# nLab pseudo-Riemannian metric

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

tangent cohesion

differential cohesion

singular cohesion

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

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Idea

The notion of pseudo-Riemannian metric is a slight variant of that of Riemannian metric.

Where a Riemannian metric is governed by a positive-definite bilinear form, a Pseudo-Riemannian metric is governed by an indefinite bilinear form.

This is equivalently the Cartan geometry modeled on the inclusion of a Lorentz group into a Poincaré group.

## Definition

A pseudo-Riemannian “metric” is a nondegenerate quadratic form on a real vector space $\mathbb{R}^n$. A Riemannian metric is a positive-definite quadratic form on a real vector space. The data of such a quadratic form may be equivalently given by a nondegenerate symmetric bilinear pairing $\langle\, , \, \rangle$ on $\mathbb{R}^n$.

A pseudo-Riemannian metric

$Q: \mathbb{R}^n \to \mathbb{R}$

can always be diagonalized: there exists a basis $e_1, \ldots, e_n$ such that

$Q(\sum_{1 \leq i \leq n} x_i e_i) = x_1^2 + \ldots + x_p^2 - x_{p+1}^2 - \ldots - x_n^2$

where the pair $(p, n-p)$ is called the signature of the form $Q$. Pseudo-Riemannian metrics on $\mathbb{R}^n$ are classified by their signatures; thus we have a standard metric of signature $(p, n-p)$ where $\{e_1, \ldots, e_n\}$ is the standard basis of $\mathbb{R}^n$.

More generally, there is a notion of pseudo-Riemannian manifold (of type $(p, n-p)$, which is an $n$-dimensional manifold $M$ equipped with a global section

$\sigma: M \to S^2(T^* M)$

of the bundle of symmetric bilinear forms over $M$, such that each $\sigma(x)$ is a nondegenerate form on the tangent space $T_x(M)$.

Certain theorems of Riemannian geometry carry over to the more general pseudo-Riemannian setting; for example, pseudo-Riemannian manifolds admit Levi-Civita connections, or in other words a unique notion of covariant differentiation of vector fields

$\nabla: (X, Y) \mapsto \nabla_X(Y)$

such that

$X \cdot \langle Y, Z\rangle = \langle \nabla_X Y, Z\rangle + \langle Y, \nabla_X Z\rangle$
$[X, Y] = \nabla_X Y - \nabla_Y X$

In that case, one may define a notion of geodesic in pseudo-Riemannian manifolds $M$, and we have a notion of “distance squared” between the endpoints along any geodesic path $\alpha: [0, 1] \to M$ (which might be a negative number of course). The term “pseudo-Riemannian metric” may refer to such distances in general pseudo-Riemannian manifolds. (I guess.)

A typical example of pseudo-Riemannian manifold is a Lorentzian manifold, where the metric is of type $(1, n-1)$. This is particularly so in the case $n = 4$, where such manifolds are the mathematical backdrop for studying general relativity and cosmological models.

• The terminology “metric” is not optimal of course: the values of the quadratic form would need to be nonnegative to avoid terminological conflict with metric as it is more commonly understood (and even in that case, the values of “metric” refer to the square of the metric rather than the metric itself). Caveat lector.
geometric contextgauge groupstabilizer subgrouplocal model spacelocal geometryglobal geometrydifferential cohomologyfirst order formulation of gravity
differential geometryLie group/algebraic group $G$subgroup (monomorphism) $H \hookrightarrow G$quotient (“coset space”) $G/H$Klein geometryCartan geometryCartan connection
examplesEuclidean group $Iso(d)$rotation group $O(d)$Cartesian space $\mathbb{R}^d$Euclidean geometryRiemannian geometryaffine connectionEuclidean gravity
Poincaré group $Iso(d-1,1)$Lorentz group $O(d-1,1)$Minkowski spacetime $\mathbb{R}^{d-1,1}$Lorentzian geometrypseudo-Riemannian geometryspin connectionEinstein gravity
anti de Sitter group $O(d-1,2)$$O(d-1,1)$anti de Sitter spacetime $AdS^d$AdS gravity
de Sitter group $O(d,1)$$O(d-1,1)$de Sitter spacetime $dS^d$deSitter gravity
linear algebraic groupparabolic subgroup/Borel subgroupflag varietyparabolic geometry
conformal group $O(d,t+1)$conformal parabolic subgroupMöbius space $S^{d,t}$conformal geometryconformal connectionconformal gravity
supergeometrysuper Lie group $G$subgroup (monomorphism) $H \hookrightarrow G$quotient (“coset space”) $G/H$super Klein geometrysuper Cartan geometryCartan superconnection
examplessuper Poincaré groupspin groupsuper Minkowski spacetime $\mathbb{R}^{d-1,1\vert N}$Lorentzian supergeometrysupergeometrysuperconnectionsupergravity
super anti de Sitter groupsuper anti de Sitter spacetime
higher differential geometrysmooth 2-group $G$2-monomorphism $H \to G$homotopy quotient $G//H$Klein 2-geometryCartan 2-geometry
cohesive ∞-group∞-monomorphism (i.e. any homomorphism) $H \to G$homotopy quotient $G//H$ of ∞-actionhigher Klein geometryhigher Cartan geometryhigher Cartan connection
examplesextended super Minkowski spacetimeextended supergeometryhigher supergravity: type II, heterotic, 11d

Last revised on February 27, 2015 at 14:57:03. See the history of this page for a list of all contributions to it.