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Abstract.This work examines the relationship between pairs of projectively equivalent Riemannian or Lorentz metrics g and g* on a manifold M that induce the same Riemannian metric on a hypersurface H. In general such metrics must be equal. In the case of distinct metrics, the structure of the metrics and the manifold are strongly determined by the set, C, of points at which g and g* are conformally related. The space (M -C, g) is locally a warped product manifold over the hypersurface H. In the Lorentz setting, C is empty. In the Riemannian case, C contains at most two points. If C is nonempty, then H is isometric to a standard sphere. Furthermore, in the case that C contains one point, natural hypotheses imply M is diffeomorphic to R". If C contains two points M is diffeomorphic to S".Introduction. This work considers the extent to which a Riemannian or Lorentz metric is determined by its geodesies. Historically, this problem has been viewed as an "inverse problem" in the calculus of variations. The problem is also related to the classical works on affine and projective connections of Cartan, Thomas, Veblen, Weyl and others.

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Context: At the Mission Design and Navigation Software Group at the Jet Propulsion Laboratory we make use of finite exponential based defect models to aid in maintenance planning and management for our widely used critical systems. However a number of pragmatic issues arise when applying defect models for a post-release system in continuous use. These include: how to utilize information from problem reports rather than testing to drive defect discovery and removal effort, practical model calibration, and alignment of model assumptions with our environment. Goal: To show how we can develop confidence in the practical applicability of our models for obtaining stable maintenance funding. Method: We describe the strong empirical and face validity we have investigated for our maintenance defect discovery and introduction models. We discuss the practical details of calibration and application within a functioning maintenance environment. Results: We find that our models, despite their simplicity, appear quite valid. Conclusions: The models are useful in justifying and obtaining stable maintenance funding.