Page - 231 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Atanothersideevery∇∈LF(M)givesrise to themap Ri(M) g→∇g∈LC(M) which isdefinedby g(Y,∇gXZ)=Xg(Y,Z)−g(∇XY,Z). IneveryRiemannianmanifold (M,g)wedefinethe followingnumerical invariants rb(M,g)= min [∇∈LF(M)] { dim(M)−rb(∇∗) } , rB(M)= min [g∈Rie(M)] { rb(M,g) } . Inevery locallyflatmanifold (M,∇)wedefinethe followingnumerical invariant rb(M,∇)= min [g∈Rie(M)] { rb(∇g) } Thenumerical invariantswe justdefinedhavenotable impacts. AppendixA.1. TheAffinelyFlatGeometry TheoremA1. Ina smoothmanifoldMthe followingassertionsare equivalent (1) rb(M)=0, (2) themanifoldMadmits locallyflat structures. AppendixA.2. TheHessianGeometry TheoremA2 (Answeraholdquestionsof [65]). InaRiemannianmanifold (M,g) the followingassertions are equivalent (1) rb(M,g)=0, (2) theRiemannianmanifold (M,g)admitsHessianstructures (M,g,∇) AComment. Assertion (2)has the followingmeaning. (i) (M,∇) is a locallyflatmanifold. (ii) everypointhasanopenneighborhoodUsupportingasystemofaffinecoordinate functions (x1,...,xm)anda local smooth functionh(x1,...,xm) such that g( ∂ ∂xi , ∂ ∂xj )= ∂2h ∂xi∂xj . AppendixA.3. TheGeometryofKoszul TheoremA3. In a locallyflatmanifold (M,∇)whoseKValgebra is denotedbyA the followingassertions are equivalent (1) rb(M,∇)=0, (2) theKVcohomologyspaceH2KV(A,R) containsametric class [g], (3) the locallyflatmanifold (M,∇)admitsHessianstructures (M,∇,g). 231