Geometric and dynamical properties of Riemannian foliations
Kim, Hobum
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https://hdl.handle.net/2142/20441
Description
Title
Geometric and dynamical properties of Riemannian foliations
Author(s)
Kim, Hobum
Issue Date
1990
Doctoral Committee Chair(s)
Bishop, Richard L.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
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.
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