# Examples of product negatively curved Riemannian manifolds

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.

• 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. Commented May 28 at 19:08
• 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. Commented May 28 at 21:08
• Cool! Do you have a reference or would you like to expand it as an answer please? Commented May 28 at 22:18

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 $$T^k\times T^{n-k}\!\times_{e^t}\!\mathbb R^m$$.