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Bishop, Richard L. Kim, Hobum 2011-05-07T12:39:17Z 2011-05-07T12:39:17Z 10000-01-01 1990 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. Made available in DSpace on 2011-05-07T12:39:17Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9114294.pdf: 1735927 bytes, checksum: afc4609014734ac2b7aa4ba1b2c2e04b (MD5) Previous issue date: 1990 Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:43:55Z Item is restricted indefinitely. Restriction data tranferred 2014-07-01T11:19:16-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission ETDs are only available to UIUC Users without author permission U of I Only AAI9114294 (UMI)AAI9114294 http://hdl.handle.net/2142/20441 eng Copyright 1990 Kim, Hobum Mathematics Geometric and dynamical properties of Riemannian foliations text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.