nLab
moduli space of Riemannian metrics

Context

Riemannian geometry

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

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

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • tangent cohesion

    • differential cohomology diagram
    • differential cohesion

      • (reduction modality \dashv infinitesimal shape modality \dashv infinitesimal flat modality)

        (&)(\Re \dashv \Im \dashv \&)

      • graded differential cohesion

        • fermionic modality\dashv bosonic modality \dashv rheonomy modality

          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous *

          \array{ && id &\dashv& id \ && \vee && \vee \ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \ && \bot && \bot \ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \ && \vee && \vee \ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \ && \bot && \bot \ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \ && \vee && \vee \ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \ && \bot && \bot \ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \ && \vee && \vee \ && \emptyset &\dashv& \ast }

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          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 gg on a given space (manifold) XX.

          Definition

          On the site CartSp of smooth Cartesian spaces consider the sheaf

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

          which sends each UCartSpU \in CartSp to the set of Riemannian metrics on UU.

          Then let XX 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=\mathbf{H} = Smooth∞Grpd.

          Write

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

          for the concretification of the internal hom: the space of metrics on XX. (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 XX.

          The group Aut(X)Aut(X) of automorphisms of XX acts on this by precomposition in the natural way

          Aut(X)×Met(X)Met(X) Aut(X) \times Met(X) \to Met(X)
          ((XϕX),g)ϕ *g. ((X \stackrel{\phi}{\to} X), g) \mapsto \phi^* g \,.

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

          The quotient (action groupoid, moduli stack)

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

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

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

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

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

          Properties

          For XX a smooth manifold, Met(X)Met(X) itself is a contractible space.

          Applications

          In the context of the theory of gravity, the moduli space of pseudo-Riemannian metrics on XX 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)

          Last revised on February 25, 2015 at 22:49:24. See the history of this page for a list of all contributions to it.