synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality $\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality $\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
The moduli space of (pseudo)-Riemannian metrics $g$ on a given space (manifold) $X$.
On the site CartSp of smooth Cartesian spaces consider the sheaf
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
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
If $X$ is a smooth manifold then $Aut(X) = Diff(X)$ is the group of diffeomorphisms of $X$.
The quotient (action groupoid, moduli stack)
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
The critical locus of this function is the moduli space of Einstein metrics.
For $X$ a smooth manifold, $Met(X)$ itself is a contractible space.
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
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.