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Title:  Projectively Equivalent Metrics Subject to Constraints 
Author(s):  Taber, William Lawrence 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
Abstract:  This thesis concerns the relationship between pairs of projectively equivalent Riemannian or Lorentz metrics that share some property along a hypersurface of a manifold. The first chapter is devoted to the construction of projectively equivalent metrics and to the recollection of classical results. In the second chapter we assume the two metrics, g and g('*), induce the same Riemannian metric on a hypersurface, H, of a manifold M. We also assume that M has dimension greater than two and that H has nondegenerate second fundamental form. Under these assumptions we establish that, unless H possesses strong symmetry with respect to M, the two metrics agree throughout M. Next, we investigate the situation in which the two metrics are, in fact, distinct. In this setting, the structure of (M,g) is strongly determined by the number of conformal points in M, that is, those points at which g and g('*) are conformally related. Under natural hypotheses, the restriction of the metrics to the complement of the set of conformal points are, locally, warped product metrics. In addition, we give conditions sufficient to insure that the restricted metrics are global warped product metrics. If g and g('*) are Lorentz metrics, then M contains no conformal points. In the Riemannian setting M can have at most two conformal points. If, in fact, M possess conformal points, then H must be a metric sphere about each of them. Furthermore, H must be isometric to a standard sphere. If M contains a single conformal point, p, and (M{p},g) is a warped product manifold, then (M,g) is diffeomorphic to (//R)('n). If M contains two conformal points, it is diffeomorphic to an nsphere. In the final chapter we generalize a result due to Ralph Alexander to higher dimensional Riemannian manifolds. Suppose (M,g) is strictly convex, and H is the smooth boundary of an open set with compact closure in M. Suppose further that d and d('*) are the distance functions determined by g and g('*). If d(p,q) = d('*)(p,q) for each pair of points, p and q, in H, then g and g('*) agree throughout M. 
Issue Date:  1980 
Type:  Text 
Language:  English 
Description:  78 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 1980. 
URI:  http://hdl.handle.net/2142/68180 
Other Identifier(s):  (UMI)AAI8108683 
Date Available in IDEALS:  20141214 
Date Deposited:  1980 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois