A minimization of a curvature functional on fiber bundles

Let B be a smooth compact orientable surface without boundary and with $\chi(B) < 0.$ We examine two types of fiber bundles M over B with fiber F. The first is a principle fiber bundle with a two-torus fiber and the second is an $S\sp2$ fiber bundle with an SO(3) group action. In each case, the t...

Full description

Bibliographic Details
Main Author: Hawkins, Christopher Ryan
Other Authors: Gao, L. Zhiyong
Format: Others
Language:English
Published: 2009
Subjects:
Online Access:http://hdl.handle.net/1911/19268
id ndltd-RICE-oai-scholarship.rice.edu-1911-19268
record_format oai_dc
spelling ndltd-RICE-oai-scholarship.rice.edu-1911-192682013-10-23T04:13:11ZA minimization of a curvature functional on fiber bundlesHawkins, Christopher RyanMathematicsLet B be a smooth compact orientable surface without boundary and with $\chi(B) < 0.$ We examine two types of fiber bundles M over B with fiber F. The first is a principle fiber bundle with a two-torus fiber and the second is an $S\sp2$ fiber bundle with an SO(3) group action. In each case, the tangent space of the bundle can be decomposed into a vertical space, those vectors tangent to fibers, and a horizontal space complementary to the vertical space and invariant under the group action. The bundle can be given a metric that is the direct sum of metrics on the vertical and horizontal spaces. Additionally, with this metric, M, is locally isometric to a product space $B\times F$ with metric $g\sb{b} + g\sb{f}.$ Here $g\sb{b}$ is any fixed metric on the base, $g\sb{f}$ is a constant curvature metric on the fiber invariant under the action of the group. We can obtain a new metric on M by scaling the horizontal component of the original by $e\sp{2u}$ and the vertical component by $f\sp2,$ where u and f are smooth functions on the base. We put certain constraints on u and f and consider the family of all such variations. In this thesis, we show, using nonlinear elliptic estimates, that among these metrics there is one for which the integral of the norm of the Ricci curvature tensor squared, $\int\sb{M}\vert Ric\vert\sp2dV,$ is minimized.Gao, L. Zhiyong2009-06-04T08:18:25Z2009-06-04T08:18:25Z1998ThesisText67 p.application/pdfhttp://hdl.handle.net/1911/19268eng
collection NDLTD
language English
format Others
sources NDLTD
topic Mathematics
spellingShingle Mathematics
Hawkins, Christopher Ryan
A minimization of a curvature functional on fiber bundles
description Let B be a smooth compact orientable surface without boundary and with $\chi(B) < 0.$ We examine two types of fiber bundles M over B with fiber F. The first is a principle fiber bundle with a two-torus fiber and the second is an $S\sp2$ fiber bundle with an SO(3) group action. In each case, the tangent space of the bundle can be decomposed into a vertical space, those vectors tangent to fibers, and a horizontal space complementary to the vertical space and invariant under the group action. The bundle can be given a metric that is the direct sum of metrics on the vertical and horizontal spaces. Additionally, with this metric, M, is locally isometric to a product space $B\times F$ with metric $g\sb{b} + g\sb{f}.$ Here $g\sb{b}$ is any fixed metric on the base, $g\sb{f}$ is a constant curvature metric on the fiber invariant under the action of the group. We can obtain a new metric on M by scaling the horizontal component of the original by $e\sp{2u}$ and the vertical component by $f\sp2,$ where u and f are smooth functions on the base. We put certain constraints on u and f and consider the family of all such variations. In this thesis, we show, using nonlinear elliptic estimates, that among these metrics there is one for which the integral of the norm of the Ricci curvature tensor squared, $\int\sb{M}\vert Ric\vert\sp2dV,$ is minimized.
author2 Gao, L. Zhiyong
author_facet Gao, L. Zhiyong
Hawkins, Christopher Ryan
author Hawkins, Christopher Ryan
author_sort Hawkins, Christopher Ryan
title A minimization of a curvature functional on fiber bundles
title_short A minimization of a curvature functional on fiber bundles
title_full A minimization of a curvature functional on fiber bundles
title_fullStr A minimization of a curvature functional on fiber bundles
title_full_unstemmed A minimization of a curvature functional on fiber bundles
title_sort minimization of a curvature functional on fiber bundles
publishDate 2009
url http://hdl.handle.net/1911/19268
work_keys_str_mv AT hawkinschristopherryan aminimizationofacurvaturefunctionalonfiberbundles
AT hawkinschristopherryan minimizationofacurvaturefunctionalonfiberbundles
_version_ 1716611060834762752