| Co-Investigator(Kenkyū-buntansha) |
NARUKI Isao Ritsumeikan Univ. Fac. Science and Engineering, Professor, 理工学部, 教授 (90027376)
NAKAJIMA Kazuhumi Ritsumeikan Univ. Fac. Science and Engineering, Professor, 理工学部, 教授 (10025489)
SHIN'YA Hitoshi Ritsumeikan Univ. Fac. Science and Engineering, Professor, 理工学部, 教授 (70036416)
KAGAWA Takaaki Ritsumeikan Univ. Fac. Science and Engineering, Assoc.Professor, 理工学部, 助教授 (90298175)
YAMADA Osanobu Ritsumeikan Univ. Fac. Science and Engineering, Professor, 理工学部, 教授 (70066744)
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| Research Abstract |
The purpose of this investigation is to consider a projective transformation of a complete (or compact) Riemannian manifold with constant scalar curvature and a holomorphically projective transformation of a complete (or compact) Kahlerian manifold with constant scalar curvature. We have studied these consideration under the additional conditions such that the transformations leaves a certain tensor invariant. The main results are following :
1. Let(M, g) be a simply connected and complete Riemannian manifold with positively constant scalar curvature. If there exists a non-affine projective change h of g such that the scalar curvature of h is constant and the change leaves their gravitational tensor fields invariant, then (M, g) is isometric to a standard sphere.
2. Let (M, g) be a simple connected and complete Riemannian manifold with positively constant scalar curvature. If there exists a non-affine projective change h of g such that the scalar curvature of h is constant and the change leaves their covariant derivatives of projective curvature tensor field invariant, then (M, g) is isometric to a standard sphere.
3. Let (M, g) be a compact Riemannian manifold with constant scalar curvature S. If there exists a non-affine projective change h of g such that the scalar curvature of h is constant and their Ricci tensors R and RィイD4〜ィエD4 satisfy
(▽ィイD4〜ィエD4ィイD2xィエD2RィイD4〜ィエD4)(Y, Z) - (▽ィイD4〜ィエD4ィイD2yィエD2RィイD4〜ィエD4)(Y, Z) = (▽ィイD2xィエD2R)(X,Z) - (▽ィイD2yィエD2R)(X,Z)
for any vector fields X, Y and Z, then S>O.
4. Concerning a holomorphically projective transformation of a Kahlerian manifold, we can obtain the results analogous to the above.
5. Let (M,J,g) be a Kahlerian manifolds with harmonic curvature. If there exists a non-affine holomorphically projective change of g, then (M, J, g) is a Einstein space.
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