Definition of arc length on manifolds without parametrization

[gel]

and $\Vert f\Vert_{\infty}$ is the measure theory notation for the minimum real number K>0 such that K>|f| almost everywhere, also known as the essential supremum of |f|. Must be what he means.

 

[mma]

I'm very sorry for you lost your detailed post. Next time don't forget to copy&paste your text somewhere before pressing the preview or submit button. Thank you for repeating it shortly.

 

[mma]

It is shown that sup(||grad a_x||) = 1 (by local geodesic calculation)

I thought originally that this "local geodesic calculation" is trivial. But now I see that I don't really know how is it.

It is clear that taking a short geodesic $$\gamma(s)$$ starting from a(x) and parametrized by its arc length, then

$$d(x,y) = a_x(y) - a_x(x) = \int_0^{d(x,y)}\dot{\gamma}(a_x)|_{\gamma(s)} ds$$

and from this follows $$\dot{\gamma}(a_x) = 1$$.

But this means only that $$g(\mathrm{grad}(a_x), \dot{\gamma}) =1$$, and of course we know that $$\Vert\dot{\gamma}\Vert$$ = 1.

But how follows $$\Vert\mathrm{grad}(a_x)\Vert = 1$$ from this?

 

[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?

 

[mma]

It is clear that taking a short geodesic $$\gamma(s)$$ starting from a(x)

Of course strarting from x and not from a(x). Sorry for the mistyping.

 

[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)"