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It is well-known that for any Riemannian manifold $M$, the spaces $$H_x(M) = \Bigl\{ \gamma \in AC\big([0, 1], M\big) ~\Big|~ \gamma(0) = x, \int_0^T|\dot{\gamma}(s)|^2 \mathrm{d}s < \infty\Bigr\}$$ a …

If you have any manifold $M$ and a vector bundle $\mathcal{V}$ over $M$, then associated to any connection $\nabla$ on $\mathcal{V}$, there is indeed a dual connection $\nabla^*$ on the dual bundle $\ …

This is a crosspost from Stackexchange, where the question did not find any attention. This is probably due to the length of this post, but I did my best to organize it as well as I could. I am thankf …

Take $V = L^2(M)$ with its usual Hilbert space topology and $W = C^\infty(M)$. Take $X = \mathrm{span}\{f\}$ for some non-smooth function. Then $X \cap W = \{0\}$, hence the kinematic tangent space of …

Given a connection on the tangent bundle, you can define the second covariant derivative $\nabla^2f$ by $$ \nabla^2f[X, Y] := \partial_X\partial_Y f - \partial_{\nabla_X Y} f.$$ Then $\nabla^2f$ is a …

Let $M$ be a (compact, let's say) Riemannian manifold, $\mathcal{V}$ a vector bundle over $M$ with covariant derivative $\nabla$ and a fiber metric. Let $L = - \mathrm{tr}(\nabla^2) + V$ with some pot …

In their paper "Conformally flat Manifolds, Kleinian Groups and Scalar Curvature", Schoen and Yau repeatedly use the term "Newtonian capacity" for a subset of $S^n$. I know the following definition: …

Suppose I have a complete, non-compact Riemannian manifold $M$ such that the volume of balls around a fixed point $p \in M$ satisfies $$\mathrm{vol}(B_R(p)) \leq v(R)$$ for some function $v$. Can we t …

You just need to check that $$ \int_M H_k(x, y) \mathrm{d}y = 1 + O(t^{k+1})$$ for all $x \in M$ and for all $k$. For example, this follows directly from the method of stationary phase (just take a ge …

Roughly, the trick is not to view $L$ as an operator on $L^2$, but on $C^0$ I will use the following version of Krein-Rutmann which is proven in "Du, Yihong: Order Structure and Topological Methods i …

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 …

The famous maximum principle of Hamilton states the following. Let $C$ be a convex $O(n)$-invariant subset of the space of algebraic curvature operators. Then if it is invariant under the ODE $$ \dot{ …

In a paper that I was reading, I stumbled across the following theorem: Let $X$ be a vector field with $$X= > a^ix^i\partial_{x^i} + > \mathcal{O}(|x|^2),$$ where $x$ is some chart and $a^i>0$. …

Let $M$ be a Riemannian manifold, and let $x, y \in M$ be non-conjugate points. Let $r, R>0$ be two numbers. I am looking for a bound on the number of geodesics between $x$ and $y$ of Length between …

Let $M$ be a Riemannian manifold. Let us look at the Riemannian exponential function $\exp_x: T_x M \supset \mathcal{D} \longrightarrow M$. The derivative of the exponential map can be expressed in T …