# nLab moduli space of Riemannian metrics

Contents

### Context

#### Riemannian geometry

Riemannian geometry

## Applications

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## 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 $X$.

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

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.

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)