Projectively Equivalent Metrics Subject to Constraints

78 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1980.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 n-sphere.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.U of I OnlyRestricted to the U of I community idenfinitely during batch ingest of legacy ETD