4
votes
1answer
117 views

Is there a field of 'real analytic geometry'?

I am wondering whether there is a field of 'real analytic geometry', and if not, why not? There are branches of geometry corresponding to increasingly large sets of functions: polynomial (algebraic ...
10
votes
2answers
167 views

Are closed geodesics the prime numbers of Riemannian manifolds?

I wonder to what extent one can support the analogy that primitive closed geodesics are the prime numbers of Riemannian manifolds? ("Primitive": traced once, as opposed to $m$-fold for $m \ge 2$.) In ...
4
votes
1answer
112 views

covariant derivative vs. exterior derivative

I have the following question. Let $M$ be a Riemannian manifold with metric $g$ and $\nabla$ the Levi-Civita connection. Let furthermore $\alpha \in \Omega^{k}(M)$ be a $k$-form such that $\nabla ...
43
votes
5answers
3k views

Exterior Derivative vs. Covariant Derivative vs. Lie Derivative

In differential geometry, there are several notions of differentiation, namely: Exterior Derivative, $d$ Covariant Derivative/Connection, $\nabla$ Lie Derivative, $\mathcal{L}$. I have listed them ...
25
votes
3answers
2k views

Why are smooth manifolds defined to be paracompact?

The way I understand things, roughly speaking, the importance of smooth manifolds is that they form the category of topological spaces on which we can do calculus. The definition of smooth manifolds ...
2
votes
1answer
125 views

Bounding projective spaces

For which $n$ does there exist a (topological, smooth, PL, complex) manifold $M^n$ such that $\partial M = \mathbb{R}\mathbb{P}^m$. Obvously, $m = n -1 $ (at least an in the real case). There are a ...