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Some geometric properties of singular pseudo-Riemannian manifolds. (Quelques propriétés géométriques des variétés pseudo-riemanniennes singulières.) (French. English summary) Zbl 0845.53044

Author’s abstract: “A pseudo-metric \(g\) on a manifold \(M\) is a field of symmetric bilinear forms on \(M\). This situation arises naturally for the immersions of manifolds into a pseudo-Riemannian manifold. The singular locus of the pseudo-metric \(g\) is the set of points where \(g\) is degenerated. In a generic context, a ‘natural’ stratification of the singular locus is given. A notion of Levi-Civita ‘dual connection’ is canonically associated to a pseudo-metric. In a generic context, some results about autoparallel sets (in particular for the singular locus), parallel translation and geodesics are established. The general notion of obstruction index for a section of a fiber bundle allows us to define a numerical invariant associated to a pseudo-metric. For a compact manifold \(M\) with a pseudo-metric \(g\), Chern-Gauss-Bonnet’s theorem can be generalized, under adequate hypotheses. Thus an integral formula whose value is the difference between the Euler-Poincaré characteristic of \(M\) and the numerical invariant associated to \(g\) is obtained. Finally, some geometric and topological properties of some manifolds admitting a flat generic pseudo-metric are pointed out”.

MSC:

53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
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