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: sage-8.9
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 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, 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 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, 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'*, 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 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::

Attachments (2)

degenerate_submanifold.py (97.9 KB) - added by Dicolevrai 7 months ago.
To create the class Degenerate Submanifold
degenerate.py (14.8 KB) - added by Dicolevrai 7 months ago.
To create the class Degenerate Manifold

Download all attachments as: .zip

Change History (17)

comment:1 Changed 11 months ago by Dicolevrai

  • Branch public/manifolds/DegenerateSubmanifold deleted
  • Commit b2523bbbf62de2e1c4a39fab1aa557f68d92e2cf deleted

comment:2 Changed 11 months ago by Dicolevrai

  • Branch set to public/manifolds/LightlikeSubmanifold
  • Commit set to 3ad4c004301268127b8c43989d33708eacb53370
  • Status changed from new to needs_info

New commits:

3ad4c00Classes DegenerateManifold and DegenerateSubmanifold

comment:3 Changed 11 months ago by git

  • Commit changed from 3ad4c004301268127b8c43989d33708eacb53370 to 9e1db14f7cb149995a0d7df91925811d3cbfe917

Branch pushed to git repo; I updated commit sha1. New commits:

9e1db14Classes DegenerateManifold and DegenerateSubmanifold

comment:4 Changed 11 months ago by egourgoulhon

  • Cc egourgoulhon added; Eric Gourgoulhon removed

comment:5 Changed 10 months ago by Dicolevrai

  • Branch public/manifolds/LightlikeSubmanifold deleted
  • Commit 9e1db14f7cb149995a0d7df91925811d3cbfe917 deleted
  • Milestone changed from sage-8.4 to sage-8.5

comment:6 Changed 10 months ago by Dicolevrai

  • Branch set to public/manifolds/LightlikeSubmanifolds
  • Commit set to 165d66a3437f45a100ace98a7b7500872b90828a
  • Status changed from needs_info to needs_review

New commits:

165d66aLightlike submanifolds an degenerate metrics

comment:7 Changed 10 months ago by git

  • Commit changed from 165d66a3437f45a100ace98a7b7500872b90828a to 69fdf2237a3a1633e979d7e8e250824a9425a2d6

Branch pushed to git repo; I updated commit sha1. New commits:

344dfafMerge branch 'develop' of https://github.com/sagemath/sage into LightlikeSubmanifolds
69fdf22Replacing the keyword 'degenerate' by 'degenerate_metric' and writing all doctests

comment:8 Changed 10 months ago by Dicolevrai

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 Dicolevrai

  • Milestone changed from sage-8.5 to sage-8.6

comment:10 Changed 7 months ago by embray

  • Milestone changed from sage-8.6 to sage-8.7

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 sage-pending or sage-wishlist.

comment:11 Changed 7 months ago by git

  • Commit changed from 69fdf2237a3a1633e979d7e8e250824a9425a2d6 to 8f7d13e008b1e6620ebe85fec1214d6024c94a41

Branch pushed to git repo; I updated commit sha1. New commits:

6499e87I don't know what is this file
6c1d969Merge branch 'public/manifolds/LightlikeSubmanifolds' of git://trac.sagemath.org/sage into develop
8f7d13eSome files were missing

Changed 7 months ago by Dicolevrai

To create the class Degenerate Submanifold

Changed 7 months ago by Dicolevrai

To create the class Degenerate Manifold

comment:12 Changed 5 months ago by embray

  • Milestone changed from sage-8.7 to sage-8.8

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 Dicolevrai

  • Branch public/manifolds/LightlikeSubmanifolds deleted
  • Commit 8f7d13e008b1e6620ebe85fec1214d6024c94a41 deleted

comment:14 Changed 4 months ago by Dicolevrai

  • Branch set to public/manifolds/DegenerateMetric
  • Commit set to d01c38a77546a11bcc00e7f442ecc34cd4091661

New commits:

f498738worksheet
d01c38aDegenerate metric

comment:15 Changed 7 weeks ago by embray

  • Milestone changed from sage-8.8 to sage-8.9

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).

Note: See TracTickets for help on using tickets.