# 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:

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)

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 $$\mathfrak{X}(U,\Phi)$$ 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


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


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
class sage.manifolds.differentiable.tensorfield_module.TensorFieldModule(vector_field_module, tensor_type)

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 $$\mathfrak{X}(U,\Phi)$$ 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


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

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

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.