Conditions for a given manifold to admit a given metric

Question

It is well-known that every smooth manifold admits a Riemannian metric and it is commonplace to study when certain manifolds admit metrics of some specific type (eg Kahler, Ricci-Flat...), but I am wondering if it is known (or a well-posed question) to ask when a specific manifold admits a specific metric. In other words, for a given manifold and its tangent bundle, is there a way to assign coordinate labels to points such that some metric, which may be a solution to a system of equations, will manifestly be admited by that manifold. I ask primarily in the context of general relativity when we solve for the metric via Einstein's Equations, but then just seem to arbitrarily set it on some manifold. It is not obvious to me anyway that whichever manifold is selected would admit the metric.

Answer 1

Let me convert my comments to an answer. I will work with Riemannian metrics since most of research is done in this setting. First of all, one needs to distinguish local existence problems (where the question is about existence of metrics satisfying some conditions on an open subset of $R^n$) from global existence problems where one can ask for some necessary or/and sufficient conditions for the existence of metrics with some properties on a class of smooth manifolds. Local problems tend to be much easier, one usually comes up with a simple example of a metric. For instance, if you ask me for the existence of a metric of constant curvature on a domain in $R^n$, I just will say "take the flat metric" (or, say, the hyperbolic metric if you insist on constant negative curvature). But sometimes even the local problem can be nontrivial, for instance, this was the case of the existence problem of Riemannian metrics with $G_2$ or $Spin(7)$ holonomy (take a look at this article). The first existence theorem was local, due to Robert Bryant

[1] "Metrics with Exceptional Holonomy", Annals of Mathematics, 1987.

This was done through a deep analysis of the systems of differential equations defining such metrics. It took until 1996 when Dominic Joyce constructed compact manifolds with $G_2$ holonomy.

Global existence problems are much harder. There are no general methods for solving problems of existence of metrics satisfying some geometric properties on a class of compact manifolds. The problems tend to present themselves in the form of a (frequently) overdetermined (if dimension of the manifold is high enough) system of nonlinear partial differential equations (PDEs) on a compact manifold. Typically, there are topological obstructions for the existence of such metric which have to be identified first.

Here is a very classical example: Which compact surfaces $S$ admit a Riemannian metric of negative curvature?

The oldest topological restriction comes in the form of the Gauss-Bonnet formula: $$ \int_S K dA= 2\pi \chi(S). $$ Here $K$ is the curvature function on $S$. Since $K$ is assumed to be negative, one gets an immediate topological restriction: $\chi(S)<0$. In the case of connected oriented surfaces, this means that the genus of the surface is $\ge 2$. Constructing such a metric on every surface of genus $\ge 2$ is not very difficult, but if you do not know such a construction, this is a nontrivial task. (One can even get a metric of constant negative curvature.) A deeper form of this existence problem/theorem comes in the

Uniformization Theorem for Riemann surfaces. Suppose that $(M,g)$ is a 2-dimensional Riemannian manifold. Then the metric $g$ is conformal (i.e. has the form $e^{u} g_0$ where $u$ is some smooth function on $M$) to a complete Riemannian metric $g_0$ of constant curvature.

Instead of choosing a background Riemannian metric $g$ one can fix a complex structure on $M$ (assuming that $M$ is oriented). Proving this theorem (which was finally done by Koebe) took several decades with several incomplete/incorrect proofs due to Klein and Poincare.

Encouraged by the success in dimension 2, you can ask about 3-dimensional manifolds:

Question. Which compact connected 3-dimensional manifolds admit metrics of negative curvature?

The answer depends on what you mean by "curvature". If you mean Ricci curvature then it turns out that every compact manifold of dimension $\ge 3$ admits a Riemannian metric of negative Ricci curvature. This is a very nontrivial theorem proved first in dimension 3 (by Robert Brooks using a combination of geometric and topological arguments) and then in all dimensions, this was proven by Joachim Lohcamp in

[2] "Metrics of Negative Ricci Curvature", Annals of Mathematics, 1994.

(You should not be surprised that both [1] and [2] appeared in Annals of Mathematics, this is widely regarded as the top mathematical journal.)

On the other hand, if you consider manifolds of positive Ricci curvature (or even scalar curvature), then this is still poorly understood (apart from dimension 3) and the existence/nonexistence results are proven by a variety of mathematical tools raging from topological (surgery theory) to hard nonlinear PDEs.

If you restrict to metrics of constant curvature of a particular sign (positive, zero, negative) then the problem is even harder. It is completely out of reach in dimensions $\ge 4$ while the 3-dimensional case was settled only recently, through the work of Gregory Perelman. The problem is known as a special and key case of Thurston's Geometrization Conjecture. The 3-dimensional Poincare Conjecture (which was open for about 100 years) was a very special case of Perelman's theorem. In contrast to [1] and [2], Perelman did not publish his work in Annals of Math, actually, he did not publish it at all. But in 2006 he was awarded for his work the Fields Medal (which is one of the most prestigious awards in mathematics), which he declined to accept. But this is another story.

Perelman's work also describes precisely which compact 3-dimensional manifolds admit Einstein metrics. For instance, for Einstein metrics with positive constant, these are precisely the manifolds with finite fundamental groups. Just to indicate the time frame: Thurston proposed the conjecture in 1970s and he proved some important special cases; a path to the solution was proposed by Hamilton in 1982 (who proved the conjecture in some important special cases); the problem was finally solved by Perelman in 2002-2003.