal transformation of g and Levi-Civita connection dual-projective transformation of , given pg, q Codazzi transform of g and α-conformal transformation of g and It is known that CPT preserves Codazzi coupling of pg, q. We wonder whether it can be further generalized while preserving Codazzi structure. James Tao (Harvard University, Cambridge MA) Jun Zhang (University of Michigan, Ann Arbor MI)Transformations and Coupling Relations for Affine Connections Conformal-projective transformation (CPT) Conformal-projective transformation (CPT) is defined (Kurose, 2002) as, for any smooth functions ψ and φ, gpX, Y q ÞÑ eψ`φ gpX, Y q X Y ÞÑ X Y ´ gpX, Y q gradg ψ ` XpφqY ` Y pφqX CPT include, as special cases, projective transformation of conformal transformation of g and Levi-Civita connection dual-projective transformation of , given pg, q Codazzi transform of g and α-conformal transformation of g and It is known that CPT preserves Codazzi coupling of pg, q. We wonder whether it can be further generalized while preserving Codazzi structure. James Tao (Harvard University, Cambridge MA) Jun Zhang (University of Michigan, Ann Arbor MI)Transformations and Coupling Relations for Affine Connections Conformal-projective transformation (CPT) Conformal-projective transformation (CPT) is defined (Kurose, 2002) as, for any smooth functions ψ and φ, gpX, Y q ÞÑ eψ`φ gpX, Y q X Y ÞÑ X Y ´ gpX, Y q gradg ψ ` XpφqY ` Y pφqX CPT include, as special cases, projective transformation of conformal transformation of g and Levi-Civita connection dual-projective transformation of , given pg, q Codazzi transform of g and α-conformal transformation of g and It is known that CPT preserves Codazzi coupling of pg, q. We wonder whether it can be further generalized while preserving Codazzi structure. James Tao (Harvard University, Cambridge MA) Jun Zhang (University of Michigan, Ann Arbor MI)Transformations and Coupling Relations for Affine Connections Conformal-projective transformation (CPT) Conformal-projective transformation (CPT) is defined (Kurose, 2002) as, for any smooth functions ψ and φ, gpX, Y q ÞÑ eψ`φ gpX, Y q X Y ÞÑ X Y ´ gpX, Y q gradg ψ ` XpφqY ` Y pφqX CPT include, as special cases, projective transformation of conformal transformation of g and Levi-Civita connection dual-projective transformation of , given pg, q Codazzi transform of g and α-conformal transformation of g and It is known that CPT preserves Codazzi coupling of pg, q. We wonder whether it can be further generalized while preserving Codazzi structure. James Tao (Harvard University, Cambridge MA) Jun Zhang (University of Michigan, Ann Arbor MI)Transformations and Coupling Relations for Affine Connections CPpV , W , Lq preserving Codazzi Structure Generalized conformal-projective transformation CPpV , W , Lq Let V and W be vector fields, and L an invertible operator. CPpV , W , Lq consists of an L-perturbation of the metric g along with a torsion-preserving transformation Dpg, W qProjp ˜V q of the connection, where ˜V is the one-form given by ˜V pXq :“ gpV , Xq for any vector field X. Proposition. (Assuming dim M ě 4) CPpV , W , Lq preserves Codazzi pairs t , gu if and only if L “ ef for some smooth function f , and V ` W “ gradg f . Take ˜V to be an arbitrary one-form, not necessarily closed, and ˜W :“ df ´ ˜V for some fixed smooth function f . CPT results when f “ φ ` ψ, in which case df “ dφ ` dψ is a natural decomposition. James Tao (Harvard University, Cambridge MA) Jun Zhang (University of Michigan, Ann Arbor MI)Transformations and Coupling Relations for Affine Connections CPpV , W , Lq preserving Codazzi Structure Generalized conformal-projective transformation CPpV , W , Lq Let V and W be vector fields, and L an invertible operator. CPpV , W , Lq consists of an L-perturbation of the metric g along with a torsion-preserving transformation Dpg, W qProjp ˜V q of the connection, where ˜V is the one-form given by ˜V pXq :“ gpV , Xq for any vector field X. Proposition. (Assuming dim M ě 4) CPpV , W , Lq preserves Codazzi pairs t , gu if and only if L “ ef for some smooth function f , and V ` W “ gradg f . Take ˜V to be an arbitrary one-form, not necessarily closed, and ˜W :“ df ´ ˜V for some fixed smooth function f . CPT results when f “ φ ` ψ, in which case df “ dφ ` dψ is a natural decomposition. James Tao (Harvard University, Cambridge MA) Jun Zhang (University of Michigan, Ann Arbor MI)Transformations and Coupling Relations for Affine Connections Recent development (Teng Fei and Jun Zhang) Let L be J (almost compatible structure) or K (almost para-complex structure): J2 “ ´id; K2 “ id. A compatible triple pg, ω, Lq satisfies: 1 gpLX, Y q ` gpX, LY q “ 0; 2 ωpLX, Y q “ ωpX, LY q; 3 ωpX, Y q “ gpLX, Y q; A manifold M is called: 1 symplectic if there exists a symplectic (skew-symmetric + non-degenerate) form ω that is closed: dω “ 0; 2 Fedosov if (i) M is symplectic and (ii) there exists a torsion-free connection parallel to ω : ω “ 0; 3 (para)K¨ahler if (i) M is symplectic and (ii) there exists an integrable L compatible with ω : ωpX, LY q “ ωpLX, Y q. James Tao (Harvard University, Cambridge MA) Jun Zhang (University of Michigan, Ann Arbor MI)Transformations and Coupling Relations for Affine Connections Recent development (Teng Fei and Jun Zhang) Let L be J (almost compatible structure) or K (almost para-complex structure): J2 “ ´id; K2 “ id. A compatible triple pg, ω, Lq satisfies: 1 gpLX, Y q ` gpX, LY q “ 0; 2 ωpLX, Y q “ ωpX, LY q; 3 ωpX, Y q “ gpLX, Y q; A manifold M is called: 1 symplectic if there exists a symplectic (skew-symmetric + non-degenerate) form ω that is closed: dω “ 0; 2 Fedosov if (i) M is symplectic and (ii) there exists a torsion-free connection parallel to ω : ω “ 0; 3 (para)K¨ahler if (i) M is symplectic and (ii) there exists an integrable L compatible with ω : ωpX, LY q “ ωpLX, Y q. James Tao (Harvard University, Cambridge MA) Jun Zhang (University of Michigan, Ann Arbor MI)Transformations and Coupling Relations for Affine Connections Codazzi Structure and (Para)-K¨ahler Structure Main Theorem Let be a torsion-free connection on M, and L denote either J (almost complex) or K (almost para-complex) operator on TM. Then, for the following three statements, any two imply the third: 1 is Codazzi-coupled with g; 2 is Codazzi-coupled with L; 3 ω “ 0. As a result, M becomes a K¨ahler or para-K¨ahler manifold. In other words, Codazzi coupling of p , Lq turns a statistical manifold or Fedosov manifold into a (para-)K¨ahler manifold, which is then both statistical and symplectic. James Tao (Harvard University, Cambridge MA) Jun Zhang (University of Michigan, Ann Arbor MI)Transformations and Coupling Relations for Affine Connections THANK YOU FOR ATTENTION!! Tao, J. and Zhang, J. (2015). Transformation and coupling relations for affine connections. Proceedings of GSI 2015. Springer. Fei, T. and Zhang, J, (in preparation). Interaction of Codazzi structur and (para)-Kahler structure. James Tao (Harvard University, Cambridge MA) Jun Zhang (University of Michigan, Ann Arbor MI)Transformations and Coupling Relations for Affine Connections