# A Differential Approach to GeometryTowards Riemannian Geometry

A Differential Approach to Geometry: Towards Riemannian Geometry [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.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

# A Differential Approach to GeometryTowards Riemannian Geometry

91 pages

/lp/springer-journals/a-differential-approach-to-geometry-towards-riemannian-geometry-OFZ7HFUBBj
Publisher
Springer International Publishing
© Springer International Publishing Switzerland 2014
ISBN
978-3-319-01735-8
Pages
253 –343
DOI
10.1007/978-3-319-01736-5_6
Publisher site
See Chapter on Publisher Site

### 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.]

Published: Aug 26, 2013

Keywords: Riemann Surface; Covariant Derivative; Gaussian Curvature; Tangent Plane; Normal Curvature