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This has nothing to do with Lie groups, I believe. Let $M$ be a Riemannian manifold. The Bochner formula on $1$-forms states that $$\nabla^* \nabla \omega = (d \delta + \delta d)\omega - \mathrm{Ric}\ …

Here is a somewhat different answer: Smooth functions always admit asymptotic expansions in each point, convergent or not. One difference between analytic functions and smooth functions is, of course …

I learned that if we are given a $C^0$ Riemannian metric on a smooth manifold $M$, geodesics (i.e. length minimizing curves) are absolutely continuous, and if the metrics is $C^{0,\alpha}$, then the g …

In Hamilton's article on the Nash-Moser Theorem, he gives the map that maps a vector field $X$ to its flow $e^{tX}$ in $\mathrm{Diff}(M)$ as an example where the implicit function theorem in Frechet s …

The Wikipedia article (http://en.wikipedia.org/wiki/Mathieu_function) is quite extensive, I believe, and in general there is NO way to explicitly calculate either eigenvalues or eigenfunctions of the …

All the answers here rely on measure-theoretic ideas to give a negative answer. I feel that this does not exactly meet the point, instead I would say: You did not quite ask the right question, but if …

Ok, thank you Misha for the comments, let me try to fill out the hints you gave myself. I try to prove the following: Let $g_n$ be a sequence of complete smooth metrics that converge in $C^0$ agains …

Let $X$ be a solution to the boundary value problem $$ X^{\prime\prime}(s) = A(s)X(s), \quad X(0) = X_0, ~~X(t) = X_1,$$ where $0 < t \leq 1$, $A$ is some matrix-valued function, defined on $[0, 1]$, …

Recall that a function $f: \mathbb{R}^d \longrightarrow \mathbb{C}$ is positive definite, iff for all numbers $N$ and $x_1, \dots, x_N \in \mathbb{R}^d$, the matrix $(a_{ij})$ with entries $$a_{ij} = …