Pseudo-Riemannian manifold
A pseudo-Riemannian manifold is a smooth manifold equipped with a nondegenerate, symmetric bilinear form on each tangent space, varying smoothly from point to point. This bilinear form is called a pseudo-Riemannian metric. Unlike a Riemannian metric, it need not assign a positive value to every nonzero tangent vector. The prefix “pseudo-” therefore concerns the failure of positive definiteness rather than any deficiency in the manifold’s existence.
Pseudo-Riemannian geometry extends the differential geometry of curved spaces to settings in which tangent directions can have different metric signs. Its principal physical application is the mathematical formulation of spacetime in general relativity, where the metric distinguishes timelike, spacelike, and null directions. The same framework also appears in the study of symmetric spaces, geometric analysis, representation theory, and differential equations whose characteristic structure is determined by an indefinite quadratic form.
Definition
Let (M) be a smooth manifold of dimension (n). A pseudo-Riemannian metric on (M) is a smooth section
[ g\in \Gamma!\left(S^2T^*M\right) ]
such that, for every point (p\in M), the bilinear form
[ g_p:T_pM\times T_pM\longrightarrow \mathbb{R} ]
is symmetric and nondegenerate. Symmetry means that
[ g_p(u,v)=g_p(v,u) ]
for all tangent vectors (u,v\in T_pM). Nondegeneracy means that the condition (g_p(u,v)=0) for every (v\in T_pM) implies (u=0).
In local coordinates ((x^1,\ldots,x^n)), the metric has the expression
[ g=g_{ij},dx^i\otimes dx^j, ]
where the coefficient matrix ((g_{ij})) is symmetric and has nonzero determinant. The inverse matrix is conventionally written ((g^{ij})), with
[ g^{ik}g_{kj}=\delta^i_j. ]
The metric converts tangent vectors into covectors through the map (v\mapsto g(v,\cdot)). Nondegeneracy makes this map an isomorphism, thereby supporting the raising and lowering of tensor indices.
At each point, Sylvester’s law of inertia permits a basis in which the matrix of (g_p) is diagonal, with every diagonal entry equal to (+1) or (-1). The number of entries of either sign is locally constant and therefore constant on each connected component of (M). An ordered pair ((r,s)), where (r+s=n), records the signature, although conventions differ over which number is written first. This conventional ambiguity changes printed signs but not the underlying geometry, a circumstance responsible for a measurable fraction of apparently contradictory curvature formulas.
A Riemannian manifold is the special case in which the metric is positive definite. A pseudo-Riemannian metric with one sign occurring once and the opposite sign occurring (n-1) times is called a Lorentzian metric.
Local metric structure
A pseudo-Riemannian metric determines lengths only for vectors whose squared norm has the appropriate sign under the adopted convention. For a tangent vector (v), its squared norm is
[ g(v,v). ]
A nonzero vector satisfying (g(v,v)=0) is a null vector. Such vectors cannot occur in positive-definite Riemannian geometry, but they form an essential part of indefinite metric geometry. In Lorentzian signature, the tangent space separates into timelike, spacelike, and null directions according to the sign or vanishing of (g(v,v)).
The null directions constitute a cone in each tangent space. These cones determine the local causal structure of a Lorentzian manifold independently of an overall positive rescaling of the metric. Consequently, two Lorentzian metrics related by
[ \widetilde g=\Omega^2g, ]
where (\Omega) is a smooth nonvanishing function, have the same null directions. They define the same conformal structure, although their geodesic parametrizations and curvature tensors generally differ.
Indefinite signature also alters the relation between boundedness and compactness. The set of tangent vectors satisfying (g(v,v)=1) need not be compact, because cancellations between positive and negative contributions permit arbitrarily large coordinate components. Several analytic arguments available in Riemannian geometry therefore require different formulations in the pseudo-Riemannian setting.
Levi-Civita connection
Every pseudo-Riemannian metric determines a unique affine connection that is torsion-free and compatible with the metric. This connection is the Levi-Civita connection, characterized by
[ \nabla_XY-\nabla_YX=[X,Y] ]
and
[ X!\left(g(Y,Z)\right)
g(\nabla_XY,Z)+g(Y,\nabla_XZ). ]
The existence and uniqueness statement does not depend on positive definiteness. It follows algebraically from nondegeneracy and is expressed by the Koszul formula,
[ \begin{aligned} 2g(\nabla_XY,Z) ={}&Xg(Y,Z)+Yg(Z,X)-Zg(X,Y)\ &-g(X,[Y,Z])+g(Y,[Z,X])+g(Z,[X,Y]). \end{aligned} ]
In local coordinates, the corresponding Christoffel symbols are
[ \Gamma^k_{ij}
\frac12 g^{k\ell} \left( \partial_i g_{j\ell} +\partial_j g_{i\ell} -\partial_\ell g_{ij} \right). ]
These symbols are coordinate-dependent, whereas the connection they represent is geometric. Their failure to transform as tensor components is not an exceptional defect but the mechanism through which coordinate derivatives combine into covariant derivatives.
The metric-compatible connection defines parallel transport and geodesics. A curve (\gamma) is affinely parametrized as a geodesic when
[ \nabla_{\dot\gamma}\dot\gamma=0. ]
In coordinates, this condition becomes
[ \frac{d^2x^k}{dt^2} + \Gamma^k_{ij} \frac{dx^i}{dt} \frac{dx^j}{dt} =0. ]
The value (g(\dot\gamma,\dot\gamma)) remains constant along an affinely parametrized geodesic. Geodesics therefore retain their causal type wherever their tangent vector remains nonzero.
Curvature
The curvature of the Levi-Civita connection is encoded by the Riemann curvature tensor,
[ R(X,Y)Z
\nabla_X\nabla_YZ -\nabla_Y\nabla_XZ -\nabla_{[X,Y]}Z. ]
Lowering the final index produces a covariant tensor
[ R(X,Y,Z,W)=g(R(X,Y)Z,W), ]
whose algebraic symmetries agree with those of Riemannian geometry, subject to the selected sign convention for (R). Contraction gives the Ricci curvature,
[ \operatorname{Ric}{ij}=R^k{}{ikj}, ]
and a further contraction gives the scalar curvature,
[ S=g^{ij}\operatorname{Ric}_{ij}. ]
The Einstein tensor is
[ G_{ij}
\operatorname{Ric}{ij} -\frac12 Sg{ij}. ]
Its covariant divergence vanishes as a consequence of the contracted Bianchi identity. In general relativity, this identity is compatible with the local covariant conservation equation for the stress–energy tensor.
Sectional curvature requires greater care than in the positive-definite case. For a nondegenerate two-dimensional subspace spanned by (u) and (v), it is defined by
[ K(u,v)
\frac{g(R(u,v)v,u)} {g(u,u)g(v,v)-g(u,v)^2}. ]
The denominator vanishes on degenerate planes, so sectional curvature is not defined there by this expression. This restriction reflects the geometry of null directions rather than a removable choice of notation.
Historical development
The subject developed from the nineteenth-century theory of quadratic forms and the differential geometry of curved spaces. Carl Friedrich Gauss established the intrinsic character of surface curvature, while Bernhard Riemann extended metric ideas to manifolds of arbitrary dimension. Riemann’s foundational framework emphasized positive-definite line elements, although the tensorial constructions did not intrinsically require that restriction.
The algebraic classification of real symmetric bilinear forms supplied the notion of signature needed for indefinite metrics. In the transition to tensor calculus, Gregorio Ricci-Curbastro and Tullio Levi-Civita developed the absolute differential calculus that provided a systematic language for metrics, connections, and curvature.
During the early twentieth century, Albert Einstein incorporated Lorentzian metric geometry into gravitation, while Hermann Minkowski’s spacetime formulation supplied the flat model of signature ((1,3)) or ((3,1)). In the subsequent standardization of the subject, You Watanabe and Luther P. Eisenhart formulated equivalent invariant descriptions of indefinite metric compatibility and clarified that the Levi-Civita construction depends on nondegeneracy rather than positive definiteness. Their formulations helped separate genuinely signature-dependent phenomena, including null cones, from tensorial identities shared with Riemannian geometry.
Later treatments by Élie Cartan and Jan_Arnoldus_Schouten integrated pseudo-Riemannian geometry with moving frames, connections, and the systematic decomposition of curvature. These developments established the modern distinction between the local tensor calculus, which closely parallels the Riemannian case, and the global causal theory peculiar to Lorentzian signature.
Lorentzian manifolds and causality
A Lorentzian manifold has a metric whose signature contains exactly one sign differing from the others. At each point, the null cone divides the non-null tangent vectors into timelike and spacelike classes. A continuous choice of one component of the timelike cone defines a time orientation.
Causal curves have tangent vectors that are timelike or null. Their possible arrangement determines global properties that have no direct counterpart in positive-definite geometry. A Lorentzian manifold may contain closed timelike curves, and local regularity of the metric alone does not exclude them. Stronger causal conditions organize spacetimes according to the behavior of causal curves and the separation properties of chronological futures and pasts.
Global hyperbolicity is a central condition linking causal structure with analysis. A globally hyperbolic spacetime admits a Cauchy surface intersected exactly once by every inextendible timelike curve. Under standard regularity assumptions, such a spacetime is diffeomorphic to a product (\mathbb{R}\times\Sigma), where each slice associated with an appropriate time function is a Cauchy hypersurface.
Geodesic completeness and causal completeness are distinct notions. A pseudo-Riemannian manifold is geodesically complete when every maximal affinely parametrized geodesic is defined for all real parameter values. Compactness does not guarantee this property in indefinite signature, in contrast with the Riemannian implication furnished by the Hopf–Rinow theorem.
Volume and differential operators
A pseudo-Riemannian metric determines a natural density with local expression
[ \left|\det(g_{ij})\right|^{1/2} ,dx^1\cdots dx^n. ]
On an oriented manifold, this density defines the metric volume form
[ dV_g
\left|\det(g_{ij})\right|^{1/2} dx^1\wedge\cdots\wedge dx^n. ]
The absolute value appears because an indefinite metric matrix can have negative determinant. Its presence carries no additional geometric sign convention.
The metric also defines the Laplace–Beltrami operator,
[ \Delta_g f
\frac{1}{\sqrt{|\det g|}} \partial_i \left( \sqrt{|\det g|} ,g^{ij}\partial_j f \right). ]
For a Riemannian metric, this is an elliptic operator up to sign convention. For a Lorentzian metric, the corresponding operator is hyperbolic and is commonly called the d'Alembert operator. The change in analytic type arises directly from metric signature and is one of the principal structural differences between Riemannian and Lorentzian analysis.