The study on differential geometrical affine hypersurfaces of an affine space by making use of Information geometry

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Let R^<n+1> be an (n+1)-dimensional affine space with torsion free affine connection D and (M,∇) an affine hypersurface of (R^<n+1>,D). Let R,Ric be the curvature tensor for M, the Ricci tensor, respectively. Act R(X,Y) to R or Ric as the derivation for every vector tangent to M and we consider the conditions of R(X,Y)・R=0 or R(X,Y)・Ric=0 for every vector tangent to M. The purpose of the presnet paper is to consider whether the weaker conditon R(X,Y)・R=0 for every vectors X,Y tanget to M than the local symmetry means the local symmetry. Also, we consider whether the equivalence of the condition of R(X,Y)・R=0 and the condition of R(X,Y) Ric=0 are equivalence. We study the nondegenerate Blaschke hypersurfaces at 2003 and the nondegenerate hypersurfaces at 2004, 2005. We can prove that either a proper affine hypersphere or an affine cylindrical is the only nondegenerate affine hypersurface of affine space with torsion free affine connection which satisfies the Ricci semi-symmetry and the results is published by Result. Math. (2005) and give the lecture with respect to those results on International symposium (ISRAMSES, 2005). Since the affine fundamental form h is nondegenerate, we can see it as the nondegenerate metric and can study by the similar way with the study of hypersurfaces of a real space form and complex Kaehler hypersurfaces of a complex space form. But, such a metric is not compatible with h. Choosing the conjugate connection ∇^^- with respect to ∇, we just become to study the statiscal manifold and information geometry. We study them, noting that ∇+∇^^- is compatible with h. On and on, we go on them and we hereafter want to study the degenerate immersion.