Tagged Questions
4
votes
1answer
53 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 ...
24
votes
5answers
1k 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 ...
23
votes
3answers
1k 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
109 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 ...
