Definition of arc length on manifolds without parametrization

[mma]

I suspect that $$\mathrm{grad}(a_x) = \dot{\gamma}$$.

Here $$a_x(y) := d(x,y)$$, the distance between x and y (I forgot to mention this in my previous post), and $$\gamma$$ is a geodesic through x, parametrized by its arc length)

So, my question is: how can I prove this?

Because the level sets of the distance function are perpedicular to the gradient vector of it, and these level sets are n-1-dimensional submanifolds, it would be enough to prove that the geodesics passing through x are always perpedicular to the level sets of the $$d(x,y)$$ distance function. Could anybody prove this?

 

[mma]

it would be enough to prove that the geodesics passing through x are always perpedicular to the level sets of the $$d(x,y)$$ distance function.

Bingo! It's http://en.wikipedia.org/wiki/Gauss%27s_lemma_(Riemannian_geometry)"