With the comforting feeling that there is indeed a variety of Riemannian manifolds out there, we shall now delve into the theory. Initially, we shall confine ourselves to infinitesimal considerations. The most important and often also least understood object of Riemannian geometry is that of the Riemannian connection. From this concept it will be possible to define curvature and more familiar items like gradients and Hessians of functions. Studying curvature is the central theme of Riemannian geometry. The idea of a Riemannian metric having curvature, while intuitively appealing and natural, is for most people the stumbling block for further progress into the realm of geometry.
KeywordsVector Field Riemannian Manifold Distance Function Covariant Derivative Sectional Curvature
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