# Why does curvature somewhat determine the topology?

I thought the topology of a manifold is defined before we define the tensor fields that live on it. A manifold can be defined by the atlas, and the atlas fixes the topology. The metric tensor is then added as an additional structure.

But apparently, the metric, or the connection, does somewhat determine the topology. For example, I read that if the curvature is positive everywhere, the topology has to be sphere-like.

But it's not possible for curvature to determine topology. It must be the other way around. It must be that the topology, defined by the atlas, places some restriction on the connection that can be defined on the manifold. Why exactly does this restriction get imposed? It should have something to do with the definition of connection.

I'd also like to know how the presence of singularities affect the answer.

Certainly the manifold is a topological space first of all (or alternatively, as you say, a set with an atlas which determines the topology). The metric tensor is an additional structure, and not at all unique. Connections on the tangent bundle are not unique either, but once a metric is fixed, there is a unique connection compatible with it in an appropriate sense, the Levi-Civita connection, whose existence is sometimes considered a miracle. When we talk about the curvature of a manifold, we mean the curvature of the Levi-Civita connection of its metric tensor, so the restriction is already on the metric.

The definition of a Riemannian (or Lorentzian) metric is pretty straightforward, so any restriction on the topology is implicit, rather than explicit, and since locally you can vary the metric as much as you want, as long as you do it in a differentiable way, the key to your question must be in how this can be done so that it remains globally differentiable (or even just continuous).

This gives a first clue as to why the topology restricts the metrics that can be supported, just like the topology restricts the continuous functions that can be supported; in other words, the existence of a metric with certain properties (like of its curvature) may say something about the topology. As a simple example around the existence of certain continuous functions, consider a manifold that admits a locally constant continuous function that is not constant. We can conclude that the manifold is not connected. If it admits an unbounded real-valued function, we conclude it cannot be compact.

Now let's go back to your specific questions. While the definition of curvature is quite abstract, it can be related in a very specific way to several more intuitive concepts, like geodesic deviation as mentioned in the comment by MBN or the relation between the volume of a geodesic ball and its radius. In all cases, the intuition is as follows: where the curvature is positive, geodesics that are initially parallel converge and the volume of a geodesic ball is smaller w.r.t. its radius than in the flat case, and where it is negative it is the other way around. The full Riemann curvature tensor makes these qualitative statements precise and computable.

To be very concrete, let's consider closed surfaces in 3-space. Picking a metric (hence a curvature) is the same as picking a specific embedding, changing it is just changing the embedding, i.e. streching and bending the surface. If you allow embeddings in higher dimensions, you actually can realize all possible metrics in this way by the Nash embedding theorem.

You can stretch out your surface to locally make it flat, or give it positive or negative curvature, but the global topology may prevent you from doing so everywhere simultaneously. For example, if we have a sphere in $$\mathbb R^3$$, we can flatten it, maybe most of it, but we will inevitably have some kind of an equator left where the curvature is higher the flatter you make the hemispheres. You could say that for this topology the curvature has nowhere to go. When you take a torus, you will have regions of positive and others of negative curvature. When going to 4-space you can actually make it flat everywhere, and likewise you can deform each closed surface so that it has constant curvature, but the amount of curvature depends strongly on the topology. This is quantified in the beautiful Gauss-Bonnet theorem:

$$\int_M KdA = 2\pi(2 - 2g)$$

where $$K$$ is the Gaussian curvature and $$g$$ is the number of holes or the genus. This says that however much you stretch or bend the surface, the average curvature will remain the same, if you decrease the curvature somewhere, it will increase elsewhere if you don't want to tear the surface.

When we allow singularities, assuming you mean by a singularity an isolated point at which the metric may not be well-defined, this changes a lot: now the curvature does have somewhere to go, namely to that point. For example, consider the sphere again, but remove the North pole: now it can be stretched out flat (by stereographic projection from the North pole for example). This defines a metric on it that is flat everywhere, except at the North pole, to which it cannot be extended smoothly.