Some informal background: a Riemannian manifold is a differentiable manifold (where the tangent space at each point has an inner product) with a positive-definite metric tensor, d(x,y) ≥ 0.

A familiar Riemannian manifold is a Euclidean manifold (where one has to add a smoothly varying inner product on the tangent space of the standard Euclidean space), with the familiar Euclidean (distance) metric (our 3-space, for example).

What is NOT a Riemannian manifold is the familiar Lorentzian manifold of general relativity (of which the Minkowskian manifold of special relativity is a special case). The Lorentzian manifold is a pseudo-Riemannian manifold, the generalization of the Riemannian manifold, such that the metric tensor need not be positive-definite. This apparently seems like a minor point to some, but pseudo-Riemannian and Riemannian manifolds are incredibly different because of this.

One of the underlying assumptions of general relativity is that spacetime can be represented by a Lorentzian manifold with signature (+,-,-,-) or (-,+,+,+) - where the signature of a metric tensor is just the number of positive and negative eigenvalues of the corresponding real symmetric matrix once it is diagonalised.

Unlike a Riemannian manifold, with a positive-definite metric, a Lorentzian manifold M, with non-positive-definite metric, g, allows tangent vectors, X, to be classified into timelike g(X,X) > 0, null g(X,X) = 0, or spacelike g(X,X) < 0.

The causal structure of relativity comes from this classification.

Interestingly, when you most often are reading a paper in a physics journal though, instead of seeing “pseudo-Riemannian” you will see the word “Riemannian”; doing a search in the Physical Review Letters this afternoon for “Riemannian Manifold” yields 526 results, while searching for “pseudo-Riemannian Manifold” only yields 51. While I am sure a few of those authors were actually are working with Riemannian manifolds (and the obvious overlap with the “pseudo-Riemannian” search), the vast majority are simply misusing the term.

Some sample offenders:

Stephen A. Fulling, “Nonuniqueness of Canonical Field Quantization in Riemannian Space-Time” (Phys. Rev. D 7, 2850 (1973), Cited 211 times) : Fulling technically means “pseudo-Riemannian space-time”, else he wouldn’t have any causal structure.

C. N. Yang, “Integral Formalism for Gauge Fields” (Phys. Rev. Lett. 33, 445 (1974), Cited 208 times). Yang starts a paragraph off with “Introduction of a Riemannian metric”, when he then must actually be introducting a pseudo-Riemannian metric. Later, when Yang is defining “Pure Spaces”, he says, “A Riemannian manifold for which the parallel-displacement gauge field is sourceless will be called a pure space.” He then asserts, “A four-dimensional Einstein space, ie. For which R

_{αβ}= 0, is a pure space.” From the definition, if he really mean a Riemannian metric, he could not conclude that “a four-dimensional Einstein space” was a pure space, because an Einstein space must have a different signature to be causal (even though with R

_{αβ}= 0 he is specifying that the metric tensor is locally isometric to a Euclidean space).

Almost anytime you see the phrase “Riemannian space-time”, they are being sloppy. There is no such thing as a Riemannian space-time.

All of these highly respected papers incorrectly refer to the spacetimes they are working in as Riemannian:

Friedrich W. Hehl, Paul von der Heyde, G. David Kerlick, and James M. Nester, “General relativity with spin and torsion: Foundations and prospects” (Rev. Mod. Phys. 48, 393 (1976), Cited 612 times)

David G. Boulware, "Quantum field theory in Schwarzschild and Rindler spaces" (Phys. Rev. D 11, 1404 (1975), Cited 117 times)

Kenneth Nordtvedt, “Equivalence Principle for Massive Bodies. II. Theory” (Phys. Rev. 169, 1017 (1968), Cited 88 times)

Leonard Parker and S. A. Fulling, “Quantized Matter Fields and the Avoidance of Singularities in General Relativity” (Phys. Rev. D 7, 2357 (1973), Cited 87 times)

M. J. Rebouças and J. Tiomno, “Homogeneity of Riemannian space-times of Gödel type” (Phys. Rev. D 28, 1251 (1983), Cited 65 times)

J. S. Dowker and Raymond Critchley, “Stress-tensor conformal anomaly for scalar, spinor, and vector fields” (Phys. Rev. D 16, 3390 (1977), Cited 59 times)

M. A. Melvin, "Dynamics of Cylindrical Electromagnetic Universes" (Phys. Rev. 139, B225 (1965), Cited 43 times)

Leonard Parker, "Conformal Energy-Momentum Tensor in Riemannian Space-Time" (Phys. Rev. D 7, 976 (1973), Cited 36 times)

A. A. Coley, N. Pelavas, and R. M. Zalaletdinov, "Cosmological Solutions in Macroscopic Gravity" (Phys. Rev. Lett. 95, 151102 (2005), Cited 32 times)

F. W. Hehl, E. A. Lord, and Y. Ne'eman, "Hypermomentum in hadron dynamics and in gravitation" (Phys. Rev. D 17, 428 (1978), Cited 20 times)

The list goes on, and on, and on…

Physicists (& Journal Editors): if you’re working in a causal spacetime (and you know you should be), don’t say “Riemannian”. Say, “Lorentzian”, or “pseudo-Riemannian”, or “non-Riemannian”, don’t be lazy.

You wouldn't say "positive" when you mean "positive, zero, or negative", so why would you say "Riemannian" when you mean "pseudo-Riemannian"?

-S.C. Kavassalis

In Riemannian geometry, a Riemannian manifold or Riemannian space (M,g) is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, in a manner which varies smoothly from point to point. The metric g is a positive definite symmetric tensor: a metric tensor. This allows one to define various notions such as angles, lengths of curves, areas (or volumes), curvature, gradients of functions and divergence of vector fields. In other words, a Riemannian manifold is a differentiable manifold in which the tangent space at each point is a finite-dimensional Euclidean space. The terms are named after German mathematician Bernhard Riemann. I am a college sophomore with a dual major in Physics and Mathematics @ University of California, Santa Barbara. By the way, i came across these excellent physics flash cards. Its also a great initiative by the FunnelBrain team. Amazing!!

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