Tangent Spaces¶

The class TangentSpace implements tangent vector spaces to a differentiable manifold.

AUTHORS:

Eric Gourgoulhon, Michal Bejger (2014-2015): initial version
Travis Scrimshaw (2016): review tweaks

REFERENCES:

Chap. 3 of [Lee2013]

class sage.manifolds.differentiable.tangent_space.TangentSpace(point: ManifoldPoint, base_ring=None)[source]¶

Bases: FiniteRankFreeModule

Tangent space to a differentiable manifold at a given point.

Let \(M\) be a differentiable manifold of dimension \(n\) over a topological field \(K\) and \(p \in M\). The tangent space \(T_p M\) is an \(n\)-dimensional vector space over \(K\) (without a distinguished basis).

INPUT:

point – ManifoldPoint; point \(p\) at which the tangent space is defined

EXAMPLES:

Tangent space on a 2-dimensional manifold:

Sage

sage: M = Manifold(2, 'M')
sage: c_xy.<x,y> = M.chart()
sage: p = M.point((-1,2), name='p')
sage: Tp = M.tangent_space(p) ; Tp
Tangent space at Point p on the 2-dimensional differentiable manifold M

Python

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M')
>>> c_xy = M.chart(names=('x', 'y',)); (x, y,) = c_xy._first_ngens(2)
>>> p = M.point((-Integer(1),Integer(2)), name='p')
>>> Tp = M.tangent_space(p) ; Tp
Tangent space at Point p on the 2-dimensional differentiable manifold M

Tangent spaces are free modules of finite rank over SymbolicRing (actually vector spaces of finite dimension over the manifold base field \(K\), with \(K=\RR\) here):

Sage

sage: Tp.base_ring()
Symbolic Ring
sage: Tp.category()
Category of finite dimensional vector spaces over Symbolic Ring
sage: Tp.rank()
2
sage: dim(Tp)
2

Python

>>> from sage.all import *
>>> Tp.base_ring()
Symbolic Ring
>>> Tp.category()
Category of finite dimensional vector spaces over Symbolic Ring
>>> Tp.rank()
2
>>> dim(Tp)
2

The tangent space is automatically endowed with bases deduced from the vector frames around the point:

Sage

sage: Tp.bases()
[Basis (∂/∂x,∂/∂y) on the Tangent space at Point p on the 2-dimensional
 differentiable manifold M]
sage: M.frames()
[Coordinate frame (M, (∂/∂x,∂/∂y))]

Python

>>> from sage.all import *
>>> Tp.bases()
[Basis (∂/∂x,∂/∂y) on the Tangent space at Point p on the 2-dimensional
 differentiable manifold M]
>>> M.frames()
[Coordinate frame (M, (∂/∂x,∂/∂y))]

At this stage, only one basis has been defined in the tangent space, but new bases can be added from vector frames on the manifold by means of the method at(), for instance, from the frame associated with some new coordinates:

Sage

sage: c_uv.<u,v> = M.chart()
sage: c_uv.frame().at(p)
Basis (∂/∂u,∂/∂v) on the Tangent space at Point p on the 2-dimensional
 differentiable manifold M
sage: Tp.bases()
[Basis (∂/∂x,∂/∂y) on the Tangent space at Point p on the 2-dimensional
 differentiable manifold M,
 Basis (∂/∂u,∂/∂v) on the Tangent space at Point p on the 2-dimensional
 differentiable manifold M]

Python

>>> from sage.all import *
>>> c_uv = M.chart(names=('u', 'v',)); (u, v,) = c_uv._first_ngens(2)
>>> c_uv.frame().at(p)
Basis (∂/∂u,∂/∂v) on the Tangent space at Point p on the 2-dimensional
 differentiable manifold M
>>> Tp.bases()
[Basis (∂/∂x,∂/∂y) on the Tangent space at Point p on the 2-dimensional
 differentiable manifold M,
 Basis (∂/∂u,∂/∂v) on the Tangent space at Point p on the 2-dimensional
 differentiable manifold M]

All the bases defined on Tp are on the same footing. Accordingly the tangent space is not in the category of modules with a distinguished basis:

Sage

sage: Tp in ModulesWithBasis(SR)
False

Python

>>> from sage.all import *
>>> Tp in ModulesWithBasis(SR)
False

It is simply in the category of modules:

Sage

sage: Tp in Modules(SR)
True

Python

>>> from sage.all import *
>>> Tp in Modules(SR)
True

Since the base ring is a field, it is actually in the category of vector spaces:

Sage

sage: Tp in VectorSpaces(SR)
True

Python

>>> from sage.all import *
>>> Tp in VectorSpaces(SR)
True

A typical element:

Sage

sage: v = Tp.an_element() ; v
Tangent vector at Point p on the
 2-dimensional differentiable manifold M
sage: v.display()
∂/∂x + 2 ∂/∂y
sage: v.parent()
Tangent space at Point p on the
 2-dimensional differentiable manifold M

Python

>>> from sage.all import *
>>> v = Tp.an_element() ; v
Tangent vector at Point p on the
 2-dimensional differentiable manifold M
>>> v.display()
∂/∂x + 2 ∂/∂y
>>> v.parent()
Tangent space at Point p on the
 2-dimensional differentiable manifold M

The zero vector:

Sage

sage: Tp.zero()
Tangent vector zero at Point p on the
 2-dimensional differentiable manifold M
sage: Tp.zero().display()
zero = 0
sage: Tp.zero().parent()
Tangent space at Point p on the
 2-dimensional differentiable manifold M

Python

>>> from sage.all import *
>>> Tp.zero()
Tangent vector zero at Point p on the
 2-dimensional differentiable manifold M
>>> Tp.zero().display()
zero = 0
>>> Tp.zero().parent()
Tangent space at Point p on the
 2-dimensional differentiable manifold M

Tangent spaces are unique:

Sage

sage: M.tangent_space(p) is Tp
True
sage: p1 = M.point((-1,2))
sage: M.tangent_space(p1) is Tp
True

Python

>>> from sage.all import *
>>> M.tangent_space(p) is Tp
True
>>> p1 = M.point((-Integer(1),Integer(2)))
>>> M.tangent_space(p1) is Tp
True

even if points are not:

Sage

sage: p1 is p
False

Python

>>> from sage.all import *
>>> p1 is p
False

Actually p1 and p share the same tangent space because they compare equal:

Sage

sage: p1 == p
True

Python

>>> from sage.all import *
>>> p1 == p
True

The tangent-space uniqueness holds even if the points are created in different coordinate systems:

Sage

sage: xy_to_uv = c_xy.transition_map(c_uv, (x+y, x-y))
sage: uv_to_xv = xy_to_uv.inverse()
sage: p2 = M.point((1, -3), chart=c_uv, name='p_2')
sage: p2 is p
False
sage: M.tangent_space(p2) is Tp
True
sage: p2 == p
True

Python

>>> from sage.all import *
>>> xy_to_uv = c_xy.transition_map(c_uv, (x+y, x-y))
>>> uv_to_xv = xy_to_uv.inverse()
>>> p2 = M.point((Integer(1), -Integer(3)), chart=c_uv, name='p_2')
>>> p2 is p
False
>>> M.tangent_space(p2) is Tp
True
>>> p2 == p
True

An isomorphism of the tangent space with an inner product space with distinguished basis:

Sage

sage: g = M.metric('g')
sage: g[:] = ((1, 0), (0, 1))
sage: Q_Tp_xy = g[c_xy.frame(),:](*p.coordinates(c_xy)); Q_Tp_xy
[1 0]
[0 1]
sage: W_Tp_xy = VectorSpace(SR, 2, inner_product_matrix=Q_Tp_xy)
sage: Tp.bases()[0]
Basis (∂/∂x,∂/∂y) on the Tangent space at Point p on the
 2-dimensional differentiable manifold M
sage: phi_Tp_xy = Tp.isomorphism_with_fixed_basis(Tp.bases()[0], codomain=W_Tp_xy)
sage: phi_Tp_xy
Generic morphism:
 From: Tangent space at Point p on the 2-dimensional differentiable manifold M
 To:   Ambient quadratic space of dimension 2 over Symbolic Ring
       Inner product matrix:
       [1 0]
       [0 1]

sage: Q_Tp_uv = g[c_uv.frame(),:](*p.coordinates(c_uv)); Q_Tp_uv
[1/2   0]
[  0 1/2]
sage: W_Tp_uv = VectorSpace(SR, 2, inner_product_matrix=Q_Tp_uv)
sage: Tp.bases()[1]
Basis (∂/∂u,∂/∂v) on the Tangent space at Point p on the
 2-dimensional differentiable manifold M
sage: phi_Tp_uv = Tp.isomorphism_with_fixed_basis(Tp.bases()[1], codomain=W_Tp_uv)
sage: phi_Tp_uv
Generic morphism:
 From: Tangent space at Point p on the 2-dimensional differentiable manifold M
 To:   Ambient quadratic space of dimension 2 over Symbolic Ring
       Inner product matrix:
       [1/2   0]
       [  0 1/2]

sage: t1, t2 = Tp.tensor((1,0)), Tp.tensor((1,0))
sage: t1[:] = (8, 15)
sage: t2[:] = (47, 11)
sage: t1[Tp.bases()[0],:]
[8, 15]
sage: phi_Tp_xy(t1), phi_Tp_xy(t2)
((8, 15), (47, 11))
sage: phi_Tp_xy(t1).inner_product(phi_Tp_xy(t2))
541

sage: Tp_xy_to_uv = M.change_of_frame(c_xy.frame(), c_uv.frame()).at(p); Tp_xy_to_uv
Automorphism of the Tangent space at Point p on the
 2-dimensional differentiable manifold M
sage: Tp.set_change_of_basis(Tp.bases()[0], Tp.bases()[1], Tp_xy_to_uv)
sage: t1[Tp.bases()[1],:]
[23, -7]
sage: phi_Tp_uv(t1), phi_Tp_uv(t2)
((23, -7), (58, 36))
sage: phi_Tp_uv(t1).inner_product(phi_Tp_uv(t2))
541

Python

>>> from sage.all import *
>>> g = M.metric('g')
>>> g[:] = ((Integer(1), Integer(0)), (Integer(0), Integer(1)))
>>> Q_Tp_xy = g[c_xy.frame(),:](*p.coordinates(c_xy)); Q_Tp_xy
[1 0]
[0 1]
>>> W_Tp_xy = VectorSpace(SR, Integer(2), inner_product_matrix=Q_Tp_xy)
>>> Tp.bases()[Integer(0)]
Basis (∂/∂x,∂/∂y) on the Tangent space at Point p on the
 2-dimensional differentiable manifold M
>>> phi_Tp_xy = Tp.isomorphism_with_fixed_basis(Tp.bases()[Integer(0)], codomain=W_Tp_xy)
>>> phi_Tp_xy
Generic morphism:
 From: Tangent space at Point p on the 2-dimensional differentiable manifold M
 To:   Ambient quadratic space of dimension 2 over Symbolic Ring
       Inner product matrix:
       [1 0]
       [0 1]

>>> Q_Tp_uv = g[c_uv.frame(),:](*p.coordinates(c_uv)); Q_Tp_uv
[1/2   0]
[  0 1/2]
>>> W_Tp_uv = VectorSpace(SR, Integer(2), inner_product_matrix=Q_Tp_uv)
>>> Tp.bases()[Integer(1)]
Basis (∂/∂u,∂/∂v) on the Tangent space at Point p on the
 2-dimensional differentiable manifold M
>>> phi_Tp_uv = Tp.isomorphism_with_fixed_basis(Tp.bases()[Integer(1)], codomain=W_Tp_uv)
>>> phi_Tp_uv
Generic morphism:
 From: Tangent space at Point p on the 2-dimensional differentiable manifold M
 To:   Ambient quadratic space of dimension 2 over Symbolic Ring
       Inner product matrix:
       [1/2   0]
       [  0 1/2]

>>> t1, t2 = Tp.tensor((Integer(1),Integer(0))), Tp.tensor((Integer(1),Integer(0)))
>>> t1[:] = (Integer(8), Integer(15))
>>> t2[:] = (Integer(47), Integer(11))
>>> t1[Tp.bases()[Integer(0)],:]
[8, 15]
>>> phi_Tp_xy(t1), phi_Tp_xy(t2)
((8, 15), (47, 11))
>>> phi_Tp_xy(t1).inner_product(phi_Tp_xy(t2))
541

>>> Tp_xy_to_uv = M.change_of_frame(c_xy.frame(), c_uv.frame()).at(p); Tp_xy_to_uv
Automorphism of the Tangent space at Point p on the
 2-dimensional differentiable manifold M
>>> Tp.set_change_of_basis(Tp.bases()[Integer(0)], Tp.bases()[Integer(1)], Tp_xy_to_uv)
>>> t1[Tp.bases()[Integer(1)],:]
[23, -7]
>>> phi_Tp_uv(t1), phi_Tp_uv(t2)
((23, -7), (58, 36))
>>> phi_Tp_uv(t1).inner_product(phi_Tp_uv(t2))
541