Pseudo-Riemannian manifold

In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the requirement of positive-definiteness is relaxed. Every tangent space of a pseudo-Riemannian manifold is a pseudo-Euclidean vector space. A special case used in general relativity is a four-dimensional Lorentzian manifold for modeling spacetime, where tangent vectors can be classified as timelike, null, and spacelike. Introduction Manifolds Manifold and Differentiable manifold In differential geometry, a differentiable manifold is a space which is locally similar to a Euclidean space. In an n-dimensional Euclidean space any point can be specified by n real numbers. These are called the coordinates of the point. An n-dimensional differentiable manifold is a generalisation of n-dimensional Euclidean space. In a manifo

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Déformations conformes des variétés de Finsler-Ehresmann

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An intrinsic approach to Finsler geometry is proposed. A concept of Finsler- Ehresmann manifold, denoted by (M,F,H), is introduced and a generalized Chern connection is built for this manifold. Conformal deformations on this manifold are considered. First, we have an analogous of Chern's theorem: we prove the existence and uniqueness of a generalized Chern connection for the manifold (M,F,H). Similarly, within an essentially koszulian formalism, we present two curvatures associated to this generalized connection, namely a R curvature and a P one. The second result is the deduction of conformal transformations laws for the generalized Chern connection and associated curvatures. The transformation of R seems to have very similar properties as that of the Riemannian curvature while that of P reveals other objects of pure Finslerian nature. Third, we construct the finsler Weyl and Schouten tensors W and S respectively and we study their conformal transformations. Furthermore, we show that for the dimension 3, the horizontal component of W for generalized Berwald manifolds is identically zero. The next result is a theorem of Weyl-Schouten type giving necessary and sufficient conditions for a Finsler-Ehresmann manifold to be conformaly R-flat. We complete this result by exploring the case of dimension 3 for Berwald spaces which gives a result very similar to the Riemannian case. In addition, we announce some necessary conditions to characterize conformal flatness of Finsler-Ehresmann manifolds.

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