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fa.useful evaluation – Taylor serie on a Riemannian manifold

I necessity some ameliorate for the next downside.

Let $M$ a riemannian manifold and $f$ a flush differential duty, then deem the next integral $$int_M Gamma(x,y)(f(y)-f(x))dV_y$$

the place $dV_y$ is the touchstone on the manifold and $Gamma$ is a optimistic duty. Now, my query is how can do a taylor growth of the integral, i.e., for instance within the illustration of $M=mathbb R$, now we have with $y=x-epsilon z$, $$int_mathbb{R}Gamma(x,x-epsilon z)(f(x-epsilon z)-f(x))dy=int_mathbb{R}Gamma(x,x-epsilon z)(-epsilon z f'(x)+(epsilon z)^2f”(x)+…)(-epsilon dz)$$

But now if I’m on a Riemannian manifold, how can I do one thing love that? I do not know any taylor succession love that. Anyway, for the illustration of the manifold I attempted to occupy the exponential map and the polar coordinates.

Thanks any thought will breathe appreciated! Thanks!

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