Abstract
This chapter focuses on Riemannian geometry. In studying the geometry of a surface in E3, it is found that some of its most important geometric properties belong to the surface itself and not the surrounding Euclidean space. Gaussian curvature is a prime example; although defined in terms of shape operators, it belongs to this intrinsic geometry, as it passes the test of isometric invariance. The dot product is but one instance of the general notion of inner product, and Riemann's idea was to replace the dot product by a quite arbitrary inner product on each tangent plane of an abstract surface M. A manifold M of arbitrary dimension furnished with a differentiable inner product on each of its tangent spaces is called a Riemannian manifold, and the resulting geometry is Riemannian geometry. Thus, a geometric surface is the same thing as a two-dimensional Riemannian manifold. Various theorems are proven in the chapter.
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