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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.

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    $\begingroup$ 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. $\endgroup$
    – Igor Belegradek
    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. $\endgroup$
    – Igor Belegradek
    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

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

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