Opened 3 years ago

Last modified 22 months ago

#26355 closed enhancement

Degenerate Metric and Degenerate Metric Manifold — at Version 28

Reported by: Dicolevrai Owned by:
Priority: major Milestone: sage-9.1
Component: geometry Keywords: Degenerate (or lightlike) submanifold
Cc: Merged in:
Authors: Hans Fotsing Tetsing Reviewers: Eric Gourgoulhon
Report Upstream: N/A Work issues:
Branch: public/manifolds/DegenerateMetricManifold (Commits, GitHub, GitLab) Commit: e9f8bba07c03da6e45bec04df721c115f4063959
Dependencies: Stopgaps:

Status badges

Description (last modified by Dicolevrai)

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_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::

Change History (30)

comment:1 Changed 3 years ago by Dicolevrai

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

comment:2 Changed 3 years 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 3 years 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 3 years ago by egourgoulhon

  • Cc egourgoulhon added; Eric Gourgoulhon removed

comment:5 Changed 3 years ago by Dicolevrai

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

comment:6 Changed 3 years 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 3 years 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 3 years 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 3 years ago by Dicolevrai

  • Milestone changed from sage-8.5 to sage-8.6

comment:10 Changed 3 years 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 3 years 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 3 years ago by Dicolevrai

To create the class Degenerate Submanifold

Changed 3 years ago by Dicolevrai

To create the class Degenerate Manifold

comment:12 Changed 3 years 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 3 years ago by Dicolevrai

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

comment:14 Changed 3 years ago by Dicolevrai

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

New commits:

f498738worksheet
d01c38aDegenerate metric

comment:15 Changed 2 years 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).

comment:16 Changed 2 years ago by Dicolevrai

  • Branch public/manifolds/DegenerateMetric deleted
  • Commit d01c38a77546a11bcc00e7f442ecc34cd4091661 deleted

comment:17 Changed 2 years ago by Dicolevrai

  • Branch set to public/manifolds/DegenerateMetrics
  • Commit set to 86fce263f3b84eaee97038545637764fcf05c471
  • Summary changed from Classes DegenerateManifold and DegenerateSubmanifold to Degenerate Metric and Degenerate Metric Manifold

Dear,

All tests passed this time. I will be happy if you would find time to review.


New commits:

86fce26Degenerate Metric

comment:18 Changed 2 years ago by git

  • Commit changed from 86fce263f3b84eaee97038545637764fcf05c471 to d5926b27de12d1c5b929e657be6c43880f6746ab

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

cb3b221Degenerate Metric 2
d5926b2Degenerate Metric 3

comment:19 Changed 2 years ago by git

  • Commit changed from d5926b27de12d1c5b929e657be6c43880f6746ab to 3456617784ad22d949d216651cbde29d9ebe8705

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

3456617rst

comment:20 Changed 2 years ago by Dicolevrai

  • Branch changed from public/manifolds/DegenerateMetrics to public/manifolds/DegenerateMetricManifold
  • Cc egourgoulhon removed
  • Commit changed from 3456617784ad22d949d216651cbde29d9ebe8705 to 089b2103a5c951707a890fa10d66d2d147014b88

New commits:

c768bc9Degenerate Metric
13c5b98Degenerate Metric 2
089b210Degenerate Metric 3

comment:21 Changed 2 years ago by egourgoulhon

  • Status changed from needs_review to needs_work

The documentation does not build (it looks like a file degenerate_metric.rst is missing in src/doc/en/reference/manifolds)

comment:22 Changed 2 years ago by git

  • Commit changed from 089b2103a5c951707a890fa10d66d2d147014b88 to 7693085d89d30cfe5f9451228b2036406bf6e82e

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

7693085Adding degenerate_metric.rst

comment:23 Changed 2 years ago by Dicolevrai

  • Status changed from needs_work to needs_review

New commits:

7693085Adding degenerate_metric.rst

comment:24 Changed 2 years ago by egourgoulhon

The documentation does not build: I get the error message

OSError: /home/eric/sage/py3/local/lib/python3.7/site-packages/sage/manifolds/differentiable/degenerate_submanifold.py:docstring of sage.manifolds.differentiable.degenerate_submanifold:7: WARNING: Inline interpreted text or phrase reference start-string without end-string.

See also the error in line 1772 of src/doc/en/reference/references/index.rst reported by the patchbot.

comment:25 Changed 2 years ago by jhpalmieri

Shouldn't "Shawrzschild" be "Schwarzschild" (appears several places).

Regarding the documentation, I skimmed it for formatting and fixed some of the problems. I am pretty sure there are more, for example cases where 'M' should be `M` (you want backquotes so the symbol will appear in math mode, not in quotes). In adapted_frame, do you want to use \xi instead of xi?

I strongly encourage running make and then opening up the reference manual (e.g. with ./sage -c 'reference()', and then proofreading for both formatting and content. I just made formatting changes, no changes to the content.

Last edited 2 years ago by jhpalmieri (previous) (diff)

comment:26 Changed 2 years ago by git

  • Commit changed from 7693085d89d30cfe5f9451228b2036406bf6e82e to 7b93c60a4397c9d028a8874b074374ca77093c3d

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

518bbdfDegenerate Metric
a4a2fe7Degenerate Metric 2
0db5aecAdding degenerate_metric.rst
7b93c60trac 26355: formatting for docstrings

comment:27 Changed 2 years ago by git

  • Commit changed from 7b93c60a4397c9d028a8874b074374ca77093c3d to e9f8bba07c03da6e45bec04df721c115f4063959

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

e9f8bbatrac 26355: formatting for docstrings

comment:28 Changed 2 years ago by Dicolevrai

  • Description modified (diff)
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