Opened 11 months ago
Last modified 7 weeks ago
#26355 needs_review enhancement
Classes DegenerateManifold and DegenerateSubmanifold
Reported by:  Dicolevrai  Owned by:  

Priority:  major  Milestone:  sage8.9 
Component:  geometry  Keywords:  Degenerate (or lightlike) submanifold 
Cc:  egourgoulhon  Merged in:  
Authors:  Hans Fotsing Tetsing  Reviewers:  Eric Gourgoulhon 
Report Upstream:  N/A  Work issues:  
Branch:  public/manifolds/DegenerateMetric (Commits)  Commit:  d01c38a77546a11bcc00e7f442ecc34cd4091661 
Dependencies:  Stopgaps: 
Description
An *embedded (resp. immersed) degenerate submanifold of a proper
pseudoRiemannian 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, you can definie any of the 4 types but most of the
methods are implemented only for degenerate hypersurfaces who belong to
r
lightlike submanifolds. However, their might be generalized to
1
lightlike submanifolds. In the litterature there is a new approach
(the rigging technique) for studying 1
lightlike submanifolds but
here we we the method of Krishan L. Duggal and Aurel Bejancu base on
the screen distribution.
Let H
be a lightlike hypersurface of a pseudoRiemannian 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 1lightlike 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, induced connection is the
linear connexion defined on H through the LeviCivitta 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'*, 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 Shawrzschild black hole. Allow us to recall that
Shawrzschild black hole is the first nontrivial 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
::
Attachments (2)
Change History (17)
comment:1 Changed 11 months ago by
 Branch public/manifolds/DegenerateSubmanifold deleted
 Commit b2523bbbf62de2e1c4a39fab1aa557f68d92e2cf deleted
comment:2 Changed 11 months ago by
 Branch set to public/manifolds/LightlikeSubmanifold
 Commit set to 3ad4c004301268127b8c43989d33708eacb53370
 Status changed from new to needs_info
comment:3 Changed 11 months ago by
 Commit changed from 3ad4c004301268127b8c43989d33708eacb53370 to 9e1db14f7cb149995a0d7df91925811d3cbfe917
Branch pushed to git repo; I updated commit sha1. New commits:
9e1db14  Classes DegenerateManifold and DegenerateSubmanifold

comment:4 Changed 11 months ago by
 Cc egourgoulhon added; Eric Gourgoulhon removed
comment:5 Changed 10 months ago by
 Branch public/manifolds/LightlikeSubmanifold deleted
 Commit 9e1db14f7cb149995a0d7df91925811d3cbfe917 deleted
 Milestone changed from sage8.4 to sage8.5
comment:6 Changed 10 months ago by
 Branch set to public/manifolds/LightlikeSubmanifolds
 Commit set to 165d66a3437f45a100ace98a7b7500872b90828a
 Status changed from needs_info to needs_review
New commits:
165d66a  Lightlike submanifolds an degenerate metrics

comment:7 Changed 10 months ago by
 Commit changed from 165d66a3437f45a100ace98a7b7500872b90828a to 69fdf2237a3a1633e979d7e8e250824a9425a2d6
comment:8 Changed 10 months ago by
The keyword value degenerate
by
degenerate_metric
for the keyword
structure
of the constructor :function:
~sage.manifolds.manifold
, and all doctests have been written.
comment:9 Changed 8 months ago by
 Milestone changed from sage8.5 to sage8.6
comment:10 Changed 7 months ago by
 Milestone changed from sage8.6 to sage8.7
Retarging tickets optimistically to the next milestone. If you are responsible for this ticket (either its reporter or owner) and don't believe you are likely to complete this ticket before the next release (8.7) please retarget this ticket's milestone to sagepending or sagewishlist.
comment:11 Changed 7 months ago by
 Commit changed from 69fdf2237a3a1633e979d7e8e250824a9425a2d6 to 8f7d13e008b1e6620ebe85fec1214d6024c94a41
comment:12 Changed 5 months ago by
 Milestone changed from sage8.7 to sage8.8
Ticket retargeted after milestone closed (if you don't believe this ticket is appropriate for the Sage 8.8 release please retarget manually)
comment:13 Changed 4 months ago by
 Branch public/manifolds/LightlikeSubmanifolds deleted
 Commit 8f7d13e008b1e6620ebe85fec1214d6024c94a41 deleted
comment:14 Changed 4 months ago by
 Branch set to public/manifolds/DegenerateMetric
 Commit set to d01c38a77546a11bcc00e7f442ecc34cd4091661
comment:15 Changed 7 weeks ago by
 Milestone changed from sage8.8 to sage8.9
Moving tickets from the Sage 8.8 milestone that have been actively worked on in the last six months to the next release milestone (optimistically).
New commits:
Classes DegenerateManifold and DegenerateSubmanifold