nLab
pseudo-Riemannian metric

Contents

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 }

          </semantics></math></div>

          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 n\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 n\mathbb{R}^n.

          A pseudo-Riemannian metric

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

          can always be diagonalized: there exists a basis e 1,,e ne_1, \ldots, e_n such that

          Q( 1inx ie i)=x 1 2++x p 2x p+1 2x n 2Q(\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,np)(p, n-p) is called the signature of the form QQ. Pseudo-Riemannian metrics on n\mathbb{R}^n are classified by their signatures; thus we have a standard metric of signature (p,np)(p, n-p) where {e 1,,e n}\{e_1, \ldots, e_n\} is the standard basis of n\mathbb{R}^n.

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

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

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

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

          such that

          XY,Z= XY,Z+Y, XZX \cdot \langle Y, Z\rangle = \langle \nabla_X Y, Z\rangle + \langle Y, \nabla_X Z\rangle
          [X,Y]= XY YX[X, Y] = \nabla_X Y - \nabla_Y X

          In that case, one may define a notion of geodesic in pseudo-Riemannian manifolds MM, and we have a notion of “distance squared” between the endpoints along any geodesic path α:[0,1]M\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,n1)(1, n-1). This is particularly so in the case n=4n = 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 GGsubgroup (monomorphism) HGH \hookrightarrow Gquotient (“coset space”) G/HG/HKlein geometryCartan geometryCartan connection
          examplesEuclidean group Iso(d)Iso(d)rotation group O(d)O(d)Cartesian space d\mathbb{R}^dEuclidean geometryRiemannian geometryaffine connectionEuclidean gravity
          Poincaré group Iso(d1,1)Iso(d-1,1)Lorentz group O(d1,1)O(d-1,1)Minkowski spacetime d1,1\mathbb{R}^{d-1,1}Lorentzian geometrypseudo-Riemannian geometryspin connectionEinstein gravity
          anti de Sitter group O(d1,2)O(d-1,2)O(d1,1)O(d-1,1)anti de Sitter spacetime AdS dAdS^dAdS gravity
          de Sitter group O(d,1)O(d,1)O(d1,1)O(d-1,1)de Sitter spacetime dS ddS^ddeSitter gravity
          linear algebraic groupparabolic subgroup/Borel subgroupflag varietyparabolic geometry
          conformal group O(d,t+1)O(d,t+1)conformal parabolic subgroupMöbius space S d,tS^{d,t}conformal geometryconformal connectionconformal gravity
          supergeometrysuper Lie group GGsubgroup (monomorphism) HGH \hookrightarrow Gquotient (“coset space”) G/HG/Hsuper Klein geometrysuper Cartan geometryCartan superconnection
          examplessuper Poincaré groupspin groupsuper Minkowski spacetime d1,1|N\mathbb{R}^{d-1,1\vert N}Lorentzian supergeometrysupergeometrysuperconnectionsupergravity
          super anti de Sitter groupsuper anti de Sitter spacetime
          higher differential geometrysmooth 2-group GG2-monomorphism HGH \to Ghomotopy quotient G//HG//HKlein 2-geometryCartan 2-geometry
          cohesive ∞-group∞-monomorphism (i.e. any homomorphism) HGH \to Ghomotopy quotient G//HG//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.