2
$\begingroup$

Let $(M, g)$ be an $n$-dimensional $C^k$ (or $C^\infty$) Riemannian manifold. On $M$ we can define metric $d_g$ as the infimum of lengths of curves that connect given two points. Fix $x \in M$ and $r>0$.

What I would like to know is, for a given choice of $r>0$, how close can $\phi \colon B_{d_g}(x, r) \to \mathbb{R}^n$ from a $C^k$ (or $C^\infty$) atlas be to an isometry on image. (Here, $B_{d_g}(x, r)$ is the ball with centre at $x$ and radius $r$). More precisely, I would like to know if there are papers that compare $d_g(y, z)$ and $d_e(\phi(y), \phi(z))$ (where $d_e$ is the Euclidean metric), where $y, z \in B_{d_g}(x, r)$, $r > 0$ is small, and $\phi$ is chosen based on $r$.

Improve this question
$\endgroup$
2
  • 1
    $\begingroup$ It suffices to compare the Riemann metric tensor of the Euclidean ball to the Riemannian metric on $B_{d_g}(x,r).$ You get the best possible result by using either almost linear coordinates or harmonic coordinates. There are papers by Stefan Peters, Greene-Wu, and Anderson-Cheeger on this. $\endgroup$
    – Deane Yang
    Commented Mar 3 at 5:48
  • 2
    $\begingroup$ Riemann normal coordinates $\endgroup$
    – Igor Khavkine
    Commented Mar 3 at 9:26

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .