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On the Gauss-Bonnet theorem for singular pseudo-metrics. (Sur le théorème de Gauss-Bonnet pour les pseudo-métriques singuliéres.) (French) Zbl 1149.53314
Séminaire de théorie spectrale et géométrie. Année 1986-1987. Chambéry: Univ. de Savoie, Fac. des Sciences, Service de Mathématiques; St. Martin d’Hères: Univ. de Grenoble I, Inst. Fourier. Sémin. Théor. Spectrale Géom., Chambéry-Grenoble 5, 99-105 (1987).
Summary: A pseudo-metric $$g$$ is a field of symmetric bilinear forms on a manifold $$M$$. If $$g$$ is nondegenerate and $$M$$ compact, then the classical theorem of Chern-Gauss-Bonnet affirms that the integral of the Pfaffian of a metric compatible with $$g$$ is equal to the Euler class of $$M$$. This property is no longer true, in general, if $$g$$ is degenerate. The object of this article is to study such a situation when the singular locus of $$g$$, that is the set of points of $$M$$ where $$g$$ is degenerate, is a submanifold of codimension 1.
For the entire collection see [Zbl 0825.00040].

##### MSC:
 53C35 Differential geometry of symmetric spaces 58A17 Pfaffian systems 53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics 53C40 Global submanifolds
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