It is an interesting and important fact that a contractible manifold (without boundary) is not necessarily homeomorphic to Euclidean space. This makes the classical Cartan-Hadamard theorem, stating that a contractible manifold equipped with a Riemannian metric of non-positive sectional curvature is diffeomorphic to Euclidean space, even more powerful.
One can ask now whether one can prove more general versions of the Cartan-Hadamard theorem by weakening the notion of non-positive curvature. A natural candidate here is the notion of curvature introduced by Alexandrov, and especially the notion of (globally) CAT(0) metric spaces. Note that the metric induced by a Riemannian metric of non-positive sectional curvature is CAT(0).
In the following we will call (topological) manifolds equipped with a CAT(0)-metric (which does not have to be induced by a Riemannian metric) CAT(0)-manifolds. Since any CAT(0) metric space is contractible, CAT(0)-manifolds are contractible. We ask now if there is a version of the Cartan-Hadamard theorem for CAT(0)-manifolds:
- By the classification of surfaces, any 2-dimensional CAT(0)-manifold is diffeomorphic to Euclidean space.
- Any 3-dimensional CAT(0)-manifold is also diffeomorphic to Euclidean space. References for this are given in the introduction of the preprint cited below.
- In dimensions 5 and higher this is wrong: Davis and Januszkiewicz constructed the first counter-examples.
The remaining case was the 4-dimensional one, and it was only recently resolved: In arXiv:2109.09438 it is proven by Lytchak, Nagano and Stadler that any 4-dimensional CAT(0)-manifold is indeed homeomorphic to Euclidean space.