Search Results

If I understand you right, you're assuming that there is already a Riemann metric chosen for you. This of course integrates to a distance function ("metric" in the sense of metric spaces) whether or …

Here's an irresponsible under-referenced response. On the one hand, this is hard! Indeed, I've read somewhere --- not handy --- that this question lurks among quantum gravity's many difficulties. Fo …

Depending (*) on the underlying degree of analyticity (**) in your calculus course, it might be just as well to start with the Stokes theorem, stating it as an existence and uniqueness theorem: Th …

As it happens, the klein hyperbolic models, the usual euclidean plane, and central projection for projective space all have geodesics that look like straight lines; on the other hand, these maps can't …

Notation: $\mathbf{R}^4$ is a smooth manifold with underlying topology $(\mathbb{R})^4$; ${\mathbb{R}}^4$ is the standard smooth structure. The two things I know best about $\mathbf{R}^4$ is that it …

This is probably making a hash of the earlier answers, but bundles are special fibrations; specifically, they are fibrations with (not canonically) isomorphic fibers. And we all like fibrations, righ …