0
$\begingroup$

Assume $g$ and $h$ are smooth Riemannian metrics on a manifold $X$ with $d_g$ and $d_h$ the induced distance functions on $X$ (via infimum of length of curves in X connecting two points).

Assume $A$ is a closed subset of $X$ such that for some constant $C>1$ one has $(1/C)\cdot g\leq h\leq C \cdot h$ on $A$. Is it then true that $(1/C')d_g\leq d_h \leq C' d_g$ on $A$, for some constant $C'>1$ ? If not, is it at least true if $A$ is compact? The problem is of course that curves connecting points in $A$ may not entirely lie in $A$.

Improve this question
$\endgroup$
2
  • $\begingroup$ If A is compact, it's trivially true. If A is not compact, and by "metric function" you mean induced metric and not inner one, it's not true; consider a plane with one metric being flat, and other obtained by multiplying flat metric by something like $\frac{cosh(x)}{cosh(y)}$, and take as $A$ small nbhd of x axis. $\endgroup$
    – Denis T
    Commented Jun 2, 2023 at 23:17
  • $\begingroup$ Thanks Denis. What about the contrary? Can one conclude from $d_g\sim d_h$ on $A$ that $g\sim h$ on $A$, if $A$ is nice enough? $\endgroup$
    – Mike_Bool
    Commented Jun 3, 2023 at 9:56

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .