tangent space

tangent space

Summary

The

tangent space

of

differential manifold

M

at a
point

x

∈

M

is the

vector space

whose

elements

are

velocities

of

trajectories

that

pass through

x

. The standard notation for the
tangent space of

M

at the point

x

is

T

x

⁢

M

.

Definition (Standard).

Let

M

be a differential manifold and

x

a point of

M

. Let

γ

i

:

I

i

→

M

,

I

i

⊂

ℝ

,

i

=

1

,

2

be
two

differentiable

trajectories passing through

x

at times

t

1

∈

I

1

,

t

2

∈

I

2

, respectively. We say that these trajectories are in

first order

contact at

x

if for all differentiable functions

f

:

U

→

ℝ

defined in some neighbourhood

U

⊂

M

of

x

, we have

(

f

∘

γ

1

)

′

⁢

(

t

1

)

=

(

f

∘

γ

2

)

′

⁢

(

t

2

)

.

First order
contact is an

equivalence relation

, and we define

T

x

⁢

M

, the

tangent space

of

M

at

x

, to be the set of corresponding

equivalence classes

.

Given a trajectory

γ

:

I

→

M

,

I

⊂

ℝ

passing through

x

at
time

t

∈

I

, we define

γ

˙

⁢

(

t

)

the

tangent vector

,
a.k.a. the

velocity

, of

γ

at time

t

, to be the
equivalence class of

γ

modulo first order contact. We endow

T

x

⁢

M

with the

structure

of a

real vector space

by identifying it
with

ℝ

n

relative to a system of local coordinates. These
identifications will differ from chart to chart, but they will all be
linearly

compatible

.

To describe this identification, consider a

coordinate chart

α

:

U

α

→

ℝ

n

,

U

α

⊂

M

,

x

∈

U

.

We call the

real vector

(

α

∘

γ

)

′

⁢

(

t

)

∈

ℝ

n

the representation of

γ

˙

⁢

(

t

)

relative to the chart

α

. It is a simple exercise
to show that two trajectories are in first order contact at

x

if and
only if their velocities have the same representation. Another simple
exercise will show that for every

𝐮

∈

ℝ

n

the trajectory

t

→

α

-

1

⁢

(

α

⁢

(

x

)

+

t

⁢

𝐮

)

has velocity

𝐮

relative to the chart

α

. Hence, every
element of

ℝ

n

represents some actual velocity, and therefore
the

mapping

T

x

⁢

M

→

ℝ

n

given by

[

γ

]

→

(

α

∘

γ

)

′

⁢

(

t

)

,

γ

⁢

(

t

)

=

x

,

is a

bijection

.

Finally if

β

:

U

β

→

ℝ

n

,

U

β

⊂

M

,

x

∈

U

β

is another chart, then for all differentiable
trajectories

γ

⁢

(

t

)

=

x

we have

(

β

∘

γ

)

′

⁢

(

t

)

=

J

⁢

(

α

∘

γ

)

′

⁢

(

t

)

,

where

J

is
the

Jacobian matrix

at

α

⁢

(

x

)

of the suitably restricted mapping

β

∘

α

-

1

:

α

⁢

(

U

α

∩

U

β

)

→

ℝ

n

. The
linearity of the above

relation

implies that the vector space structure of

T

x

⁢

M

is

independent

of the
choice of coordinate chart.

Definition (Classical).

Historically, tangent vectors were
specified as elements of

ℝ

n

relative to some system of

coordinates

, a.k.a. a coordinate chart. This point of view naturally
leads to the definition of a tangent space as

ℝ

n

modulo changes
of coordinates.

Let

M

be a differential manifold represented as a

collection

of
parameterization domains

{

V

α

⊂

ℝ

n

:

α

∈

𝒜

}

indexed by

labels
belonging to a set

𝒜

, and

transition function

diffeomorphisms

σ

α

⁢

β

:

V

α

⁢

β

→

V

β

⁢

α

,

α

,

β

∈

𝒜

,

V

α

⁢

β

⊂

V

α

Set

M

^

=

{

(

α

,

x

)

∈

𝒜

×

ℝ

n

:

x

∈

V

α

}

,

and recall that a points of the

manifold

are represented by elements
of

M

^

modulo an equivalence relation imposed by the transition functions
[see Manifold — Definition (Classical)].
For a transition function

σ

α

⁢

β

, let

J

⁢

σ

α

⁢

β

:

V

α

⁢

β

→

Mat

n

,

n

(

ℝ

)

denote the corresponding Jacobian matrix of

partial derivatives

. We
call a triple

(

α

,

x

,

𝐮

)

,

α

∈

𝒜

,

x

∈

V

α

,

𝐮

∈

ℝ

n

the representation of a tangent vector at

x

relative to

coordinate
system

α

, and make the identification

(

α

,

x

,

𝐮

)

≃

(

β

,

σ

α

⁢

β

⁢

(

x

)

,

[

J

⁢

σ

α

⁢

β

]

⁢

(

x

)

⁢

(

𝐮

)

)

,

α

,

β

∈

𝒜

,

x

∈

V

α

⁢

β

,

𝐮

∈

ℝ

n

.

to arrive at the definition of a tangent vector at

x

.

Notes.

The notion of tangent space derives from the observation
that there is no natural way to relate and compare velocities at
different points of a manifold. This is already evident when we
consider objects moving on a

surface

in 3-space, where the velocities
take their value in the

tangent planes

of the surface. On a general
surface, distinct points correspond to distinct tangent planes, and
therefore the velocities at distinct points are not commensurate.

The situation is even more complicated for an

abstract manifold

, where
absent an ambient

Euclidean

setting there is, apriori, no obvious
“tangent plane” where the velocities can reside. This point of view
leads to the definition of a velocity as some sort of equivalence
class.

See also:

tangent bundle,

connection

, parallel translation

Title

tangent space

Canonical name

TangentSpace

Date of creation

2013-03-22 12:21:04

Last modified on

2013-03-22 12:21:04

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

5

Author

rmilson (146)

Entry type

Definition

Classification

msc 53-00

Synonym

tangent vector

Related topic

Manifold

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