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:

[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 $(k,l)$ along a differentiable manifold $U$ with values on a parallelizable manifold $M$ , via a differentiable map $U \rightarrow M$ .

Given two non-negative 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 $\mathcal{X}(U,\Phi)$ 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

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

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

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

sage: isinstance(T20, FiniteRankFreeModule)
True

because $M = \RR^3$ 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 $T^{(2,0)}(\RR^3)$ 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)
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 $\RR^3$ 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

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 $(k,l)$ along a differentiable manifold $U$ with values on a differentiable manifold $M$ , via a differentiable map $U \rightarrow M$ .

Given two non-negative 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 $\mathcal{X}(U,\Phi)$ 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

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

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

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

sage: isinstance(T20, FiniteRankFreeModule)
False

because $M = S^2$ is not parallelizable:

sage: 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)
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 $T^{(2,0)}(M)$ 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

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 $M$ 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

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

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 which self 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

tensor_type()¶

Return the tensor type of self.

OUTPUT:

pair $(k,l)$ of non-negative 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)

zero()¶

Return the zero of self.

TESTS:

sage: M = Manifold(2, 'M')
sage: U = M.open_subset('U'); V = M.open_subset('V')
sage: c_xy.<x,y> = U.chart(); c_uv.<u,v> = V.chart()
sage: M.declare_union(U,V)
sage: T20 = M.tensor_field_module((2,0))
sage: T20.zero()
Tensor field zero of type (2,0) on the
 2-dimensional differentiable manifold M

Tensor Field Modules¶

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