EDG Ch.7 -- Riemannian Geometry

Introduction to Chapter 7: Riemannian Geometry

In studying the geometry of a surface in R³ we found that some of its most important geometric properties belong to the surface itself and not to the surrounding Euclidean space. Gaussian curvature is a prime example; although defined in terms of shape operators, it belongs to this intrinsic geometry since it passes the test of isometric invariance.

As long ago as 1854, Riemann drew the correct conclusion: There must exist a geometrical theory of surfaces completely independent of R³, a geometry built from the start solely of isometric invariants. In this chapter we describe the fundamentals of the resulting theory, concentrating on its dominant features: Gaussian curvature and geodesics. Our constant guides will be the two special cases that led to its discovery: the intrinsic geometry of surfaces in R³, and Euclidean geometry---particularly that of the plane R².

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