# nLab Lorentzian geometry

Contents

### Context

#### Riemannian geometry

Riemannian geometry

# Contents

## Idea

Lorentzian geometry is the geometry of Minkowski spacetime, hence essentially of a Euclidean space, but equipped not with the standard Euclidean Riemannian metric of spacetime signature $(+,+,+,\ldots,+)$ (which yields Euclidean geometry) but with the pseudo-Riemannian metric of signature $(-,+,+,\ldots,+)$. This is in particular the context of the theory of physics called “theory of special relativity”, and it is locally the context of the “theory of general relativity”.

Note that while Lorentzian geometry is analogous to Euclidean geometry (as Minkowski space is analogous to a Euclidean space), a Lorentzian manifold is analogous to a Riemannian manifold. Thus, one might use ‘Lorentzian geometry’ analogously to Riemannian geometry (and insist on Minkowski geometry for our topic here), but usually one skips all the way to pseudo-Riemannian geometry (which studies pseudo-Riemannian manifolds, including both Riemannian and Lorentzian manifolds).

## Lorentzian Cartan geometry and first order gravity

Since the isometry group of Minkowski spacetime is the Poincaré group, and since Minkowski spacetime is the quotient of the Poincaré group by the Lorentz group-subgroup, Lorentzian geometry and the study of Lorentzian manifolds is to a large extent the Cartan geometry of the Poincaré group.

Promoting this perspective from global to local symmetry yields the first order formulation of gravity. Promoting this perspective form the Poincaré group to the super Poincaré group yields supergravity. Promoting it further to the Lie n-algebra extensions of the super Poincaré group (from the brane scan/brane bouquet) yields type II supergravity, heterotic supergravity and 11-dimensional supergravity in higher Cartan geometry-formulation (D'Auria-Fré formulation of supergravity).

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

Introductions and surveys include

• Javayoles, Sánchez, An introduction to Lorentzian Geometry and its application, 2010 (pdf)

• Christian Bär, N. Ginaux, Frank Pfäffle, I. Lorentzian geometry (pdf)

• Graciela Birman, Katsumi Nomizu, Trigonometry in Lorentzian geometry, The American Mathematical Monthly Vol. 91, No. 9 (Nov., 1984), pp. 543-549 (JSTOR)