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This is a crosspost from math.stackexchange Given a Riemannian manifold $M$, let $c(t) = \exp_p(tX)$ be the geodesic emanating from $p \in M$ with initial value $X$. Let $t_0$ be small enough, then w …

In Chern-Weil Theory, characteristic classes appear as certain closed differential forms associated to vector bundles with connections (constructed via the curvature form of the connection). When yo …

This is crosspost from math.stackexchange https://math.stackexchange.com/questions/311213/heat-kernel-asymptotics-on-manifold-with-boundary where the question did not yield any answer On a closed Rie …

Ad 1: Yes, there is. The formula is $$\nabla^*(X^\flat \otimes u) = - \nabla_X u -\mathrm{div}(X) \cdot u,$$ as can easily seen by local computation. Here, $X$ is a vector field and $X^\flat$ is the d …

A hands down proof not using the theory of Hamiltonian systems can be done by just proving that the Jacobian determinant of the transformation is zero. We have $$ TSM \cong \pi^* TM \oplus VSM,$$ wh …

Let $M$ be an $4m$-dimensional Riemannian manifold. We can then form the Pontryagin classes $p_k(TM)$ of the tangent bundle using Chern-Weil theory. For any sequence of numbers $k_1, \dots, k_l$ such …

Let $M$ be a compact Riemannian manifold and let $V \in C^\infty(M)$. Consider the operator $\Delta + V$ and let $p_t(x, y)$ be the corresponding heat kernel. If $\Delta + V$ is a positive operator su …

If $M$ is a spin manifold, then any submanifold of codimension 1 is also a spin manifold. This yields a lot of examples, for example, that $S^n$ is spin etc. (I may not have understood your point com …

This post is related to this and this post. It is known that on a complete Riemannian manifold, the space $C^\infty_c(M)$ is generally not dense in the Sobolev spaces $W^{k, p}(M)$ ($1 \leq p < \inft …

The cotangent bundle of a manifold has a canonical symplectic form and if we choose a riemannian metric on $M$, we can give it an almost complex structure. Is this structure integrable, and if it is …

Theorem. Let V be a $C^\infty$ function on a riemannian manifold $M$ and $p$ be a nondegenerate local minimum with $V(p)=0$. Then there is a unique positive function $\varphi \in C^\infty(U)$ such tha …

My stackexchange post was somewhat unsatisfactory (also because I may not have stated clear enough what my interest was). So here it goes! Let $M$ be a compact Riemannian manifold and $\Delta$ be the …

I am looking for references (with proof) for the following statement: Let $(M, g)$ be a Riemannian manifold with bounded curvature and let $p_t(x , y)$ be the heat kernel of $M$. Let $K$ be compac …

Consider on the circle $S^1$ the operator $$L := - \frac{\partial^2}{\partial \theta^2} + c$$ for some constant $c \in \mathbb{R}$. What is its $\zeta$-regularized determinant? This should be well- …

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 …