nLab
moduli space of Riemannian metrics

Idea

The moduli space of (pseudo)-Riemannian metrics $g$ on a given space (manifold) $X$.

Definition

On the site CartSp of smooth Cartesian spaces consider the sheaf

$$\mathrm{Met}:{\mathrm{CartSp}}^{\mathrm{op}}\to \mathrm{Set}$$Met : CartSp^{op} \to Set

which sends each $U\in \mathrm{CartSp}$ to the set of Riemannian metrics on $U$.

Then let $X$ be a smooth manifold, or more generally a diffeological space, or more generally a Lie groupoid or more generally a smooth ∞-groupoid, all regarded in $H=$ Smooth∞Grpd.

Write

$$\mathrm{Met}(X):=\mathrm{Conc}[X,\mathrm{Met}]$$Met(X) := Conc [X,Met]

for the concretification of the internal hom: the space of metrics on $X$. (Various variations and extensions of this statement are of interest and can easily be written out. The above direct statement works for possibly degenerate metrics.)

A point in this space is a single (pseudo-)Riemannian metric on $X$.

The group $\mathrm{Aut}(X)$ of automorphisms of $X$ acts on this by precomposition in the natural way

$$\mathrm{Aut}(X)\times \mathrm{Met}(X)\to \mathrm{Met}(X)$$Aut(X) \times Met(X) \to Met(X)

$$((X\stackrel{\varphi }{\to }X),g)\mapsto {\varphi }^{*}g.$$((X \stackrel{\phi}{\to} X), g) \mapsto \phi^* g \,.

If $X$ is a smooth manifold then $\mathrm{Aut}(X)=\mathrm{Diff}(X)$ is the group of diffeomorphisms of $X$.

The quotient (action groupoid, moduli stack)

$$\mathrm{Met}(X)//\mathrm{Diff}(X)\in \mathrm{Smooth}\mathrm{\infty }\mathrm{Grpd}$$Met(X)//Diff(X) \in Smooth\infty Grpd

is the moduli space of (pseudo-)Riemannian metrics on $X$.

Various variations of this are of interest. For instance there one consider the Einstein-Hilbert action

$$S:\mathrm{Met}(X)//\mathrm{Diff}(X)\to ℝ.$$S : Met(X)//Diff(X) \to \mathbb{R} \,.

The critical locus of this function is the moduli space of Einstein metrics.

Properties

For $X$ a smooth manifold, $\mathrm{Met}(X)$ itself is a contractible space.

Applications

In the context of the theory of gravity, the moduli space of pseudo-Riemannian metrics on $X$ is the configuration space of the field theory of gravity (general relativity). The moduli space of Einstein metrics is then called the covariant phase space: this is the subspace of solutions of the Einstein equations.

References

Textbook references include

• Mikhail Gromov, Metric structures for Riemannian and non-Riemannian spaces Birkhäuser (1999)

chapter 4 of

• A. L. Besse, Einstein Manifolds , Ergeb. Math. Grenzgeb. 10, Springer-Verlag, New York, (1987) MR 88f:53087 Zbl 0613.53001

• M. E. Shanks, The Space of Metrics on a Compact Metrizable Space , American Journal of Mathematics Vol. 66, No. 3, Jul. (1944) (JSTOR)

• Halldor Eliasson, On variations of metrics Math. Scand. 29 (1971) 317-327 (pdf)

• F. Farrell, Pedro Ontaneda, The Teichmüller space of pinched negatively curved metrics on a hyperbolic manifold is not contractible (pdf)

• Boris Botvinnik, Bernhard Hanke, Thomas Schick, and Mark Walsh, Homotopy groups of the space of metrics of positive scalar curvature (pdf)