0
$\begingroup$

Suppose X,g is a complete negative sectional curvature Riemannian manifold. And $C \subset X$ is compact submanifold. What are the minimal conditions on $C$ so that we can deform g on the restriction to $X-C$ to a complete negatively curved metric on $X-C$.

The comments give one important obstruction, in that $C$ should have codimension 2.

Improve this question
$\endgroup$
9
  • 5
    $\begingroup$ If $X$ is the hyperbolic $n$-space with $n>2$ and $C$ is a point, then then $X-C$ is not aspherical, so has no complete negatively curved metric (by Cartan-Hadamard). $\endgroup$
    – Igor Belegradek
    Commented May 25 at 11:52
  • $\begingroup$ You should assume that $C$ has codimension 2. $\endgroup$
    – Moishe Kohan
    Commented May 25 at 13:19
  • 1
    $\begingroup$ Even if $C$ has codimension 2, I am not aware of any such deformation. $\endgroup$
    – Igor Belegradek
    Commented May 25 at 13:29
  • 2
    $\begingroup$ @MoisheKohan: the OP makes no assumption about $C$. Say, what if $C$ is a codimension 2 sphere? If the curvature is constant near $C$, and $C$ is totally geodesic of codimension 2, the deformation exists and is quite straightforward. There are similar (less straightforward) results when $X$ is locally symmetric and $C$ is totally geodesic of codimension 2; in this case the normal bundle of $C$ won't be trivial, but this is irrelevant. For the most difficult case see arxiv.org/abs/1609.01974 which has references to easier cases. $\endgroup$
    – Igor Belegradek
    Commented May 25 at 13:53
  • 1
    $\begingroup$ Please edit your question so that it doesmake sense. $\endgroup$
    – Deane Yang
    Commented May 25 at 19:23

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .