0

Consider a smooth manifold. Its atlas uniquely specifies the manifold topology. Now suppose we want to define a (pseudo-)Riemannian metric tensor on such manifold. Are there any constraints on the properties of the metric tensor, coming from the underlying manifold topology?

As an example, consider a smooth manifold with manifold topology $S^2\times R^2$. Am I allowed to specify ANY (pseudo-)Riemannian metric on it?

Didier
  • 19,132
Frank
  • 63
  • 1
    What do you mean with "allowed to specify"? If you can write it down then it exists. But for example you cannot have a Lorentz metric (pseudo-Riemannian metric of signature $(1,n-1)$) on $S^2$ or on any other compact manifold with Euler characteristic $\neq0$. – s.harp Dec 27 '20 at 10:13
  • I use "allowed to specify" in the following sense: I am not allowed to specify a Lorentzian metric on $S^2$. Thanks for the answer! Can you point me to any references where I can study these topics? – Frank Dec 27 '20 at 10:38
  • $S^2\times R^2$ admits pseudo-Riemannian metrics of all possible signatures: (4.0), (3,1), (2,2). And $S^2$ does not admit a metric of signature (1,1), see here. – Moishe Kohan Dec 27 '20 at 22:16
  • Thanks! Is the manifold topology related to the direct sum decomposition of the tangent bundle? – Frank Dec 28 '20 at 07:46
  • @Frank: Of course. – Moishe Kohan Dec 28 '20 at 16:41
  • Can you please point me to a reference where this is explained? – Frank Dec 29 '20 at 11:28

1 Answers1

3

There is a huge interaction between the Riemannian metric of a manifold and its topology. This interaction appears through its curvature.

A first example is the well-known Gauss-Bonnet formula. Given a Riemannian surface $(\Sigma,g)$, denote by $K$ its Gaussian curvature. Then:

$$\int_\Sigma K = 2\pi\chi(\Sigma)$$ Thus, the metric tensor is forced to have a global behaviour controlled by the topology of the surface. As an application, there does not exist any metric with negative curvature on the sphere or on the torus.

Another well-known interaction is the following. If $(M,g)$ is a Riemannian manifold with non-positive sectionnal curvature, then its universal cover is diffeomorphic to $\mathbb{R}^n$. This implies that a $n$ dimensional sphere does not have any metric of non-positive sectionnal curvature.

A last interaction is more subtle: the bahaviour of the curvature affects the aglebraic properties of the fundamental group of a compact manifold. On some positive assumptions on the curvature, the fundamental group has to have polynomial growth, while with some negative assumptions, it has at least exponential growth.

Note that every manifold can be endowed a Riemannian metric. The case of pseudo-Riemannian metrics is not the same because one cannot glue local pseudo-Riemannian metrics with partition of unity and expect the result to be a pseudo-Riemannian metric. For example, this theorem shows that the Lorentzian case is restricted to a few examples: every noncompact connected smooth manifold admits a Lorentz metric, and a compact connected smooth manifold admits a Lorentz metric if and only if its Euler characteristic is zero.

In your example, the manifold $S^2\times\mathbb{R}^2$ is simply connected and is not diffeomorphic to $\mathbb{R}^4$. Hence, it does not admit a metric with negative curvature.

Didier
  • 19,132
  • Thank you for the answer! I am specifically interested in how the manifold topology affects the possible (pseudo-Riemannian) metrics we can define (not the other way around, for now), and the Gauss-Bonnet formula is one example of that. Do you know any references that I can study that treat such topics, both for Riemannian and pseudo-Riemannian metrics? – Frank Dec 27 '20 at 10:35
  • 1
    There is no "other way around". All properties I stated show that topology affects the possible metrics: they are obstruction to the existence of some particular metrics. For example, a compact manifold whose fundamental group has polynomial growth does not admit a metric with sectionnal curvature. There is a section in Riemannian geometry, Gallot, Hulin, Lafontaine, that is dedicated to shuch interactions. – Didier Dec 27 '20 at 10:38
  • Thank you for the clarification, and the reference! – Frank Dec 27 '20 at 10:39
  • Hi again, I was only able to find results for compact (pseudo-)Riemannian manifolds, or for 2-dimensional manifolds. Do you know any references that treats these issues for paracompact (pseudo-)Riemannian manifolds, possibly of dimension up to 4? – Frank Dec 27 '20 at 14:09