Tangent space ( $T_{p} M$ )

The tangent space

T_{p} M

is the vector space of tangent vectors to a smooth manifold

M

at

p

. For an embedded manifold

M \subseteq R^{N}

, it is

T_{p} M = {γ^{'} (0) : γ : (- ε, ε) \to M is smooth and γ (0) = p} .

For a regular parametrized surface

X (u, v)

, it is spanned by

X_{u}

and

X_{v}

.

Table of contents

Tangent space by point derivations ( $T_{x} M = Der_{x} (C_{x}^{\infty} (M), R)$ )

A tangent vector is a linear map on germs of smooth functions satisfying

D (ab) = a (x) D (b) + b (x) D (a)

. In a manifold chart, first-order expansion shows

D h = \sum_{i} D (u^{i}) \partial_{i} h (x)

. Thus the coordinate derivations form a basis, and the dimension equals the manifold dimension. Composition of germs with a smooth map defines its differential of a smooth map without choosing charts.

Cotangent space ( $T_{p}^{*} M$ )

The cotangent space at

p

is the dual vector space

T_{p}^{*} M = (T_{p} M)^{*}

. Its elements are covectors on the tangent space. In a manifold chart

x

, the basis

d x^{i} ∣_{p}

is dual to

\partial_{i} ∣_{p}

. Under a coordinate change to

y

, the coefficients of

α = a_{i} d x^{i} = b_{j} d y^{j}

obey

b_{j} = \sum_{i} a_{i} \partial x^{i} / \partial y^{j}

. The disjoint union of these spaces is the cotangent bundle.

Inner product on exterior powers of a cotangent space ( $⟨ ξ_{1} \land \dots \land ξ_{p}, ζ_{1} \land \dots \land ζ_{p} ⟩ = det (⟨ ξ_{i}, ζ_{j} ⟩)$ )

A Riemannian metric induces a dual inner product on each cotangent space. On decomposable covectors define

⟨ ξ_{1} \land \dots \land ξ_{p}, ζ_{1} \land \dots \land ζ_{p} ⟩ = det (⟨ ξ_{i}, ζ_{j} ⟩)

and extend bilinearly. Increasing wedges of an orthonormal coframe form an orthonormal basis. This positive-definite inner product is the one used to define the Hodge star operator.

Tangent space from a local parametrization

If

ϕ : U \subseteq R^{d} \to M \subseteq R^{n}

is a local parametrization with

ϕ (u) = p

, then

T_{p} M = im D ϕ_{u} .

The derivative has rank

d

, so this is a

d

-dimensional vector subspace. A change of parametrization composes

D ϕ_{u}

with an invertible coordinate-change derivative, leaving its image unchanged.

Closedness of tangent spaces under convergent base points

For an embedded smooth manifold

M \subseteq R^{n}

, if

p_{i} \to p

,

w_{i} \in T_{p_{i}} M

, and

w_{i} \to w

, then

w \in T_{p} M

. In a local graph representation

M = {(u, g (u))}

, write

w_{i} = (v_{i}, D g (u_{i}) v_{i})

and use continuity of

D g

.

Coordinate basis ( $\partial / \partial x^{μ}$ )

Local coordinates

x^{μ}

induce the coordinate-basis tangent vectors

\partial_{μ} = \partial / \partial x^{μ}

. Under a coordinate change they transform by the chain rule.

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Tangent space by

Wikipedia Bot  1

In mathematics, particularly in differential geometry, the concept of tangent space is fundamental to understanding the local properties of differentiable manifolds. ### Definition The **tangent space** at a point on a manifold is a vector space that consists of the tangent vectors at that point. Intuitively, you can think of it as the space of all possible directions in which you can tangentially pass through a given point on the manifold. ### Formal Construction 1.

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Tangent space by

Ciro Santilli  40 Updated 2025-07-16

TODO what's the point of it.

Bibliography:

www.youtube.com/watch?v=j1PAxNKB_Zc Manifolds #6 - Tangent Space (Detail) by WHYB maths (2020). This is worth looking into.
- www.youtube.com/watch?v=oxB4aH8h5j4 actually gives a more concrete example. Basically, the vectors are defined by saying "we are doing the Directional derivative of any function along this direction".
  One thing to remember is that of course, the most convenient way to define a function $f$ and to specify a direction, is by using one of the coordinate charts.
  We can then just switch between charts by change of basis.
jakobschwichtenberg.com/lie-algebra-able-describe-group/ by Jakob Schwichtenberg
math.stackexchange.com/questions/1388144/what-exactly-is-a-tangent-vector/2714944 What exactly is a tangent vector? on Stack Exchange

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