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My question has two parts, one concerning the state of the art of the subject, and the other the usefulness.

1. State of the art. Can someone provide references reflecting the state of the art in semi-Riemannian geometry with degenerate metric? I am aware of the research of Demir Kupeli, dealing with the case when the signature of the metric is constant. I am mostly interested in the cases when the signature is allowed to vary.

2. Usefulness. Is there a "market" for results extended from the nondegenerate semi-Riemannian geometry to the degenerate one? Would the community of mathematicians and physicists be interested in possible applications, for example to the singularities encountered in General Relativity?

Sorry if these questions seem odd, but I am interested to know whether is worth investing time and resources in doing research in this field. I would like to hear as many opinions as possible.

Update As a matter of fact, I already invested much time and resources, but as I told in an answer to a different question, I would like to make sure I am not reinventing the wheel.

There are also some results concerning cosmological models which start as Riemannian, then, on a hypersurface, the metric becomes Lorentzian.

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This is called sub-Riemannian geometry. See, for example, en.wikipedia.org/wiki/Sub-Riemannian_manifold –  Deane Yang Dec 1 '10 at 22:37
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@Deane Yang: Actually, sub-Riemannian manifolds are (sort of) complementary to the singular (semi-)Riemannian manifolds, but they are not the same (See for example springerlink.com/content/k768420845277744). For the sub-Riemannian manifolds, the metric is restricted at each point to a subspace of the tangent space; for the singular (semi-)Riemannian manifolds, the metric cancels on a subspace of the tangent space (the degenerate subspace). –  Cristi Stoica Dec 2 '10 at 6:05
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@Cristi: I have no real information to give you, but I do have a general comment that may be relevant to your second question. I think that most professional mathematicians would agree that attempting to generalize a classic subject by some unmotivated weakening of the definition of its basic structure is not likely to lead to fruitful mathematics. Good generalizations most often arise from making some minimal change necessary to treat a problem that comes up naturally and "just misses" fitting into an existing theory. So I guess I am saying you are "putting the cart before the horse". –  Dick Palais Feb 6 '11 at 18:11
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@Dick Palais: thank you, I think your are right in general. In this particular case, I am interested in the singularities in General Relativity, so I do have a motivation. But I would like to know of other applications as well. Especially in mathematics, because this problem is mostly mathematical, and it may be more accessible to a mathematician. I think that knowing more applications can always increase the motivation. –  Cristi Stoica Feb 6 '11 at 21:16
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@Cristi The singularity theorems prove that general relativity predicts on a globally hyperbolic space-time subject to certain energy conditions the space-time is singular. But, a singular space-time need not possess any infinite, metric-derived quantities, curvature polynomials and the like. In particular, a singular space-time need not have a degenerate metric. A singular space-time is simply an inextendible space-time on which there exist incomplete geodesics. So, saying that a singular space-time must have a degenerate metric or must have some quantity that's infinite is incorrect. –  Kelly Davis Feb 10 '11 at 21:11
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4 Answers

There are almost two years since I asked this question. At that time I already had developed part of the formalism, but wanted to learn more about other approaches.

I would like to answer my own question, by presenting my own research which took place in the meantime.

I considered smooth metrics, which are allowed to be degenerate and change signature. The main problem was that the covariant derivative and the curvature need in their definition the inverse of the metric, which is not defined or singular. The first step was to define an invariant contraction between covariant indices. I did this in Tensor Operations on Degenerate Inner Product Spaces (http://arxiv.org/abs/1112.5864)

This allowed me to find cases in which we can define covariant derivative for differential forms, and construct smooth Riemann tensor $R_{abcd}$. The tensor $R^a{}_{bcd}$ is equivalent with it only for non-degenerate metrics, otherwise is not defined or singular. I did this in On Singular Semi-Riemannian Manifolds (http://arxiv.org/abs/1105.0201). I gave some examples which showed that this kind of metrics actually exist. I also found a densitized version of Einstein's equation, which is equivalent with Einstein's for non-degenerate metrics, but also works at this kind of singularities.

To construct a new class of examples and applications to physics, I used warped products with warping function which may vanish: Warped Products of Singular Semi-Riemannian Manifolds (http://arxiv.org/abs/1105.3404). I also found the Cartan's Structural Equations for Degenerate Metric (http://arxiv.org/abs/1111.0646).

From the warped products introduced above, I could show that the Friedmann-Lemaitre-Robertson-Walker model is of this type: Big Bang singularity in the Friedmann-Lemaitre-Robertson-Walker spacetime (http://arxiv.org/abs/1112.4508), Beyond the Friedmann-Lemaitre-Robertson-Walker Big Bang singularity (http://arxiv.org/abs/1203.1819, Commun. Theor. Phys. 58(4) (2012), 613-616).

The black hole singularities apparently are not of this type, because the metric has components which become singular, so it is not smooth. But appropriate coordinate changes make their metric analytic, as shown in Schwarzschild Singularity is Semi-Regularizable (http://arxiv.org/abs/1111.4837, Eur. Phys. J. Plus (2012) 127: 83), Analytic Reissner-Nordstrom Singularity (http://arxiv.org/abs/1111.4332, Phys. Scr. 85 (2012) 055004), Kerr-Newman Solutions with Analytic Singularity and no Closed Timelike Curves (http://arxiv.org/abs/1111.7082). This allows finding globally hyperbolic spacetimes with singularities Spacetimes with Singularities (http://www.anstuocmath.ro/mathematics/pdf26/Art16.pdf).

I showed that there is also an alternative way to write an Einstein equation at singularities (http://arxiv.org/abs/1203.2140). This allows finding a general class of Big Bang singularities, which may be anysotropic and inhomogeneous, and which satisfy the Weyl Curvature Hypothesis (http://arxiv.org/abs/1203.3382) of Penrose. An interesting consequence is the existence of a dimensional reduction, which may reopen the possibility of quantizing gravity by perturbative methods: Quantum Gravity from Metric Dimensional Reduction at Singularities (http://arxiv.org/abs/1205.2586). An overview of this research is given in my seminary held at JINR, Dubna: An Exploration of the Singularities in General Relativity (http://arxiv.org/abs/1207.5303).

A more accessible introduction is given in the essay Did God Divide by Zero?.

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It might be difficult to define singular semi-Riemannian manifolds applied to general relativity and that might be the reason why in quantum field theory and string theory topological smooth manifolds seem to suffice. The idea might be that in cosmologcial terms every singularity gets smoothened no matter how bad it is whether caught in a big bang or what naught.

There are of course non-smooth manifolds with degenerate metrics such as in all kinds of black holes.

So, as far as general relativity is concerned it appears that the idea is both useful and state of the art alright.

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Here are a lot of examples of different metrics. I don't know if they are what you are looking for but they might help?

http://panda.unm.edu/Courses/Finley/p570.html

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See the paper: MR2598628 Reviewed Steinbauer, Roland: A note on distributional semi-Riemannian geometry. Novi Sad J. Math. 38 (2008), no. 3, 189–199.

See also the book: MR1883263 Reviewed Grosser, Michael; Kunzinger, Michael; Oberguggenberger, Michael; Steinbauer, Roland Geometric theory of generalized functions with applications to general relativity. Mathematics and its Applications, 537. Kluwer Academic Publishers, Dordrecht, 2001. xvi+505 pp.

There singular semi-Riemannian metrics are studied in the sense of distributions. But there is the need to multiply distributions in order to compute curvature and check Einstein's equation. So this uses an extension of distributions where you can multiply, but loose some properties.

Edit: You find many papers in this directions by looking here

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Thank you very much for the references, I will try to find them and read them. –  Cristi Stoica Oct 20 '12 at 21:44
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