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Totally geodesic maps into manifolds with no focal points
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Title
Totally geodesic maps into manifolds with no focal points
Name
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Dibble
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James
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1982-
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James Dibble
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author
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Rong
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Xiaochun
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Xiaochun Rong
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Advisory Committee
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chair
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Luo
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Feng
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Feng Luo
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Advisory Committee
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internal member
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(type = family)
Han
NamePart
(type = given)
Zheng-Chao
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Zheng-Chao Han
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Advisory Committee
Role
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(authority = RULIB)
internal member
Name
(type = personal)
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Croke
NamePart
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Christopher
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Christopher Croke
Affiliation
Advisory Committee
Role
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outside member
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Rutgers University
Role
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(authority = RULIB)
degree grantor
Name
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Graduate School - New Brunswick
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school
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Text
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theses
OriginInfo
DateCreated
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2014
DateOther
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2014-10
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2014
Place
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xx
Language
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eng
Abstract
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The space of totally geodesic maps in each homotopy class [F] from a compact Riemannian manifold M with non-negative Ricci curvature into a complete Riemannian manifold N with no focal points is path-connected. If [F] contains a totally geodesic map, then each map in [F] is energy-minimizing if and only if it is totally geodesic. When N is compact, each map from a product W x M into N is homotopic to a smooth map that's totally geodesic on the M-fibers. These results generalize the classical theorems of Eells-Sampson and Hartman about manifolds with non-positive sectional curvature and are proved using neither a geometric flow nor the Bochner identity. They can be used to extend to the case of no focal points a number of splitting theorems proved by Cao-Cheeger-Rong about manifolds with non-positive sectional curvature and, in turn, to generalize a theorem of Heintze-Margulis about collapsing. The results actually require only an isometric splitting of the universal covering space of M and other topological properties that, by the Cheeger-Gromoll splitting theorem, hold when M has non-negative Ricci curvature. The flat torus theorem is combined with a theorem about the loop space of a manifold with no conjugate points to show that the space of totally geodesic maps in [F] is path-connected. A center-of-mass method due to Cao-Cheeger-Rong is used to construct a homotopy to a totally geodesic map when M is compact. The asymptotic norm of a Z^m-equivariant metric is used to show that the energies of C^1 maps in [F] are bounded below by a constant involving the energy of an affine surjection from a flat Riemannian torus onto a flat semi-Finsler torus, with equality for a given map if and only if it is totally geodesic. This builds on work of Croke-Fathi. It is also shown that the ratio of convexity radius to injectivity radius can be made arbitrarily small over the class of compact Riemannian manifolds of any fixed dimension at least two. This uses Gulliver's examples of manifolds with focal points but no conjugate points.
Subject
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Topic
Mathematics
Subject
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Topic
Geodesics (Mathematics)
Subject
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Topic
Geometry, Riemannian
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Rutgers University Electronic Theses and Dissertations
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ETD_5993
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electronic resource
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application/pdf
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text/xml
Extent
1 online resource (vii, 136 p. : ill.)
Note
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Ph.D.
Note
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Includes bibliographical references
Note
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by James Dibble
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Title
Graduate School - New Brunswick Electronic Theses and Dissertations
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rucore19991600001
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NjNbRU
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doi:10.7282/T36Q1VQB
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(authority = ExL-Esploro)
ETD doctoral
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Rights
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The author owns the copyright to this work.
RightsHolder
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Name
FamilyName
Dibble
GivenName
James
Role
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RightsEvent
Type
Permission or license
DateTime
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2014-10-06 16:29:59
AssociatedEntity
Name
James Dibble
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Copyright holder
Affiliation
Rutgers University. Graduate School - New Brunswick
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Author Agreement License
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I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.
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2014-10-31
DateTime
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2015-10-31
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Embargo
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Access to this PDF has been restricted at the author's request. It will be publicly available after October 31st, 2015.
Copyright
Status
Copyright protected
Availability
Status
Open
Reason
Permission or license
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windows xp
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