1
$\begingroup$

It is a well known theorem of Anderson that any vector bundle over a negative sectional curvature Riemannian manifold admits a metric of negative sectional curvature.

Q: Are there examples of complete negative sectional curvature manifolds of the form $X \times Y$, for $Y$ closed and $X$ non-compact and not simply connected?

It should also be noted that X being non-compact is essential, as a theorem of Preissmann precludes negative sectional curvature metrics on compact products.

$\endgroup$
3
  • 1
    $\begingroup$ The condition "negative sectional curvature" is difficult to work with for open manifolds. Pinched negative curvature is a much easier condition, e.g. in a discrete isometry group of a negatively pinched Hadamard manifold the centralizer of any infinite order element is virtually nilpotent, so your $X\times Y$ cannot be negatively pinched. The best result I know is that if $X\times Y$ is negatively curved, the deck group of its universal cover fixes a point at infinity, see Corollary 11.6 in arxiv.org/pdf/1306.1256, which is a survey of open negatively curved manifolds. $\endgroup$ Commented May 28 at 19:08
  • $\begingroup$ Actually I take it back. There are examples where say $Y$ is a torus and $X$ is the product of a torus and a Euclidean space. Then $X\times Y$ has a complete metric of constant negative curvature. What one does not have is examples where $Y$ is closed negatively curved. $\endgroup$ Commented May 28 at 21:08
  • $\begingroup$ Cool! Do you have a reference or would you like to expand it as an answer please? $\endgroup$
    – Yasha
    Commented May 28 at 22:18

1 Answer 1

3
$\begingroup$

To expand on my second comment: Consider the warped product $T^n\!\times_{e^t}\!\mathbb R$, where $T^n$ is the Riemannian product of $n$ circles. The warped product is a complete Riemannian manifold of constant sectional curvature $-1$. Then $T^n\!\times_{e^t}\!\mathbb R$ is diffeomorphic to $T^k\times T^{n-k}\!\times_{e^t}\!\mathbb R$.

Also if $N$ is a manifold of constant curvature $-1$, then so is the warped product $N\times_{\sinh t}\mathbb R$. In this way one can build hyperbolic manifolds diffeomorphic to $T^k\times T^{n-k}\!\times_{e^t}\!\mathbb R^m$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.