Applications of Manifolds not embedded in Euclidean Space In the first few pages of his book Introduction to Smooth Manifolds, Lee writes: 

But
  for more sophisticated applications, it is an undue restriction to require
  smooth manifolds to be subsets of some ambient Euclidean space.

A way to motivate why we develop tools to do calculus in manifolds is through examples in physics. For instance, one might want to do calculus on the function that maps the surface of the earth to its temeprature on the real line, we might want to study electromagnetic properties of a torus, and so forth. In each of these cases, it is easy to study these surfaces as subsets (or submanifolds) of Euclidean space. 
What are some "sophisticated applications" of manifolds outside of math - as Lee writes - that do not allow us to work with manifolds that are embedded in Euclidean space? 
Edit: I'm obviously looking for examples other than the one Lee gives himself (and perhaps the most popular here): Looking at space-time as a four dimensional manifold, where it doesn't make sense to embed it in an ambient space. 
 A: In my opinion, the point here is that one can think of manifolds intrinsically, without need of seeing them as subsets of some Euclidean space, and this provides a much cleaner theory.
Indeed, any smooth manifold can be smoothly embedded into Euclidean space, as proven by the Whitney embedding theorem (and other results if you impose additional structure, e.g. the Nash embedding theorem).
A: A fundamental application for which it is undue to consider a manifold as a subset of some Euclidean space is the whole field of Riemannian geometry (as you know), but also the whole field of Pseudoriemannian geometries, like Lorentzian Geometry.
An example is the hyperbolic model with the Minkowski metric. The Minkowski metric is a particular kind of Lorentzian metric, meaning a metric that is no more positive definite, and may have positive and negative eigenvalues. The Minkowski metric has three positive eigenvalues and one negative. This has enormously different properties than the standard rules of calculus in Euclidean spaces: for example for some vectors it even holds a reverse Cauchy-Schwartz inequality and reverse triangular inequality, in the opposite direction of the usual. This has lots of implications in special relativity, giving rise to the famous cone structure. If we considered an hyperboloid in the Euclidean space (meaning with the standard positive definite scalar product), then we would never be able to describe these physical phenomena.
