Degenerate manifolds#
- class sage.manifolds.differentiable.degenerate.DegenerateManifold(n, name, metric_name=None, signature=None, base_manifold=None, diff_degree=+Infinity, latex_name=None, metric_latex_name=None, start_index=0, category=None, unique_tag=None)#
Bases:
DifferentiableManifold
Degenerate Manifolds
A degenerate manifold (or a null manifold) is a pair \((M,g)\) where \(M\) is a real differentiable manifold (see
DifferentiableManifold
) and \(g\) is a field of degenerate symmetric bilinear forms on \(M\) (seeDegenerateMetric
).INPUT:
n
– positive integer; dimension of the manifoldname
– string; name (symbol) given to the manifoldmetric_name
– (default:None
) string; name (symbol) given to the metric; ifNone
,'g'
is usedsignature
– (default:None
) signature \(S\) of the metric as a tuple: \(S = (n_+, n_-, n_0)\), where \(n_+\) (resp. \(n_-\), resp. \(n_0\)) is the number of positive terms (resp. negative terms, resp. zero tems) in any diagonal writing of the metric components; ifsignature
is not provided, \(S\) is set to \((ndim-1, 0, 1)\), being \(ndim\) the manifold’s dimensionambient
– (default:None
) if notNone
, must be a differentiable manifold; the created object is then an open subset ofambient
diff_degree
– (default:infinity
) degree \(k\) of differentiabilitylatex_name
– (default:None
) string; LaTeX symbol to denote the manifold; if none is provided, it is set toname
metric_latex_name
– (default:None
) string; LaTeX symbol to denote the metric; if none is provided, it is set tometric_name
start_index
– (default: 0) integer; lower value of the range of indices used for “indexed objects” on the manifold, e.g. coordinates in a chartcategory
– (default:None
) to specify the category; ifNone
,Manifolds(RR).Differentiable()
(orManifolds(RR).Smooth()
ifdiff_degree
=infinity
) is assumed (see the categoryManifolds
)unique_tag
– (default:None
) tag used to force the construction of a new object when all the other arguments have been used previously (withoutunique_tag
, theUniqueRepresentation
behavior inherited fromManifoldSubset
, viaDifferentiableManifold
andTopologicalManifold
, would return the previously constructed object corresponding to these arguments).
EXAMPLES:
A degenerate manifold is constructed via the generic function
Manifold()
, using the keywordstructure
:sage: M = Manifold(3, 'M', structure='degenerate_metric') sage: M 3-dimensional degenerate_metric manifold M sage: M.parent() <class 'sage.manifolds.differentiable.degenerate.DegenerateManifold_with_category'>
The metric associated with
M
is:sage: g = M.metric() sage: g degenerate metric g on the 3-dimensional degenerate_metric manifold M sage: g.signature() (0, 2, 1)
Its value has to be initialized either by setting its components in various vector frames (see the above examples regarding the 2-sphere and Minkowski spacetime) or by making it equal to a given field of symmetric bilinear forms (see the method
set()
of the metric class). Both methods are also covered in the documentation of methodmetric()
below.REFERENCES:
- metric(name=None, signature=None, latex_name=None, dest_map=None)#
Return the metric giving the null manifold structure to the manifold, or define a new metric tensor on the manifold.
INPUT:
name
– (default:None
) name given to the metric; ifname
isNone
or matches the name of the metric defining the null manifold structure ofself
, the latter metric is returnedsignature
– (default:None
; ignored ifname
isNone
) signature \(S\) of the metric as a tuple: \(S = (n_+, n_-, n_0)\), where \(n_+\) (resp. \(n_-\), resp. \(n_0\)) is the number of positive terms (resp. negative terms, resp. zero tems) in any diagonal writing of the metric components; ifsignature
is not provided, \(S\) is set to \((ndim-1, 0, 1)\), being \(ndim\) the manifold’s dimensionlatex_name
– (default:None
; ignored ifname
isNone
) LaTeX symbol to denote the metric; ifNone
, it is formed fromname
dest_map
– (default:None
; ignored ifname
isNone
) instance of classDiffMap
representing the destination map \(\Phi:\ U \rightarrow M\), where \(U\) is the current manifold; ifNone
, the identity map is assumed (case of a metric tensor field on \(U\))
OUTPUT:
instance of
DegenerateMetric
EXAMPLES:
Metric of a 3-dimensional degenerate manifold:
sage: M = Manifold(3, 'M', structure='degenerate_metric', start_index=1) sage: X.<x,y,z> = M.chart() sage: g = M.metric(); g degenerate metric g on the 3-dimensional degenerate_metric manifold M
The metric remains to be initialized, for instance by setting its components in the coordinate frame associated to the chart
X
:sage: g[1,1], g[2,2] = -1, 1 sage: g.display() g = -dx⊗dx + dy⊗dy sage: g[:] [-1 0 0] [ 0 1 0] [ 0 0 0]
Alternatively, the metric can be initialized from a given field of degenerate symmetric bilinear forms; we may create the former object by:
sage: X.coframe() Coordinate coframe (M, (dx,dy,dz)) sage: dx, dy = X.coframe()[1], X.coframe()[2] sage: b = dx*dx + dy*dy sage: b Field of symmetric bilinear forms dx⊗dx+dy⊗dy on the 3-dimensional degenerate_metric manifold M
We then use the metric method
set()
to makeg
being equal tob
as a symmetric tensor field of type(0,2)
:sage: g.set(b) sage: g.display() g = dx⊗dx + dy⊗dy
Another metric can be defined on
M
by specifying a metric name distinct from that chosen at the creation of the manifold (which isg
by default, but can be changed thanks to the keywordmetric_name
inManifold()
):sage: h = M.metric('h'); h degenerate metric h on the 3-dimensional degenerate_metric manifold M sage: h[1,1], h[2,2], h[3,3] = 1+y^2, 1+z^2, 1+x^2 sage: h.display() h = (y^2 + 1) dx⊗dx + (z^2 + 1) dy⊗dy + (x^2 + 1) dz⊗dz
The metric tensor
h
is distinct from the metric entering in the definition of the degenerate manifoldM
:sage: h is M.metric() False
while we have of course:
sage: g is M.metric() True
Providing the same name as the manifold’s default metric returns the latter:
sage: M.metric('g') is M.metric() True
- open_subset(name, latex_name=None, coord_def={})#
Create an open subset of
self
.An open subset is a set that is (i) included in the manifold and (ii) open with respect to the manifold’s topology. It is a differentiable manifold by itself. Moreover, equipped with the restriction of the manifold metric to itself, it is a null manifold. Hence the returned object is an instance of
DegenerateManifold
.INPUT:
name
– name given to the open subsetlatex_name
– (default:None
) LaTeX symbol to denote the subset; if none is provided, it is set toname
coord_def
– (default: {}) definition of the subset in terms of coordinates;coord_def
must a be dictionary with keys charts in the manifold’s atlas and values the symbolic expressions formed by the coordinates to define the subset.
OUTPUT:
instance of
DegenerateManifold
representing the created open subset
EXAMPLES:
Open subset of a 3-dimensional degenerate manifold:
sage: M = Manifold(3, 'M', structure='degenerate_metric', start_index=1) sage: X.<x,y,z> = M.chart() sage: U = M.open_subset('U', coord_def={X: [x>0, y>0]}); U Open subset U of the 3-dimensional degenerate_metric manifold M sage: type(U) <class 'sage.manifolds.differentiable.degenerate.DegenerateManifold_with_category'>
We initialize the metric of
M
:sage: g = M.metric() sage: g[1,1], g[2,2] = -1, 1
Then the metric on
U
is determined as the restriction ofg
toU
:sage: gU = U.metric(); gU degenerate metric g on the Open subset U of the 3-dimensional degenerate_metric manifold M sage: gU.display() g = -dx⊗dx + dy⊗dy sage: gU is g.restrict(U) True
- class sage.manifolds.differentiable.degenerate.TangentTensor(tensor, embedding, screen=None)#
Bases:
TensorFieldParal
Let
S
be a lightlike submanifold embedded in a pseudo-Riemannian manifold(M,g)
withPhi
the embedding map. LetT1
be a tensor onM
alongS
or not.TangentTensor(T1, Phi)
returns the restrictionT2
ofT1
alongS
that in addition can be applied only on vector fields tangent toS
, whenT1
has a covariant part.INPUT:
tensor
– a tensor field on the ambient manifoldembedding
– the embedding mapPhi
EXAMPLES:
Section of the lightcone of the Minkowski space with a hyperplane passing through the origin:
sage: M = Manifold(4, 'M', structure="Lorentzian") sage: X.<t,x,y,z> = M.chart() sage: S = Manifold(2, 'S', ambient=M, structure='degenerate_metric') sage: X_S.<u,v> = S.chart() sage: Phi = S.diff_map(M, {(X_S, X): [sqrt(u^2+v^2), u, v, 0]}, ....: name='Phi', latex_name=r'\Phi') sage: Phi_inv = M.diff_map(S, {(X, X_S): [x, y]}, name='Phi_inv', ....: latex_name=r'\Phi^{-1}') sage: S.set_immersion(Phi, inverse=Phi_inv); S.declare_embedding() sage: g = M.metric() sage: g[0,0], g[1,1], g[2,2], g[3,3] = -1, 1, 1, 1 sage: V = M.vector_field(0,0,0,1) sage: S.set_transverse(rigging=t, normal=V) sage: xi = M.vector_field(sqrt(x^2+y^2+z^2), x, y, 0) sage: U = M.vector_field(0, -y, x, 0) sage: Sc = S.screen('Sc', U, xi); sage: T1 = M.tensor_field(1,1).along(Phi); T1[0,0] = 1 sage: V1 = M.vector_field().along(Phi); V1[0] = 1; V1[1]=1 sage: T1(V1).display() ∂/∂t sage: from sage.manifolds.differentiable.degenerate_submanifold import TangentTensor sage: T2 = TangentTensor(T1, Phi) sage: T2 Tensor field of type (1,1) along the 2-dimensional degenerate submanifold S embedded in 4-dimensional differentiable manifold M with values on the 4-dimensional Lorentzian manifold M sage: V2 = S.projection(V1) sage: T2(V2).display() u/sqrt(u^2 + v^2) ∂/∂t
Of course \(T1\) and \(T2\) give the same output on vector fields tangent to S:
sage: T1(xi.along(Phi)).display() sqrt(u^2 + v^2) ∂/∂t sage: T2(xi.along(Phi)).display() sqrt(u^2 + v^2) ∂/∂t
- extension()#
Return initial tensor
EXAMPLES:
Section of the lightcone of the Minkowski space with a hyperplane passing through the origin:
sage: M = Manifold(4, 'M', structure="Lorentzian") sage: X.<t,x,y,z> = M.chart() sage: S = Manifold(2, 'S', ambient=M, structure='degenerate_metric') sage: X_S.<u,v> = S.chart() sage: Phi = S.diff_map(M, {(X_S, X): [sqrt(u^2+v^2), u, v, 0]}, ....: name='Phi', latex_name=r'\Phi') sage: Phi_inv = M.diff_map(S, {(X, X_S): [x, y]}, name='Phi_inv', ....: latex_name=r'\Phi^{-1}') sage: S.set_immersion(Phi, inverse=Phi_inv); S.declare_embedding() sage: g = M.metric() sage: g[0,0], g[1,1], g[2,2], g[3,3] = -1,1,1,1 sage: V = M.vector_field(); V[3] = 1 sage: S.set_transverse(rigging=t, normal=V) sage: xi = M.vector_field(); xi[0] = sqrt(x^2+y^2+z^2); xi[1] = x; xi[2] = y sage: U = M.vector_field(); U[1] = -y; U[2] = x sage: Sc = S.screen('Sc', U, xi); sage: T1 = M.tensor_field(1,1).along(Phi); T1[0,0] = 1 sage: from sage.manifolds.differentiable.degenerate_submanifold import TangentTensor sage: T2 = TangentTensor(T1, Phi); T3 = T2.extension() sage: T3 is T2 False sage: T3 is T1 True