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

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

which sends each $U \in CartSp$ 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 $\mathbf{H} =$ Smooth∞Grpd.

Write

$$Met(X) := Conc [X,Met]$$

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 $Aut(X)$ of automorphisms of $X$ acts on this by precomposition in the natural way

$$Aut(X) \times Met(X) \to Met(X)$$

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

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

The quotient (action groupoid, moduli stack)

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

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

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

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

Properties

For $X$ a smooth manifold, $Met(X)$ 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.

Related concepts

moduli spaces

• moduli space of complex structures

• moduli space of conformal structures

• moduli space of curves

• moduli space of Riemannian metrics

• moduli stack of elliptic curves

• moduli space of Calabi-Yau spaces

• moduli stack of tori

• moduli stack of formal groups

• moduli space of connections, moduli space of flat connections

• Hitchin moduli space

• moduli space of monopoles

• moduli space of framed manifolds

• moduli space of bundles

• moduli space of (higher) line bundles

• quiver variety

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)