MathOverflow

Let $M$ be a manifold. Let $G$ be a Lie group and $\mathfrak{g}$ be its Lie algebra.
Let $P(M,G)$ be a principal bundle. Recall that, a connection on $P(M,G)$ is a distribution $\mathcal{H}\subseteq TP$ satisfying certain conditions. Equivalently, a $\mathfrak{g}$-valued differential forms on $P$ satisfying certain conditions....

Let $T$ and $T'$ be triangles in the hyperbolic plane $\mathbb{H}^2$, denote by $A, B, C$ and$A', B', C'$ their vertices respectively. Let $f : T \to T'$ be the unique "barycentric interpolation" that maps $A \mapsto A'$, $B \mapsto B'$, $C \mapsto C...

I am working on a problem-solving frameworkbased on gradient flow dynamics and haveproved the following result:
Given G: S → R≥0 satisfying:
Lower semicontinuity

Let $M$ be a smooth (Hausdorff, paracompact) and connected $n$-dimensional manifold, $n\in\mathbb{N}$. Given $p\in M$ and $g$ a smooth Riemannian metric on $M$, let $V^g_p\! M\subset T_pM$ be the maximal domain of the exponential map $\exp^g_p$ around...