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On a Riemannian manifold $(M,g)$, let $F(s)=\int_{\partial B_s(x_0)}udS$ where

  • $u$ is a smooth function,
  • ${\partial B_s(x_0)}$ is the (geodesical) sphere of center $x_0$ and radius $r$
  • dS is the standard (hyper)surface measure (induced by $g$ on submanifold)

My question is how to explicitly compute, for small $s$, the derivative $F^\prime(s) $.

I would really appreciate if someone can point to me a precise reference.

PS: It should answer the same question with ball instead of sphere (dv instead of dS).

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In the article "The volume of a small geodesic ball of a Riemannian manifold" Alfred Gray gives explicit power series expansions for the volume of the geodesic ball and the sphere in an analytic Riemannian manifold, see Theorem 3.1.

This should be enough to find the derivatives you seek.

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Thanks for this useful reference. I don't have time now; but it seems that included a addiotnnal function u should not be problematic. – aximab Apr 4 '13 at 15:19

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