| Co-Investigator(Kenkyū-buntansha) |
NUMATA Minoru Faculty of Education, Iwate University Professor, 教育学部, 教授 (50028255)
FUMIO Nakajima Faculty of Education, Iwate University Professor, 教育学部, 教授 (20004484)
KOJIMA Hisashi Faculty of Education, Iwate University Professor, 教育学部, 教授 (90146118)
AKIO Miyai Faculty of Education, Iwate University Assistant, 教育学部, 助手 (70003960)
KAWADA Koichi Faculty of Education, Iwate University Ass.Professor, 教育学部, 助教授 (70271830)
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| Research Abstract |
(a) We get the following result :
Let (M,F,g) be a codimension-q bundle-like foliation on a closed Riemannian manifold of positive curvature. (1) If q is even, then F has a compact leaf. (2) If q is odd, then F has a leaf whose closure is a closed codimsnsion-(q-1) submanifold.
As a corollary, we extend Berger's famous result :
Any Killing vector field on a closed Riemannian manifold with positive sectional curvature admits a zero point or a closed orbit.
(b) We study the dual 1-form to the mean curvature vector of a foliation. We give a characterization of such 1-forms for codimension-one foliations. We also have a simple characterization when the foliation is a bundle foliation, and when the dual 1-form is basic.
(c) We get the following result :
Let (M,F,g) be a codimension-1 minimal foliation on a complete Riemannian manifold of non-negative Ricci curvature. If the growth of F is not greater than 2, then F is totally geodesic. Further, (M,g) is locally a Riemannina product.
As a byproduct, we get a simple proof of Mirand's result on minimal graphs, and foliated version of the result by Alencar and do Carmo on constant mean curvature hypersurfaces.
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