Changes between Initial Version and Version 28 of Ticket #26355
 Timestamp:
 10/04/19 17:51:33 (2 years ago)
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Ticket #26355

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new
toneeds_review
 Property Cc Eric Gourgoulhon removed

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Summary
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Classes DegenerateManifold and DegenerateSubmanifold
toDegenerate Metric and Degenerate Metric Manifold

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public/manifolds/DegenerateSubmanifold
topublic/manifolds/DegenerateMetricManifold

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sage8.4
tosage8.9

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b2523bbbf62de2e1c4a39fab1aa557f68d92e2cf
toe9f8bba07c03da6e45bec04df721c115f4063959

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Ticket #26355 – Description
initial v28 25 25 `(M,g)`. Then the normal bundle `T^\perp H` intersect the tangent 26 26 bundle `TH`. The radical distribution is defined as 27 'Rad(TH)=TH\cap T^\perp H'. In case of hypersurfaces, and more27 `Rad(TH)=TH\cap T^\perp H`. In case of hypersurfaces, and more 28 28 generally `1`lightlike submanifolds, this is a rank 1 distribution. 29 29 A screen distribution `S(TH)` is a complementary of `Rad(TH)` in `TH`. … … 39 39 N = \frac{1}{g(\xi, v)}\left(v\frac{g(v,v)}{2g(xi, v)}\xi\right) 40 40 41 Tensors on the ambient manifold 'M' are projected on 'H' along 'N'41 Tensors on the ambient manifold `M` are projected on `H` along `N` 42 42 to obtain induced objects. For instance, induced connection is the 43 linear connexion defined on H through the LeviCivitta connection of 44 'g' along `N`. 43 linear connexion defined on `H` through the LeviCivitta connection of `g` along `N`. 45 44 46 45 To work on a degenerate submanifold, after defining `H` as an instance 47 46 of :class:`~sage.manifolds.differentiable.manifold.DifferentiableManifold`, 48 with the keyword *structure='degenerate'*, you have to set a transvervector 49 `v` and screen distribution together with the radical distribution. 47 with the keyword *structure='degenerate_metric'*, you have to set a transvervector `v` and screen distribution together with the radical distribution. 50 48 51 49 An example of degenerate submanifold from General Relativity is the 52 horizon of the Shawrzschild black hole. Allow us to recall that 53 Shawrzschild black hole is the first nontrivial solution of Einstein's 54 equations. It describes the metric inside a star of radius `R = 2m`, 55 being `m` the inertial mass of the star. It can be seen as an open 56 ball in a Lorentzian manifold structure on `\RR^4`:: 50 horizon of the Schwarzschild black hole. Allow us to recall that 51 Schwarzschild 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`::