Graduate Thesis Or Dissertation
 

Singular geometry and geometric singularities

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https://ir.library.oregonstate.edu/concern/graduate_thesis_or_dissertations/9w0325678

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  • In this thesis the author extends the standard techniques of differential geometry to the case of distributional (in the sense of Schwartz) pseudoriemannian structures. A new definition of flows for distributional vector fields is developed using generalized paths which are essentially nonlinear-distributional measures, and an existence theorem is proved when the coefficients are distribution extensions of real meromorphic functions. Conjugate points are defined by means of a new characterization in the smooth case, as the caustic set of the geodesic flow. The theory is applied to obtain a new interpretation of general relativity in which the Einstein equations do not break down at spacetime singularities. Fourier Integral Operator theory is used to study the propagation of spacetime singularities and the solvability of the Einstein equations in a class of algebraically special spacetimes. Explicit computations for the Schwarzschild structure show it is (modulo constants) a partial fundamental solution of the Einstein equations in the sense of partial differential equations.
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  • File scanned at 300 ppi (Monochrome) using Capture Perfect 3.0.82 on a Canon DR-9080C in PDF format. CVista PdfCompressor 4.0 was used for pdf compression and textual OCR.
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