Degenerate Metric and Degenerate Metric Manifold #26355
Description
An embedded (resp. immersed) degenerate submanifold of a proper
pseudo-Riemannian manifold (M,g) is an embedded (resp. immersed)
submanifold H of M as a differentiable manifold such that pull
back of the metric tensor g via the embedding (resp. immersion)
endows H with the structure of a degenerate manifold.
Degenerate submanifolds are study in many fields of mathematics and physics,
for instance in Differential Geometry (especially in geometry of
lightlike submanifold) and in General Relativity. In geometry of lightlike
submanifolds, according to the dimension r of the radical distribution
(see below for definition of radical distribution), degenerate submanifolds
have been classify into 4 subgroups: r-lightlike submanifolds, Coisotropic
submanifolds, Isotropic submanifolds and Totally lightlike submanifolds.
(See the book of Krishan L. Duggal and Aurel Bejancu in REFERENCES.)
In the present module, one can define any of the 4 types but most of the methods are implemented only for degenerate hypersurfaces who belong to
r-lightlike submanifolds. However, there might be generalized to
1-lightlike submanifolds. In the literature, there is a new approach (the rigging technique) for studying 1-lightlike submanifolds but
here we the method of Krishan L. Duggal and Aurel Bejancu base on
the screen distribution.
Let H be a lightlike hypersurface of a pseudo-Riemannian manifold
(M,g). Then the normal bundle T^\perp H intersect the tangent
bundle TH. The radical distribution is defined as
Rad(TH)=TH\cap T^\perp H. In case of hypersurfaces, and more
generally 1-lightlike submanifolds, this is a rank 1 distribution.
A screen distribution S(TH) is a complementary of Rad(TH) in TH.
Giving a screen distribution S(TH) and a null vector field \xi
locally defined and spanning Rad(TH), there exists a unique
transversal null vector field 'N' locally defined and such that
g(N,\xi)=1. From a transverse vector 'v', the null rigging 'N'
is giving by the formula
.. MATH::
N = \frac{1}{g(\xi, v)}\left(v-\frac{g(v,v)}{2g(xi, v)}\xi\right)
Tensors on the ambient manifold M are projected on H along N
to obtain induced objects. For instance, the induced connection is the linear connexion defined on H through the Levi-Civitta connection of g along N.
To work on a degenerate submanifold, after defining H as an instance of :class:~sage.manifolds.differentiable.manifold.DifferentiableManifold, with the keyword structure='degenerate_metric', you have to set a transvervector v and screen distribution together with the radical distribution.
An example of degenerate submanifold from General Relativity is the
horizon of the Schwarzschild black hole. Allow us to recall that
Schwarzschild black hole is the first non-trivial solution of Einstein's equations. It describes the metric inside a star of radius R = 2m, being m the inertial mass of the star. It can be seen as an open ball in a Lorentzian manifold structure on \RR^4::
Component: geometry
Keywords: Degenerate (or lightlike) submanifold
Author: Hans Fotsing Tetsing
Branch/Commit: ede7f5d
Reviewer: Eric Gourgoulhon
Issue created by migration from https://trac.sagemath.org/ticket/26355