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2ND WORKSHOP OF THE SAO PAULO JOURNAL OF MATHEMATICAL SCIENCES: JEAN-LOUS KOSZUL IN SAO PAULO, HIS WORK AND LEGACY UNIVERSITY OF SAO PAULO BRASIL NOVEMBER 2019 MICHEL NGUIFFO BOYOM 1. PART A: SEMINAR OF GEOMETRY. EXISTENCE DEFECTS OF GEOMETRIC-TOPOLOGICAL STRUCTURES IN DIFFERENTIAL MANIFOLDS 8 N ovember 2019 1991 Mathematics Subject Classification. Primaries 53B05 , 53C12, 53C16, 22F50 . Secondarie 18G60. Key words and phrases. Lie algebroids, KV cohomology, canonical characteristic class, Koszul geometry, functor of Amari, locally flat manifolds, complex systems. 1

2 MICHEL NGUIFFO BOYOM 2. EXISTENCE OF GEOMETRIC STRUCUTRES versus GLOBAL ANALYSIS AND HOMOLOGY 2.1. Some open problems In Finite dimensional Di ff erential Geometry, Riemanniann structure ( M, g ) and gauge struc- ture ( M, r ) are examples of Geometric Structure which exists in every di ff erential manifold M . Here r is a Koszul connection in the tangent bundle of M . For many important Geometric structures the question whether a given di ff erential mani- fold M does admit a given Geometric structure S is widely known to be an open di ffi cult problem. Examples of those open problems are. (1.1) The existence of symplectic structures in a gven maniflod M . (1.2) The existence of left invariant symplectic structure in a given Lie group G . (1.3) The existence of two-sided invariant Riemannian structure in a given Lie group G . Similar open problems are met in the gauge geometry of tangent vector bundles of smooth manifolds. (2.1) The existence of locally flat Koszul connections in the tangent bundle of a given man- ifold M . (2.2) The existence of left invariant locally flat connections in a given Lie group G . (2.3) The existence of two-sided invariant Koszul connections in a given Lie group G . Mutatis mutandis one faces open existence problems in the Di ff erential Toplogy. (3.1) The existence of regular Riemannian foliations in a given manifold M (3.2) The existence of regular symplectic foliations in a given manifold M . (3.3) The existence of foliations with a prescrbed structure for leaves. 2.2. Motivations Throughout this talk a Riemannian structure in a smooth manifold M is a couple ( M, g ) formed of M and a non degnerate symmetric bilinear form g . A foliation is called regular if the dimension of leaves in constant. I go to focus on the question whether a given manifold M does admit (eventually sin- gular) foliations the leaves of which leaves carry a prescribed structures S . To this aim, I go to overview some matherials which will be used.

2ND WORKSHOP OF THE SAO PAULO JOURNAL OF MATHEMATICAL SCIENCES:JEAN-LOUS KOSZUL IN SAO PAULO, HIS WORK AND LEGACYUNIVERSITY 3. H omological materials Let us recall that a locaaly flat structure in a smooth manifold M is a couple ( M, r ) formed of M and a locally flat Koszul connection r . The local flatness means the following identities [ X, Y ] = r X Y � r Y X, r X ( r Y Z ) � r Y ( r X Z ) = r [ X,Y ] Z. Here X, Y, Z are smooth vector fields and [ X, Y ] is the Poisson braacket. The vector space of smooth vector in M and the vector associative algebra of real valued smooth functions in M are denoted by A and by C 1 ( M ) respectively. For � = X 1 ⌦ .. ⌦ X q + 1 one put � i � = .. ⌦ ˆ X i ⌦ .. r X i ( � i � ) = Σ j , i .. ⌦ ˆ X i ⌦ .. ⌦ r X i X j ⌦ .. I go the involve the (positively) graded di ff erential vector spaces ( � q C q ( r ) , � KV ) , ( � q C q ( r ) , � � ) . Here C q ( r ) = Hom R ( A ⌦ q , C 1 ( M )) , the di ff erentials � KV ; � � : C q ( r ) ! C q + 1 ( r ) are defined as it follows, given f 2 C q ( r ) and � as above ( 2 . 1 ) � KV f ( � ) = Σ i  q ( � 1 ) i [ d ( f ( � i � ))( X i ) � f ( r X i ( � i � ))] ( 2 . 2 ) �� f ( � ) = Σ i  q + 1 ( � 1 ) i [( d ( f ( � i � ))( X i ) � f ( r X i � i � )] The operators � KV and � � satisfy the following identities � � � = 0 , � � � � � = 0 . The derived cohomology spaces are denoted by H KV ( r ) = � q H q KV ( r ) , H � ( r ) = � q H q � ( r ) .

4 MICHEL NGUIFFO BOYOM 4. F undamental E quations To handle some between the open problems which have been raised, I go to assign two di ff erential operators to every pair of Koszul Connections defined in the tangent bundle TM . Now ( r , r * ) is a pair of Koszul connections (defined in the same tangent bundle TM ). 4.1. The Hessian equation of r The Hessian di ff erential operator of r assigns a (2,1)-tensor to every vector field X , namely r 2 X which is defined by ( r 2 X )( Y, Z ) = r Y ( r Z X ) � r r Y Z X. Let x = ( x 1 , .., x n ) be local coordinate functions and let 1 X k � X = Σ m � x i � � = Σ k Γ k r � ij � x j � x k � xi Let one evalue the principal symbol of X ! r 2 X , ( r 2 X )( � , � � ) = Σ � Ω � . ij � x i � x j � x � Here � Γ � ij = � 2 X � � X k � X k � X � jk Ω � + Σ k [ Γ � + Γ � � Γ k ] + Σ k [ + Σ m ( Γ m jk Γ k im � Γ m ij Γ � mk )] . ik jk ij � x i � x j � x j � x i � x k � x i This expression looks awful, nevertheless from the viewpoint of both the Syernberg Ge- ometry and the Spencer formalism, it allow to see that the d ff erential operator X ! r 2 X is of type 2 and is involutive. Since the involutivity yields the formal integrability, the equation r 2 X = O is formally integrable. Lemma 4.1. The sheaf J ( r ) of solutions of the equation r 2 X = 0 is a sheaf of real associative algbera whose product is defined by r Let KOSS be the convex set of symmetric Koszul connections in TM . At x 2 M let J r ( x ) ⇢ T x M be the vector which spanned by the valued at x of sections of J ( r ). One define the following numerical geometric invariants r b ( M, r ) = max x 2 M { dim ( M ) � dim ( J r ( x ) } , and r b ( M ) = r2 KOSS r b ( r ) max

2ND WORKSHOP OF THE SAO PAULO JOURNAL OF MATHEMATICAL SCIENCES:JEAN-LOUS KOSZUL IN SAO PAULO, HIS WORK AND LEGACYUNIVERSITY 4.2. The Gauge equation The space (2,1)- tensors in a smooth manifold M is denoted by T 2 1 ( M ). The vector bundle of infinitesimal gauge transformations of TM is denoted by G ( TM ). For every pair of Koszul connections, ( r , r * ), the T 2 1 ( M )-valued di ff erential operator G ( TM ) 3 � ! D rr * ( � ) is defined by D rr * � = r * � � � � � r Let X, Y be vector fields, D r r * � ( X, Y ) = r * X � ( Y ) � � ( r X Y ) We denote by J ( rr * ) the sheaf of solutions of the equation D r r * � = 0 . 4.3. The Amari-Chentsov Formalism We have raised open (existence) problm in the di ff erental topology. Remind that a Rie- mannian foliation in M is a couple ( M, g ) where g is a symmetric bilinear form subject to the following requirements. (r.1) The rank of g is constant. (r.2) If a vector filed X is a section of the kernel of g then L X g = 0 , Here L X g is the Lie derivative of g in the direction X . Mutatis mutandis a symplectic foliation in M is a couple ( M ? � ) where � is a closed di ff er- ential 2-form subject to the following requirements (s.1) The rank of � is constant. (s.2) If a vector field X is a section ofthe kernel of � then L X � = 0 . According to the Amari-Rao-Chentsov formalism evry Riemannian metric tensor g is a symmetry of the a ffi ne space of the convex set of Koszul connections in TM . Given such a connection r its image r g under the metric tensor g is defined by g ( r g X Y, Z ) = Xg ( Y, Z ) � g ( Y, r X Z ) We go to focus on global sections of the sheaf J ( rr g ) If both r and r g are symmetric, viz torsion free, the triple ( M, g, r ) is called a statistical manifold.

6 MICHEL NGUIFFO BOYOM Let � be an infinitesimal gauge transformation of the vector bundle TM . To � one assigns two other infinitesimal gauge transformations of TM , namely Ψ and Ψ * which are defined as it follow. g ( Ψ ( X ) , Y ) = 1 2 ( g ( � ( X ) , Y ) + g ( X, � ( Y ))) . g ( Ψ * ( X ) , Y ) = 1 2 ( g ( � ( X ) , Y ) � g ( X, � ( Y ))) Theorem 4.2. If � is a section of the sheaf J ( rr g then so are Ψ and Ψ * . Further if g is positive definite one has the g -orthogonal decoposition TM = Ker ( Ψ ) � Im ( Ψ ) . TM = Ker ( Ψ * ) � Im ( Ψ * ) . Now we involve global sections of the sheaf J ( rr g ) to introduce new numerical invari- ants ( 3 . 2 . 1 ) ( r d )( g, r ) = � 2 J ( rr g ) [max max x 2 M { dim ( M ) � rank ( Ψ ( x )) } ] ( 3 . 2 . 2 ) r d ( r ) = max � r d ( r , g ) g n o ( 3 . 2 . 3 ) s d ( g, r ) = � 2 J ( rr g ) [max max dim ( M ) � rank ( Ψ * ( x )) ] x 2 M ( 3 . 2 . 4 ) s d ( r ) = max � s d ( g, r ) g ( 3 . 2 . 5 ) s d ( M ) = max � s d ( r ) r2 KOSS ( M )

2ND WORKSHOP OF THE SAO PAULO JOURNAL OF MATHEMATICAL SCIENCES:JEAN-LOUS KOSZUL IN SAO PAULO, HIS WORK AND LEGACYUNIVERSITY 4.4. Links with the de Rham algebra Henceforth we will be concerned with global section of the sheaf J ( rr g . We go to point out some exact sequences which are linked with some between the open existence problems that I have listed. Given a Koszul connection r in TM the vector sheaf of r -parallel symmetric (2,0)-tensors is denoted by S r 2 ( M ) , The sheaf of r -parallel skew symmetric (2,0)-tensors is denoted by 2 ( M ) Ω r Definition 4.3. A Hessian cocycle in a locally flat structure ( M, r ) is a non degenerate symmetric 2-cocyle in C ( r , � KV ) A compact locally flat structure ( M, r ) is called hyperbolic if C ( r ) , � KV contains a positive definite exact 2-cocycle g , viz g = � KV � with � 2 C 1 ( r )