Tangent Space

A tangent space is a vector space associated with a point of a smooth manifold, representing the possible instantaneous directions and velocities of motion through that point. Written $T_pM$ for a manifold $M$ and a point $p$, it generalizes the tangent line to a curve and the tangent plane to a surface. If $M$ has dimension $n$, then $T_pM$ is an $n$-dimensional real vector space. Tangent spaces provide the local linear structures needed to define differentiation on manifolds. (web.stanford.edu)

Geometric interpretation

For a smooth submanifold $M$ of Euclidean space $\mathbb{R}^N$, the tangent space at $p$ can be identified with the collection of velocity vectors $\gamma'(0)$ of smooth curves lying in $M$ and satisfying $\gamma(0)=p$. This collection is a linear subspace of $\mathbb{R}^N$, with dimension equal to that of $M$, rather than necessarily that of the surrounding space. A local parametrization $\phi$, with $\phi(0)=p$, identifies it with the image of $d\phi_0$. (web.stanford.edu)

The vector space $T_pM$ must be distinguished from the translated affine space $p+T_pM$, which is the tangent line or plane drawn through $p$. The former contains the zero vector; the latter passes through the point of tangency. For an abstract manifold, an ambient Euclidean space is unnecessary: tangent vectors are defined intrinsically from the smooth structure. (ocw.mit.edu)

Definition using curves

On a smooth manifold without boundary, consider smooth curves

Two such curves represent the same tangent vector if their coordinate velocities agree:

where $\varphi$ is a coordinate chart around $p$. The chain rule ensures that agreement in one chart implies agreement in every chart. A tangent vector is thus an equivalence class of curves with the same first-order behavior at $p$. (math.stanford.edu)

Coordinates transport ordinary vector addition and scalar multiplication to these classes. Every coordinate velocity is realized by a local curve, so the resulting space has dimension $n$. Importantly, a tangent vector records velocity, not merely the unparametrized path: replacing $\gamma(t)$ by $\gamma(at)$ multiplies its velocity by $a$. Curves may therefore trace the same path while representing different tangent vectors. (math.stanford.edu)

Definition using derivations

An equivalent definition treats a tangent vector as a directional differentiation operator. Let $C^\infty(M)$ denote the algebra of smooth real-valued functions on $M$. A derivation at $p$ is a real linear map

satisfying the pointwise product rule

The vector space of these derivations is naturally identified with $T_pM$. One may equivalently work with functions defined near $p$, identifying functions that agree on a neighborhood of $p$. (math.stanford.edu)

The curve representing $v$ acts on a smooth function by

This connects geometric velocity with the directional derivative. The derivation formulation is particularly useful because it defines tangent vectors without reference to an embedding or a chosen parametrization. (math.stanford.edu)

Coordinates and differentials

Local coordinates $x^1,\ldots,x^n$ determine a basis

of $T_pM$. Consequently, every tangent vector has a unique expression

Its action on a function is computed using the partial derivatives of that function’s coordinate expression. Under a change to coordinates $y^j$, its components transform as

The components change, but the underlying vector does not. (math.mit.edu)

A smooth map $F:M\to N$ induces its differential, or pushforward,

It maps a curve velocity to the velocity of the composed curve $F\circ\gamma$. In coordinates, this linear map is represented by the Jacobian matrix. Differentials obey

extending ordinary differentiation to manifolds. (math.mit.edu)

Embedded examples and constraints

For the unit sphere $S^{n-1}\subset\mathbb{R}^n$,

Thus its tangent space is the orthogonal complement of the radial direction, using the Euclidean inner product. At the north pole of $S^2$, it consists of vectors $(a,b,0)$, whereas the affine tangent plane consists of points $(a,b,1)$. (math.ucla.edu)

More generally, if $M=F^{-1}(c)$ for a smooth map $F:\mathbb{R}^N\to\mathbb{R}^k$ and $c$ is a regular value, then

The tangent space is therefore the kernel of the linearized constraints. Its dimension is $N-k$, by the rank–nullity theorem. The regularity assumption matters: singular zero sets need not be manifolds, and the kernel of a defining differential need not describe their curve velocities. (web.stanford.edu)

Tangent bundles and additional structure

The disjoint union

forms the tangent bundle, a vector bundle over $M$. For an $n$-dimensional manifold its total space has dimension $2n$; locally it is identified with $U\times\mathbb{R}^n$. A smooth vector field assigns a tangent vector to every point smoothly. Local identifications do not imply a canonical global identification of all tangent spaces. (math.ucla.edu)

The dual space $T_p^*M$ is called the cotangent space. For a smooth scalar function, $df_p$ belongs to this dual space. A Riemannian metric tensor supplies an inner product on each tangent space, allowing lengths and angles to be measured and defining the gradient by

These metric notions require additional structure beyond the tangent space itself. (math.stanford.edu)