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I need some help for the following problem. Let $M$ a riemannian manifold and $f$ a smooth differential function, then consider the following integral $$\int_M \Gamma(x,y)(f(y)-f(x))dV_y$$ where $dV_y$...

This question pertains to Lemma 3.5 of this article. Let $M$ be a smooth Riemannian manifold and $\gamma$ some geodesic with respect to the Levi-Civita connection $\nabla$. For any $C^2$ vector field $...

Let $(M,g)$ be an $n$-dimensional Riemannian manifold. Let $p\in M$, and let $\{x^i\}_{i=1}^n$ be normal coordinates centered around $p$. Using Jacobi field, one can show that the metric $g$ has the ...

I have posted this question in mathSE a few weeks ago (and proposed a bounty) but so far got no response. In the book Riemanian geometry (by do-Carmo), the following result is proved (Corollary 2.9 ...