Tensor Field Modules¶
The set of tensor fields along a differentiable manifold with values on
a differentiable manifold
via a differentiable map
(possibly
and
) is a module over the algebra
of differentiable scalar fields on
. It is a free module if
and only if
is parallelizable. Accordingly, two classes are devoted
to tensor field modules:
TensorFieldModule
for tensor fields with values on a generic (in practice, not parallelizable) differentiable manifold,
TensorFieldFreeModule
for tensor fields with values on a parallelizable manifold.
AUTHORS:
- Eric Gourgoulhon, Michal Bejger (2014-2015): initial version
- Travis Scrimshaw (2016): review tweaks
REFERENCES:
- [KN1963]
- [Lee2013]
- [ONe1983]
-
class
sage.manifolds.differentiable.tensorfield_module.
TensorFieldFreeModule
(vector_field_module, tensor_type)¶ Bases:
sage.tensor.modules.tensor_free_module.TensorFreeModule
Free module of tensor fields of a given type
along a differentiable manifold
with values on a parallelizable manifold
, via a differentiable map
.
Given two non-negative integers
and
and a differentiable map
the tensor field module
is the set of all tensor fields of the type
(where
is the tensor bundle of type
over
) such that
for all
, i.e.
is a tensor of type
on the tangent vector space
. Since
is parallelizable, the set
is a free module over
, the ring (algebra) of differentiable scalar fields on
(see
DiffScalarFieldAlgebra
).The standard case of tensor fields on a differentiable manifold corresponds to
and
; we then denote
by merely
. Other common cases are
being an immersion and
being a curve in
(
is then an open interval of
).
Note
If
is not parallelizable, the class
TensorFieldModule
should be used instead, foris no longer a free module.
INPUT:
vector_field_module
– free moduleof vector fields along
associated with the map
tensor_type
– pairwith
being the contravariant rank and
the covariant rank
EXAMPLES:
Module of type-
tensor fields on
:
sage: M = Manifold(3, 'R^3') sage: c_xyz.<x,y,z> = M.chart() # Cartesian coordinates sage: T20 = M.tensor_field_module((2,0)) ; T20 Free module T^(2,0)(R^3) of type-(2,0) tensors fields on the 3-dimensional differentiable manifold R^3
is a module over the algebra
:
sage: T20.category() Category of finite dimensional modules over Algebra of differentiable scalar fields on the 3-dimensional differentiable manifold R^3 sage: T20.base_ring() is M.scalar_field_algebra() True
is a free module:
sage: isinstance(T20, FiniteRankFreeModule) True
because
is parallelizable:
sage: M.is_manifestly_parallelizable() True
The zero element:
sage: z = T20.zero() ; z Tensor field zero of type (2,0) on the 3-dimensional differentiable manifold R^3 sage: z[:] [0 0 0] [0 0 0] [0 0 0]
A random element:
sage: t = T20.an_element() ; t Tensor field of type (2,0) on the 3-dimensional differentiable manifold R^3 sage: t[:] [2 0 0] [0 0 0] [0 0 0]
The module
coerces to any module of type-
tensor fields defined on some subdomain of
:
sage: U = M.open_subset('U', coord_def={c_xyz: x>0}) sage: T20U = U.tensor_field_module((2,0)) sage: T20U.has_coerce_map_from(T20) True sage: T20.has_coerce_map_from(T20U) # the reverse is not true False sage: T20U.coerce_map_from(T20) Conversion map: From: Free module T^(2,0)(R^3) of type-(2,0) tensors fields on the 3-dimensional differentiable manifold R^3 To: Free module T^(2,0)(U) of type-(2,0) tensors fields on the Open subset U of the 3-dimensional differentiable manifold R^3
The coercion map is actually the restriction of tensor fields defined on
to
.
There is also a coercion map from fields of tangent-space automorphisms to tensor fields of type
:
sage: T11 = M.tensor_field_module((1,1)) ; T11 Free module T^(1,1)(R^3) of type-(1,1) tensors fields on the 3-dimensional differentiable manifold R^3 sage: GL = M.automorphism_field_group() ; GL General linear group of the Free module X(R^3) of vector fields on the 3-dimensional differentiable manifold R^3 sage: T11.has_coerce_map_from(GL) True
An explicit call to this coercion map is:
sage: id = GL.one() ; id Field of tangent-space identity maps on the 3-dimensional differentiable manifold R^3 sage: tid = T11(id) ; tid Tensor field Id of type (1,1) on the 3-dimensional differentiable manifold R^3 sage: tid[:] [1 0 0] [0 1 0] [0 0 1]
-
Element
¶ alias of
TensorFieldParal
-
class
sage.manifolds.differentiable.tensorfield_module.
TensorFieldModule
(vector_field_module, tensor_type)¶ Bases:
sage.structure.unique_representation.UniqueRepresentation
,sage.structure.parent.Parent
Module of tensor fields of a given type
along a differentiable manifold
with values on a differentiable manifold
, via a differentiable map
.
Given two non-negative integers
and
and a differentiable map
the tensor field module
is the set of all tensor fields of the type
(where
is the tensor bundle of type
over
) such that
for all
, i.e.
is a tensor of type
on the tangent vector space
. The set
is a module over
, the ring (algebra) of differentiable scalar fields on
(see
DiffScalarFieldAlgebra
).The standard case of tensor fields on a differentiable manifold corresponds to
and
; we then denote
by merely
. Other common cases are
being an immersion and
being a curve in
(
is then an open interval of
).
Note
If
is parallelizable, the class
TensorFieldFreeModule
should be used instead.INPUT:
vector_field_module
– moduleof vector fields along
associated with the map
tensor_type
– pairwith
being the contravariant rank and
the covariant rank
EXAMPLES:
Module of type-
tensor fields on the 2-sphere:
sage: M = Manifold(2, 'M') # the 2-dimensional sphere S^2 sage: U = M.open_subset('U') # complement of the North pole sage: c_xy.<x,y> = U.chart() # stereographic coordinates from the North pole sage: V = M.open_subset('V') # complement of the South pole sage: c_uv.<u,v> = V.chart() # stereographic coordinates from the South pole sage: M.declare_union(U,V) # S^2 is the union of U and V sage: xy_to_uv = c_xy.transition_map(c_uv, (x/(x^2+y^2), y/(x^2+y^2)), ....: intersection_name='W', restrictions1= x^2+y^2!=0, ....: restrictions2= u^2+v^2!=0) sage: uv_to_xy = xy_to_uv.inverse() sage: W = U.intersection(V) sage: T20 = M.tensor_field_module((2,0)); T20 Module T^(2,0)(M) of type-(2,0) tensors fields on the 2-dimensional differentiable manifold M
is a module over the algebra
:
sage: T20.category() Category of modules over Algebra of differentiable scalar fields on the 2-dimensional differentiable manifold M sage: T20.base_ring() is M.scalar_field_algebra() True
is not a free module:
sage: isinstance(T20, FiniteRankFreeModule) False
because
is not parallelizable:
sage: M.is_manifestly_parallelizable() False
On the contrary, the module of type-
tensor fields on
is a free module, since
is parallelizable (being a coordinate domain):
sage: T20U = U.tensor_field_module((2,0)) sage: isinstance(T20U, FiniteRankFreeModule) True sage: U.is_manifestly_parallelizable() True
The zero element:
sage: z = T20.zero() ; z Tensor field zero of type (2,0) on the 2-dimensional differentiable manifold M sage: z is T20(0) True sage: z[c_xy.frame(),:] [0 0] [0 0] sage: z[c_uv.frame(),:] [0 0] [0 0]
The module
coerces to any module of type-
tensor fields defined on some subdomain of
, for instance
:
sage: T20U.has_coerce_map_from(T20) True
The reverse is not true:
sage: T20.has_coerce_map_from(T20U) False
The coercion:
sage: T20U.coerce_map_from(T20) Conversion map: From: Module T^(2,0)(M) of type-(2,0) tensors fields on the 2-dimensional differentiable manifold M To: Free module T^(2,0)(U) of type-(2,0) tensors fields on the Open subset U of the 2-dimensional differentiable manifold M
The coercion map is actually the restriction of tensor fields defined on
to
:
sage: t = M.tensor_field(2,0, name='t') sage: eU = c_xy.frame() ; eV = c_uv.frame() sage: t[eU,:] = [[2,0], [0,-3]] sage: t.add_comp_by_continuation(eV, W, chart=c_uv) sage: T20U(t) # the conversion map in action Tensor field t of type (2,0) on the Open subset U of the 2-dimensional differentiable manifold M sage: T20U(t) is t.restrict(U) True
There is also a coercion map from fields of tangent-space automorphisms to tensor fields of type-
:
sage: T11 = M.tensor_field_module((1,1)) ; T11 Module T^(1,1)(M) of type-(1,1) tensors fields on the 2-dimensional differentiable manifold M sage: GL = M.automorphism_field_group() ; GL General linear group of the Module X(M) of vector fields on the 2-dimensional differentiable manifold M sage: T11.has_coerce_map_from(GL) True
Explicit call to the coercion map:
sage: a = GL.one() ; a Field of tangent-space identity maps on the 2-dimensional differentiable manifold M sage: a.parent() General linear group of the Module X(M) of vector fields on the 2-dimensional differentiable manifold M sage: ta = T11.coerce(a) ; ta Tensor field Id of type (1,1) on the 2-dimensional differentiable manifold M sage: ta.parent() Module T^(1,1)(M) of type-(1,1) tensors fields on the 2-dimensional differentiable manifold M sage: ta[eU,:] # ta on U [1 0] [0 1] sage: ta[eV,:] # ta on V [1 0] [0 1]
-
Element
¶ alias of
TensorField
-
base_module
()¶ Return the vector field module on which
self
is constructed.OUTPUT:
- a
VectorFieldModule
representing the module on whichself
is defined
EXAMPLES:
sage: M = Manifold(2, 'M') sage: T13 = M.tensor_field_module((1,3)) sage: T13.base_module() Module X(M) of vector fields on the 2-dimensional differentiable manifold M sage: T13.base_module() is M.vector_field_module() True sage: T13.base_module().base_ring() Algebra of differentiable scalar fields on the 2-dimensional differentiable manifold M
- a
-
tensor_type
()¶ Return the tensor type of
self
.OUTPUT:
- pair
of non-negative integers such that the tensor fields belonging to this module are of type
EXAMPLES:
sage: M = Manifold(2, 'M') sage: T13 = M.tensor_field_module((1,3)) sage: T13.tensor_type() (1, 3) sage: T20 = M.tensor_field_module((2,0)) sage: T20.tensor_type() (2, 0)
- pair
-
zero
()¶ Return the zero of
self
.