1
$\begingroup$

I'm trying to understand basic differential geometry (my background is in mathematical logic), and I'm having a bit of difficulty with a basic point. Frequently we want to consider the set of metrics on a given manifold modulo some equivalence relation (generally, isometry), but I can't visualize metrics very easily. Is there some example of a reasonably simple manifold, and two "very different" metrics on it, that would help me understand what spaces of possible metrics can look like?

Thanks in advance

  • 0
    One very simple example: the usual metric, $d(x,y)=|x-y|$, on $\mathbb{R}$, and the bounded metric $d_1(x,y)=\frac{d(x,y)}{1+d(x,y)}=\frac{|x-y|}{1+|x-y|}\;.$2012-02-18

0 Answers 0