| Co-Investigator(Kenkyū-buntansha) |
YAMAMOTO Makoto Chuo University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (10158305)
MIYOSHI Shigeaki Chuo University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (60166212)
OHARU Shinnosuke Chuo University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (40063721)
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| Budget Amount *help |
¥2,400,000 (Direct Cost: ¥2,400,000)
Fiscal Year 2005: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2004: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2003: ¥1,000,000 (Direct Cost: ¥1,000,000)
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| Research Abstract |
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.
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