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 Title: Geometric and dynamical properties of Riemannian foliations Author(s): Kim, Hobum Doctoral Committee Chair(s): Bishop, Richard L. Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Mathematics Abstract: Given a Riemannian foliation ${\cal F}$ on a Riemannian manifold M with a bundle-like metric, geometric and dynamical properties of geodesics orthogonal to the leaves of the foliation are studied.In one line of work, the concepts of ${\cal F}$-Jacobi fields and ${\cal F}$-Jacobi tensors are introduced. Using these concepts, an upper bound for the index of a focal point of a leaf is obtained, when the orthogonal complement of the foliation is involutive. In particular, it is proved that there is no focal point of a leaf of a Riemannian foliation of codimension one. Moreover, it is also proved that if M is a complete Riemannian manifold of nonnegative sectional curvature, and if the norm of the integrability tensor is small compared with the sectional curvature of M, then ${\cal F}$ is totally geodesic.In another line of work, it is proved that: (1) ${\cal F}$ is harmonic if and only if the geodesic flow preserves the corresponding Riemannian volume form on the normal bundle corresponding to a Sasaki-type metric, in case ${\cal F}$ is transversally flat.(2) There is no Riemannian foliation on a compact Riemannian manifold of negative sectional curvature. The proof uses Oseledec's multiplicative ergodic theorem. Issue Date: 1990 Type: Text Language: English URI: http://hdl.handle.net/2142/20441 Rights Information: Copyright 1990 Kim, Hobum Date Available in IDEALS: 2011-05-07 Identifier in Online Catalog: AAI9114294 OCLC Identifier: (UMI)AAI9114294
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