Tensor Field Modules

The set of tensor fields along a differentiable manifold \(U\) with values on a differentiable manifold \(M\) via a differentiable map \(\Phi: U \rightarrow M\) (possibly \(U = M\) and \(\Phi = \mathrm{Id}_M\)) is a module over the algebra \(C^k(U)\) of differentiable scalar fields on \(U\). It is a free module if and only if \(M\) 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 \(M\),

  • TensorFieldFreeModule for tensor fields with values on a parallelizable manifold \(M\).

AUTHORS:

  • Eric Gourgoulhon, Michal Bejger (2014-2015): initial version

  • Travis Scrimshaw (2016): review tweaks

REFERENCES:

class sage.manifolds.differentiable.tensorfield_module.TensorFieldFreeModule(vector_field_module, tensor_type)[source]

Bases: TensorFreeModule

Free module of tensor fields of a given type \((k,l)\) along a differentiable manifold \(U\) with values on a parallelizable manifold \(M\), via a differentiable map \(U \rightarrow M\).

Given two nonnegative integers \(k\) and \(l\) and a differentiable map

\[\Phi:\ U \longrightarrow M,\]

the tensor field module \(T^{(k,l)}(U, \Phi)\) is the set of all tensor fields of the type

\[t:\ U \longrightarrow T^{(k,l)} M\]

(where \(T^{(k,l)}M\) is the tensor bundle of type \((k,l)\) over \(M\)) such that

\[t(p) \in T^{(k,l)}(T_{\Phi(p)}M)\]

for all \(p \in U\), i.e. \(t(p)\) is a tensor of type \((k,l)\) on the tangent vector space \(T_{\Phi(p)}M\). Since \(M\) is parallelizable, the set \(T^{(k,l)}(U,\Phi)\) is a free module over \(C^k(U)\), the ring (algebra) of differentiable scalar fields on \(U\) (see DiffScalarFieldAlgebra).

The standard case of tensor fields on a differentiable manifold corresponds to \(U = M\) and \(\Phi = \mathrm{Id}_M\); we then denote \(T^{(k,l)}(M,\mathrm{Id}_M)\) by merely \(T^{(k,l)}(M)\). Other common cases are \(\Phi\) being an immersion and \(\Phi\) being a curve in \(M\) (\(U\) is then an open interval of \(\RR\)).

Note

If \(M\) is not parallelizable, the class TensorFieldModule should be used instead, for \(T^{(k,l)}(U,\Phi)\) is no longer a free module.

INPUT:

  • vector_field_module – free module \(\mathfrak{X}(U,\Phi)\) of vector fields along \(U\) associated with the map \(\Phi: U \rightarrow M\)

  • tensor_type – pair \((k,l)\) with \(k\) being the contravariant rank and \(l\) the covariant rank

EXAMPLES:

Module of type-\((2,0)\) tensor fields on \(\RR^3\):

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

>>> from sage.all import *
>>> M = Manifold(Integer(3), 'R^3')
>>> c_xyz = M.chart(names=('x', 'y', 'z',)); (x, y, z,) = c_xyz._first_ngens(3)# Cartesian coordinates
>>> T20 = M.tensor_field_module((Integer(2),Integer(0))) ; T20
Free module T^(2,0)(R^3) of type-(2,0) tensors fields on the
 3-dimensional differentiable manifold R^3

\(T^{(2,0)}(\RR^3)\) is a module over the algebra \(C^k(\RR^3)\):

sage: T20.category()
Category of tensor products 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

>>> from sage.all import *
>>> T20.category()
Category of tensor products of finite dimensional modules over
 Algebra of differentiable scalar fields on the 3-dimensional differentiable manifold R^3
>>> T20.base_ring() is M.scalar_field_algebra()
True

\(T^{(2,0)}(\RR^3)\) is a free module:

sage: from sage.tensor.modules.finite_rank_free_module import FiniteRankFreeModule_abstract
sage: isinstance(T20, FiniteRankFreeModule_abstract)
True

>>> from sage.all import *
>>> from sage.tensor.modules.finite_rank_free_module import FiniteRankFreeModule_abstract
>>> isinstance(T20, FiniteRankFreeModule_abstract)
True

because \(M = \RR^3\) is parallelizable:

sage: M.is_manifestly_parallelizable()
True

>>> from sage.all import *
>>> 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]

>>> from sage.all import *
>>> z = T20.zero() ; z
Tensor field zero of type (2,0) on the 3-dimensional differentiable
 manifold R^3
>>> 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]

>>> from sage.all import *
>>> t = T20.an_element() ; t
Tensor field of type (2,0) on the 3-dimensional differentiable
 manifold R^3
>>> t[:]
[2 0 0]
[0 0 0]
[0 0 0]

The module \(T^{(2,0)}(\RR^3)\) coerces to any module of type-\((2,0)\) tensor fields defined on some subdomain of \(\RR^3\):

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)
Coercion 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

>>> from sage.all import *
>>> U = M.open_subset('U', coord_def={c_xyz: x>Integer(0)})
>>> T20U = U.tensor_field_module((Integer(2),Integer(0)))
>>> T20U.has_coerce_map_from(T20)
True
>>> T20.has_coerce_map_from(T20U)  # the reverse is not true
False
>>> T20U.coerce_map_from(T20)
Coercion 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 \(\RR^3\) to \(U\).

There is also a coercion map from fields of tangent-space automorphisms to tensor fields of type \((1,1)\):

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

>>> from sage.all import *
>>> T11 = M.tensor_field_module((Integer(1),Integer(1))) ; T11
Free module T^(1,1)(R^3) of type-(1,1) tensors fields on the
 3-dimensional differentiable manifold R^3
>>> 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
>>> 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]

>>> from sage.all import *
>>> id = GL.one() ; id
Field of tangent-space identity maps on the 3-dimensional
 differentiable manifold R^3
>>> tid = T11(id) ; tid
Tensor field Id of type (1,1) on the 3-dimensional differentiable
 manifold R^3
>>> tid[:]
[1 0 0]
[0 1 0]
[0 0 1]
Element[source]

alias of TensorFieldParal

class sage.manifolds.differentiable.tensorfield_module.TensorFieldModule(vector_field_module, tensor_type, category=None)[source]

Bases: UniqueRepresentation, ReflexiveModule_tensor

Module of tensor fields of a given type \((k,l)\) along a differentiable manifold \(U\) with values on a differentiable manifold \(M\), via a differentiable map \(U \rightarrow M\).

Given two nonnegative integers \(k\) and \(l\) and a differentiable map

\[\Phi:\ U \longrightarrow M,\]

the tensor field module \(T^{(k,l)}(U,\Phi)\) is the set of all tensor fields of the type

\[t:\ U \longrightarrow T^{(k,l)} M\]

(where \(T^{(k,l)} M\) is the tensor bundle of type \((k,l)\) over \(M\)) such that

\[t(p) \in T^{(k,l)}(T_{\Phi(p)}M)\]

for all \(p \in U\), i.e. \(t(p)\) is a tensor of type \((k,l)\) on the tangent vector space \(T_{\Phi(p)} M\). The set \(T^{(k,l)}(U,\Phi)\) is a module over \(C^k(U)\), the ring (algebra) of differentiable scalar fields on \(U\) (see DiffScalarFieldAlgebra).

The standard case of tensor fields on a differentiable manifold corresponds to \(U = M\) and \(\Phi = \mathrm{Id}_M\); we then denote \(T^{(k,l)}(M,\mathrm{Id}_M)\) by merely \(T^{(k,l)}(M)\). Other common cases are \(\Phi\) being an immersion and \(\Phi\) being a curve in \(M\) (\(U\) is then an open interval of \(\RR\)).

Note

If \(M\) is parallelizable, the class TensorFieldFreeModule should be used instead.

INPUT:

  • vector_field_module – module \(\mathfrak{X}(U,\Phi)\) of vector fields along \(U\) associated with the map \(\Phi: U \rightarrow M\)

  • tensor_type – pair \((k,l)\) with \(k\) being the contravariant rank and \(l\) the covariant rank

EXAMPLES:

Module of type-\((2,0)\) 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

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M') # the 2-dimensional sphere S^2
>>> U = M.open_subset('U') # complement of the North pole
>>> c_xy = U.chart(names=('x', 'y',)); (x, y,) = c_xy._first_ngens(2)# stereographic coordinates from the North pole
>>> V = M.open_subset('V') # complement of the South pole
>>> c_uv = V.chart(names=('u', 'v',)); (u, v,) = c_uv._first_ngens(2)# stereographic coordinates from the South pole
>>> M.declare_union(U,V)   # S^2 is the union of U and V
>>> xy_to_uv = c_xy.transition_map(c_uv, (x/(x**Integer(2)+y**Integer(2)), y/(x**Integer(2)+y**Integer(2))),
...                 intersection_name='W', restrictions1= x**Integer(2)+y**Integer(2)!=Integer(0),
...                 restrictions2= u**Integer(2)+v**Integer(2)!=Integer(0))
>>> uv_to_xy = xy_to_uv.inverse()
>>> W = U.intersection(V)
>>> T20 = M.tensor_field_module((Integer(2),Integer(0))); T20
Module T^(2,0)(M) of type-(2,0) tensors fields on the 2-dimensional
 differentiable manifold M

\(T^{(2,0)}(M)\) is a module over the algebra \(C^k(M)\):

sage: T20.category()
Category of tensor products 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

>>> from sage.all import *
>>> T20.category()
Category of tensor products of modules over Algebra of differentiable scalar fields
 on the 2-dimensional differentiable manifold M
>>> T20.base_ring() is M.scalar_field_algebra()
True

\(T^{(2,0)}(M)\) is not a free module:

sage: from sage.tensor.modules.finite_rank_free_module import FiniteRankFreeModule_abstract
sage: isinstance(T20, FiniteRankFreeModule_abstract)
False

>>> from sage.all import *
>>> from sage.tensor.modules.finite_rank_free_module import FiniteRankFreeModule_abstract
>>> isinstance(T20, FiniteRankFreeModule_abstract)
False

because \(M = S^2\) is not parallelizable:

sage: M.is_manifestly_parallelizable()
False

>>> from sage.all import *
>>> M.is_manifestly_parallelizable()
False

On the contrary, the module of type-\((2,0)\) tensor fields on \(U\) is a free module, since \(U\) is parallelizable (being a coordinate domain):

sage: T20U = U.tensor_field_module((2,0))
sage: isinstance(T20U, FiniteRankFreeModule_abstract)
True
sage: U.is_manifestly_parallelizable()
True

>>> from sage.all import *
>>> T20U = U.tensor_field_module((Integer(2),Integer(0)))
>>> isinstance(T20U, FiniteRankFreeModule_abstract)
True
>>> 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]

>>> from sage.all import *
>>> z = T20.zero() ; z
Tensor field zero of type (2,0) on the 2-dimensional differentiable
 manifold M
>>> z is T20(Integer(0))
True
>>> z[c_xy.frame(),:]
[0 0]
[0 0]
>>> z[c_uv.frame(),:]
[0 0]
[0 0]

The module \(T^{(2,0)}(M)\) coerces to any module of type-\((2,0)\) tensor fields defined on some subdomain of \(M\), for instance \(T^{(2,0)}(U)\):

sage: T20U.has_coerce_map_from(T20)
True

>>> from sage.all import *
>>> T20U.has_coerce_map_from(T20)
True

The reverse is not true:

sage: T20.has_coerce_map_from(T20U)
False

>>> from sage.all import *
>>> T20.has_coerce_map_from(T20U)
False

The coercion:

sage: T20U.coerce_map_from(T20)
Coercion 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

>>> from sage.all import *
>>> T20U.coerce_map_from(T20)
Coercion 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 \(M\) to \(U\):

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

>>> from sage.all import *
>>> t = M.tensor_field(Integer(2),Integer(0), name='t')
>>> eU = c_xy.frame() ; eV = c_uv.frame()
>>> t[eU,:] = [[Integer(2),Integer(0)], [Integer(0),-Integer(3)]]
>>> t.add_comp_by_continuation(eV, W, chart=c_uv)
>>> 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
>>> T20U(t) is t.restrict(U)
True

There is also a coercion map from fields of tangent-space automorphisms to tensor fields of type-\((1,1)\):

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

>>> from sage.all import *
>>> T11 = M.tensor_field_module((Integer(1),Integer(1))) ; T11
Module T^(1,1)(M) of type-(1,1) tensors fields on the 2-dimensional
 differentiable manifold M
>>> GL = M.automorphism_field_group() ; GL
General linear group of the Module X(M) of vector fields on the
 2-dimensional differentiable manifold M
>>> 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]

>>> from sage.all import *
>>> a = GL.one() ; a
Field of tangent-space identity maps on the 2-dimensional
 differentiable manifold M
>>> a.parent()
General linear group of the Module X(M) of vector fields on the
 2-dimensional differentiable manifold M
>>> ta = T11.coerce(a) ; ta
Tensor field Id of type (1,1) on the 2-dimensional differentiable
 manifold M
>>> ta.parent()
Module T^(1,1)(M) of type-(1,1) tensors fields on the 2-dimensional
 differentiable manifold M
>>> ta[eU,:]  # ta on U
[1 0]
[0 1]
>>> ta[eV,:]  # ta on V
[1 0]
[0 1]
Element[source]

alias of TensorField

base_module()[source]

Return the vector field module on which self is constructed.

OUTPUT:

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

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M')
>>> T13 = M.tensor_field_module((Integer(1),Integer(3)))
>>> T13.base_module()
Module X(M) of vector fields on the 2-dimensional differentiable
 manifold M
>>> T13.base_module() is M.vector_field_module()
True
>>> T13.base_module().base_ring()
Algebra of differentiable scalar fields on the 2-dimensional
 differentiable manifold M
tensor_type()[source]

Return the tensor type of self.

OUTPUT: pair \((k,l)\) of nonnegative integers such that the tensor fields belonging to this module are of type \((k,l)\)

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)

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M')
>>> T13 = M.tensor_field_module((Integer(1),Integer(3)))
>>> T13.tensor_type()
(1, 3)
>>> T20 = M.tensor_field_module((Integer(2),Integer(0)))
>>> T20.tensor_type()
(2, 0)
zero()[source]

Return the zero of self.