Grant-in-Aid for Scientific Research (C)
|Allocation Type||Single-year Grants|
|Research Institution||Iwate University|
OSHIKIRI Gen-ichi Faculty of Education, Iwate University Professor, 教育学部, 教授 (70133931)
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)
小宮山 晴夫 岩手大学, 教育学部, 講師 (90042762)
|Project Period (FY)
1998 – 2000
Completed(Fiscal Year 2000)
|Budget Amount *help
¥3,200,000 (Direct Cost : ¥3,200,000)
Fiscal Year 2000 : ¥700,000 (Direct Cost : ¥700,000)
Fiscal Year 1999 : ¥800,000 (Direct Cost : ¥800,000)
Fiscal Year 1998 : ¥1,700,000 (Direct Cost : ¥1,700,000)
|Keywords||Foliation / Bundle-like foliation / Mean curvature function / Minimal foliation / Constant mean curvature / Totally geodesic / Minimal graph / Bernstein問題 / 葉の成長度 / 平均曲率ベクトル / 束葉層 / ベーシック 1-形式 / リーマン葉層 / コンパクト葉 / 断面曲率正 / キリングベクトル場|
(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.