Give a smooth Riemannian manifold $(M,g)$, (i) how can one compute Taylor expansion of the square of the Riemannian distance function $d^2(x,x_0)$ at $(x',x_0)$?
I've tried to use $dist=\int (g_{\mu\nu} \frac{dx^\mu}{dt} \frac{dx^\nu}{dt} )^{1/2} dt$ to expand $dist^2$, but it was so messy. BTW, I've found that a similar question has been asked at mathoverflow.net. The answer given is actually very nice but it goes a little too technical and quite cryptic to me... In particular, the final result seems to a generalised Cosine law, right?!
(ii) Doesn't it make a sense to write $d^2(x,x_0)=d^2(x'+(x-x'),x_0)$ and take it as a scalar function?! Under what conditions can one assume that the square of the distance function is a polynomial?
I would appreciate answers not demanding an all profound background...