I thought about the Taylor expansion on a Riemannian manifold and guess the Taylor expansion of ##f## around point ##x=x_0## on the Riemannian manifold ##(M,g)## should be something similar to:(adsbygoogle = window.adsbygoogle || []).push({});

[itex]f(x) = f(x_0) +(x^\mu - x_0^\mu) \partial_\mu f(x)|_{x=x_0} + \frac{1}{2} (x^\mu - x_0^\mu) (x^\nu - x_0^\nu) H_{\mu\nu}(f)|_{x=x_0}+ \dots[/itex]

here H is Hessian

[itex]H_{\mu\nu} (f)=\nabla_\mu\partial_\nu f=\partial_\mu\partial_\nu f-\Gamma_{\mu\nu}^\rho\partial_\rho f[/itex]

Is this generalization true? Unfortunately, I couldn't find any proper references.

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# Riemannin generalization of the Taylor expansion

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