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Towards Riemannian Geometry

In: A Differential Approach to Geometry

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  • Francis Borceux

    (Université catholique de Louvain)

Abstract

The first purpose of this chapter is to provide a deep intuition of formal notions like the metric tensor, the Christoffel symbols, the Riemann tensor, vector fields, the covariant derivative, and so on: an intuition based on the consideration of surfaces in the three dimensional real space. We switch next to the study of geodesics, geodesic curvature and systems of geodesic coordinates. As an example of a Riemann surface, we develop the study of the Poincaré half plane and prove that it is a model of non-Euclidean geometry. We conclude with the Gauss–Codazzi–Mainardi equations and the related question of the embeddability of Riemann surfaces in the three dimensional real space.

Suggested Citation

  • Francis Borceux, 2014. "Towards Riemannian Geometry," Springer Books, in: A Differential Approach to Geometry, edition 127, chapter 0, pages 253-343, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-01736-5_6
    DOI: 10.1007/978-3-319-01736-5_6
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