Taylor series on a Riemannian manifold

Question

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 $dV_y$ is the measure on the manifold and $\Gamma$ is a positive function. Now, my question is how can do a taylor expansion of the integral, i.e., for example in the case of $M=\mathbb R$, 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 something like that? I don't know any taylor series like that. Anyway, for the case of the manifold I tried to occupy the exponential map and the polar coordinates.

Thanks any idea will be appreciated! Thanks!

Answer 1

If you fix $x\in M$, you can set up a Riemann normal coordinate chart $(z^i)$ centered at $x$. Then you can just do the usual Taylor series expansion in $z$. Semi-globally, in a normal convex neighborhood of the diagonal $\Delta \subset \mathcal{U} \subset M\times M$, you can define the geodesic distance function $r(x,y)$ between $x,y \in \mathcal{U}$. The function $\sigma(x,y) = \frac{1}{2} r(x,y)^2$ is sometimes called (Synge's) world function. It's gradient in the $y$ variable $\sigma^j(x,y) = g^{jk}(y) \nabla_k \sigma(x,y)$ is a good substitute for the normal coordinate difference $z^j = y^j - x^j$ and is defined on the whole neighborhood $\mathcal{U}$ of the diagonal $\Delta$. A Taylor expansion of the form $$ f(y) = f(x) + f_i(x) \sigma^i(x,y) + \frac{1}{2} f_{ij}(x) \sigma^i(x,y) \sigma^j(x,y) + \cdots $$ is sometimes called the covariant Taylor series for $f$. Similar expansions can be done also for tensor fields, not just scalars.

Covariant Taylor series have been extensively used for some specific applications in theoretical and mathematical physics (see for instance the PhD thesis of Avramidi from 1986).