Terse Notes on Riemannian Geometry

These notes cover the basics of Riemannian geometry, Lie groups, and symmetric spaces. This is just a listing of the basic definitions and theorems with no in-depth discussion or proofs. Some exercises are included at the end of each section to give you something to think about. See the references cited within for more complete coverage of these topics. Many geometric entities are representable as Lie groups or symmetric spaces. Transformations of Euclidean spaces such as translations, rotations, scalings, and affine transformations all arise as elements of Lie groups. Geometric primitives such as unit vectors, oriented planes, and symmetric, positive-definite matrices can be seen as points in symmetric spaces. This chapter is a review of the basic mathematical theory of Lie groups and symmetric spaces. The various spaces that are described throughout these notes are all generalizations, in one way or the other, of Euclidean space, R. Euclidean space is a topological space, a Riemannian manifold, a Lie group, and a symmetric space. Therefore, each section will use R as a motivating example. Also, since the study of geometric transformations is stressed, the reader is encouraged to keep in mind that R can also be thought of as a transformation space, that is, as the set of translations on R itself.