Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such ...

learn more… | top users | synonyms (1)

1
vote
0answers
34 views
+50

Can you determine the length of a curve by the lengths of its projections onto planes?

If $\Gamma \subset \mathbb R^n$ is 1-rectifiable, then its Hausdorff measure is equal to its integralgeometric measure. That is, $$\mathcal H^1(\Gamma) = \int\limits_{G(1,\mathbb R^n)} \int\limits_K ...
11
votes
0answers
95 views
+100

Why is a PDE a submanifold (and not just a subset)?

I struggle a bit with understanding the idea behind the definition of a PDE on a fibred manifold. Let $\pi: E \to M$ be a smooth locally trivial fibre bundle. In Gromovs words a partial differential ...