Tangent space

The tangent space of a differentiable manifold M is a vector space at a point p on the manifold whose elements are the tangent vectors (or velocities) to the curves passing through that point p. The tangent space at this point p is usually denoted $T_{p}M$.

The tangent space is necessary for a manifold because it offers a way in which tangent vectors at different points on the manifold can be compared (via an affine connection). If the manifold is a submanifold of $\scriptstyle \mathbb {R} ^{n}$, then the tangent space at a point can be thought of as an n-dimensional hyperplane at that point. However, this ambient Euclidean space is unnecessary to the definition of the tangent space.

The tangent space at a point has the same dimension as the manifold, and the union of all the tangent spaces of a manifold is called the tangent bundle, which itself is a manifold of twice the dimension of M.

Definition

Although it is tempting to define a tangent space as a "space where tangent vectors live", without a definition of a tangent space there is no definition of a tangent vector. There are various ways in which a tangent space can be defined, the most intuitive of which is in terms of directional derivatives; the space $T_{p}M$ is the space identified with directional derivatives at p.

Directional derivative

A curve on the manifold is defined as a differentiable map $\scriptstyle \gamma :(a,b)\rightarrow M$. Let $\scriptstyle \gamma (t_{0})\,=\,p$. If one defines $\scriptstyle {\mathcal {F}}_{p}$ to be all the functions $\scriptstyle f:M\rightarrow \mathbb {R} ^{n}$ that are differentiable at the point p, then one can interpret

$\gamma '(t_{0}):\,{\mathcal {F}}_{p}\rightarrow \mathbb {R}$

to be an operator such that

$\gamma '(t_{0})(f)=(f\circ \gamma )'(t_{0})=\lim _{h\rightarrow 0}{\frac {f(\gamma (t_{0}+h))-f(\gamma (t_{0}))}{h}}$

and is a directional derivative of f in the direction of the curve $\scriptstyle \gamma$. This operator can be interpreted as a tangent vector.

The tangent space is then the set of all directional derivatives of curves at the point p.