Math 2.0

Differential geometry Definition

Pseudo-Riemannian manifold

Builds on Smooth manifoldTangent bundleTensor fieldSymmetric bilinear formNondegenerate bilinear formSignature of a quadratic form

Definition

Let MM be a smooth manifold of dimension nn, with tangent spaces TpMT_pM. A pseudo-Riemannian metric on MM is a smooth symmetric (0,2)(0, 2)-tensor field gg on MM such that:

  1. for each p∈Mp \in M, the map gp ⁣:TpM×TpM→ℝg_p \colon T_pM \times T_pM \to \R is a symmetric bilinear form;
  2. for each p∈Mp \in M, the form gpg_p is nondegenerate: if gp(v,w)=0g_p(v, w) = 0 for all w∈TpMw \in T_pM, then v=0v = 0;
  3. the signature of gpg_p is the same for all p∈Mp \in M.

A pseudo-Riemannian manifold is a pair (M,g)(M, g) of a smooth manifold MM and a pseudo-Riemannian metric gg on MM.

Remarks

Example: on ℝ2\R^2 with coordinates (t,x)(t, x), let g=−dt2+dx2g = -dt^2 + dx^2. At each point its matrix is diag⁡(−1,1)\operatorname{diag}(-1, 1), which is symmetric with nonzero determinant, so gg is nondegenerate with constant signature: (ℝ2,g)(\R^2, g) is pseudo-Riemannian.

Non-example: on ℝ2\R^2, let g=dt2+t dx2g = dt^2 + t \, dx^2. Its matrix diag⁡(1,t)\operatorname{diag}(1, t) is degenerate where t=0t = 0, so clause 2 fails on the line t=0t = 0.

On a connected manifold, clause 3 follows from clause 2 and continuity, since the signature can only change where the determinant of gg vanishes. A Riemannian metric is the case where gpg_p is positive definite at every point. In local coordinates (x1,…,xn)(x^1, \ldots, x^n) one writes g=∑i,jgij dxi dxjg = \sum_{i,j} g_{ij} \, dx^i \, dx^j with a symmetric invertible matrix (gij)(g_{ij}).

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Nondegenerate bilinear form, concept, no evidence levelNondegenerate bilinea…conceptSignature of a quadratic form, concept, no evidence levelSignature of a quadra…conceptSmooth manifold, concept, no evidence levelSmooth manifoldconceptSymmetric bilinear form, concept, no evidence levelSymmetric bilinear fo…conceptTangent bundle, concept, no evidence levelTangent bundleconceptTensor field, concept, no evidence levelTensor fieldconceptPseudo-Riemannian manifold, definition, no evidence levelPseudo-Riemannian man…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Nondegenerate bilinear form concept
  2. Signature of a quadratic form concept
  3. Smooth manifold concept
  4. Symmetric bilinear form concept
  5. Tangent bundle concept
  6. Tensor field concept
  7. Pseudo-Riemannian manifold definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Pseudo-Riemannian_manifold
About this page
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Kinddefinition
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MSC53B30, 53C50
DefinesPseudo-Riemannian manifold
UsesSmooth manifold, Tangent bundle, Tensor field, Symmetric bilinear form, Nondegenerate bilinear form, Signature of a quadratic form
Changed2 November 2026

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