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In coordinates, the Laplace-Beltrami operator on a Riemannian manifold $(M,g)$ can be written as: $$ \Delta_g = g^{ij}\partial_{ij} - g^{jk}\Gamma^\ell_{jk}\partial_\ell $$ The second term: $$ \mu^\...

first of all, I am not sure if this question fits here. I asked this question on math.stackexchange also but didn't get an answer so far. In Isaac Chavel's book Eigenvalues in Riemannian Geometry, ...

In $\mathbb{R}^n$, if $\gamma$ is a line segment between $x_0 = \gamma (0)$ and $x = \gamma (t)$, one has the following formula: $$\frac {\mathbb{e}^{- \frac{1}{4} \int_0^t <\dot{\gamma}, \dot{\...

Consider a smooth $n$-dimensional Riemannian manifold $M$ with sufficiently nice geometric and topological properties such that there exist a unique heat kernel $(t,x,y) \mapsto p(t,x,y)$ on it ($t>...

Let (M,g) be a complete Riemannian manifold. It is well known that the Laplace operator is essentially self-adjoint on $C^\infty_c(M)$. This extends to the (de Rahm) Laplace operator on forms. Thus in ...