THE GEOMETRY OF HESSIAN STRUCTURES
CONTENTS 1 A?NE SPACES AND CONNECTIONS A?NE SPACES CONNECTIONS VECTOR BUNDLES 2 HESSIAN STRUCTURES HESSIAN STRUCTURES HESSIAN STRUCTURES AND K¨AHLERIAN STRUCTURES DUAL HESSIAN STRUCTURES DIVERGENCES FOR HESSIAN STRUCTURES CODAZZI STRUCTURES 3 CURVATURES FOR HESSIAN STRUCTURES HESSIAN CURVATURE TENSORS AND KOSZUL FORMS HESSIAN SECTIONAL CURVATURE 4 REGULAR CONVEX CONES REGULAR CONVEX CONES HOMOGENEOUS SELF-DUAL CONES 5 HESSIAN STRUCTURES AND A?NE DI?ERENTIAL GEOMETRY A?NE HYPERSURFACES LEVEL SURFACES OF POTENTIAL FUNCTIONS LAPLACIANS OF GRADIENT MAPPINGS 6 HESSIAN STRUCTURES AND INFORMATION GEOMETRY DUAL CONNECTIONS ON SMOOTH FAMILIES OF PROBABILITY DISTRI- BUTIONS HESSIAN STRUCTURES INDUCED BY NORMAL DISTRIBUTIONS 7 COHOMOLOGY ON ?AT MANIFOLDS (P, Q)-FORMS ON ?AT MANIFOLDS LAPLACIANS ON ?AT MANIFOLDS KOSZUL’S VANISHING THEOREM LAPLACIANS ON HESSIAN MANIFOLDS LAPLACIAN ?L A?NE CHERN CLASSES OF ?AT MANIFOLDS 8 COMPACT HESSIAN MANIFOLDS A?NE DEVELOPMENTS AND EXPONENTIAL MAPPINGS FOR ?AT MANIFOLDS CONVEXITY OF HESSIAN MANIFOLDS KOSZUL FORMS ON HESSIAN MANIFOLDS 9 SYMMETRIC SPACES WITH INVARIANT HESSIAN STRUCTURES INVARIANT ?AT CONNECTIONS AND A?NE REPRESENTATIONS INVARIANT HESSIAN STRUCTURES AND A?NE REPRESENTATIONS SYMMETRIC SPACES WITH INVARIANT HESSIAN STRUCTURES 10 HOMOGENEOUS SPACES WITH INVARIANT HESSIAN STRUCTURES SIMPLY TRANSITIVE TRIANGULAR GROUPS HOMOGENEOUS REGULAR CONVEX DOMAINS AND CLANS PRINCIPAL DECOMPOSITIONS OF CLANS AND REAL SIEGEL DOMAINS 11 HOMOGENEOUS SPACES WITH INVARIANT PROJECTIVELY ?AT CONNECTIONS INVARIANT PROJECTIVELY ?AT CONNECTIONS INVARIANT CODAZZI STRUCTURES OF CONSTANT CURVATURE CHAPTER A?NE SPACES IN THIS SECTION WE GIVE A BRIEF OUTLINE OF THE CONCEPTS OF A?NE SPACES, A?NE TRANSFORMATIONS AND A?NE REPRESENTATIONS WHICH ARE NECESSARY FOR AN UN- DERSTANDING OF THE CONTENTS OF SUBSEQUENT CHAPTERS OF THIS BOOK DE?NITION 1 1 LET V BE AN N-DIMENSIONAL VECTOR SPACE AND ?A NON- EMPTY SET ENDOWED WITH A MAPPING, (P, Q) ??× ???? ? PQ ?V, SATISFYING THE FOLLOWING CONDITIONS (1) FOR ANY P, Q, R ??WE HAVE ? ? PR = ? ? PQ + ? ? QR (2) FOR ANY P ??AND ANY V ?V THERE EXISTS A UNIQUE Q ??SUCH THAT V = ? ? PQ THEN ?IS SAID TO BE AN N-DIMENSIONAL A?NE SPACE ASSOCIATED WITH V GEOMETRY OF HESSIAN STRUCTURES EXAMPLE 1 1 LET V BE AN N-DIMENSIONAL VECTOR SPACE WE DE?NE A MAP- PING (P, Q) ?V × V ??? ? PQ = Q ?P ?V THEN THE SET V IS AN N-DIMENSIONAL A?NE SPACE ASSOCIATED WITH THE VECTOR SPACE V EXAMPLE 1 2 AN A?NE SPACE ASSOCIATED WITH THE STANDARD VECTOR SPACE RN = {P = (P1, , PN) | PI ?R} IS SAID TO BE THE STANDARD A?NE SPACE A PAIR {O; E1, , EN} OF A POINT O ??AND A BASIS {E1, , EN} OF V IS SAID TO BE AN A?NE FRAME OF ?WITH ORIGIN O AN A?NE FRAME {O; E1, , EN} DE?NES AN N-TUPLE OF FUNCTIONS {X1, , XN} ON ?BY ? ? OP = X I XI(P)EI, P ??, WHICH IS CALLED AN A?NE COORDINATE SYSTEM ON ?WITH RESPECT TO THE A?NE FRAME LET {¯X1, , ¯XN} BE ANOTHER A?NE COORDINATE SYSTEM WITH RESPECT TO AN A?NE FRAME {¯O; ¯E1, , ¯EN} IF EJ = P I AI J¯EI, ? ? ¯OO = P I AI¯EI, THEN ¯XI = X J AI JXJ + AI REPRESENTING THE COLUMN VECTORS [XI], [¯XI] AND [AI] BY X = [XI], ¯X = [¯XI] AND A = [AI] RESPECTIVELY, AND THE MATRIX [AI J] BY A = [AI J], WE HAVE ¯X = AX + A, OR ¯X = A A 0