Today we cover some foundational results in Riemannian Geometry.

A Riemannian metric is a smooth choice of inner product on each tangent space. We show that a Riemannian metric $m$ on a smooth manifold $M$ determines a (normal) metric on $M$, and then introduce the exponential map of a compact Riemannian manifold.

Although Riemannian Geometry is a deep and fascinating subject in its own right, for our purposes the Riemannian metric should simply be regarded as part of the background structure. This is analogous to the way that the dynamics of a map on a vector space did not depend on the choice of norm.

Next lecture we get back to dynamics, and introduce hyperbolic sets for differentiable dynamical systems on manifolds.



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