Tangent Spaces and Vector Fields

On a smooth manifold there is no ambient space to point arrows into — the manifold is all there is. Yet we still want directions: which way does a particle move, which way does temperature increase, which way is "downhill"? The answer at each point p is a whole vector space of directions, the tangent space T_pM. Glue one such space to every point and let it vary smoothly and you get a vector field — the object behind fluid flow, force fields, and the differential equations of motion.

The subtlety is defining "a direction" without ever leaving the surface. There are three standard ways to do it, and — pleasingly — they all describe the same space.

Three pictures of one tangent vector

(i) Velocities of curves. The most physical picture. Take a smooth curve \gamma : (-\varepsilon, \varepsilon) \to M that passes through p at time 0, so \gamma(0) = p. Its velocity there, \gamma'(0), is a tangent vector. Every direction you can leave p in is the velocity of some curve; two curves define the same tangent vector when they agree to first order.

(ii) Derivations. The most algebraic picture. A tangent vector is an operator that eats a smooth function and returns the rate of change of that function in its direction — a directional derivative. Abstractly, a derivation at p is a linear map v on smooth functions obeying the Leibniz product rule

v(fg) = f(p)\,v(g) + g(p)\,v(f).

Anything satisfying linearity and this rule is a tangent vector — no picture of an arrow required.

(iii) The coordinate basis. The most computational picture. Pick a chart with local coordinates (x^1, \dots, x^n) near p. The partial-derivative operators

\left. \frac{\partial}{\partial x^1} \right|_p, \ \dots, \ \left. \frac{\partial}{\partial x^n} \right|_p

form a basis of T_pM. Every tangent vector is a unique combination of them,

v = \sum_{i=1}^{n} v^i \, \frac{\partial}{\partial x^i} \Bigg|_p ,

with real components v^1, \dots, v^n. There are exactly n basis vectors — one per coordinate — which is why the tangent space has dimension n.

At each point p of a smooth n-manifold M:

Worked examples

The plane. For M = \mathbb{R}^2 with coordinates (x, y), the tangent space at any point has basis \partial/\partial x,\ \partial/\partial y, so T_p\mathbb{R}^2 \cong \mathbb{R}^2. Here every tangent space looks like a copy of the plane itself — which is exactly why flat space lets you get away with thinking of vectors as arrows you can slide around freely.

The sphere. For M = S^2 and a point p, the tangent space T_pS^2 is the plane just grazing the sphere at p — the flat sheet touching it there. It is 2-dimensional, matching \dim S^2 = 2, and it tilts as p moves.

A vector field. A vector field X assigns to each point p a tangent vector X_p \in T_pM, varying smoothly. In coordinates,

X = \sum_{i=1}^{n} X^i(x) \, \frac{\partial}{\partial x^i},

with smooth component functions X^i. On the plane, the spinning field X = -y\,\partial/\partial x + x\,\partial/\partial y gives an arrow at every point circulating counter-clockwise — the velocity field of a rigid rotation. On the circle S^1, the unit tangent field points the same way all the way round and never vanishes.

See a tangent vector and a field

Take the circle S^1. At a point p the tangent space T_pS^1 is the line just touching the circle there — a 1-dimensional space. A curve running around the circle has a velocity \gamma'(0) lying in that line. Do this at every point and the velocities assemble into a smooth vector field that never vanishes — something a sphere, by the hairy-ball theorem, can never manage. Step through: the tangent line, then the velocity vector, then the whole field.

Moving vectors between points — the differential

A smooth map F : M \to N carries curves to curves, and so carries their velocities to velocities. This is the pushforward (or differential)

dF_p : T_pM \longrightarrow T_{F(p)}N,

a linear map between tangent spaces. If v = \gamma'(0) then dF_p(v) = (F \circ \gamma)'(0): push the curve forward, then take its velocity. In coordinates dF_p is exactly the Jacobian matrix of F at p. It is the manifold version of "the derivative is the best linear approximation" — the derivative of a map between curved spaces is a linear map between their tangent spaces.

A vector field on S^2 is a way of laying a hair flat at every point of a sphere. The hairy ball theorem (Poincaré, Brouwer) says this is impossible without a bald spot: every continuous tangent vector field on an even-dimensional sphere must vanish somewhere. Comb a coconut and you always get at least one cowlick.

The circle escapes this — you can comb S^1 smoothly all the way round, which is why the figure above has an unbroken field. The obstruction on S^2 is topological, measured by the Euler characteristic (\chi(S^2) = 2 \neq 0); the torus, with \chi(T^2) = 0, can be combed flat. Meteorologically, the theorem promises that somewhere on Earth the horizontal wind is always exactly zero — the calm eye of a cyclone that must exist.

A tangent vector belongs to one point. The velocity of a curve at p lives in T_pM; a velocity at a different point q lives in a different vector space T_qM. These are separate spaces, and there is no canonical way to add a vector at p to a vector at q, or even to ask whether they are "the same direction".

In flat \mathbb{R}^n we cheat: we slide arrows around and compare them, because all the tangent spaces are secretly identified with one another. On a curved manifold that identification does not exist for free. Comparing tangent vectors at different points requires extra structure — a connection (parallel transport) — and the answer generally depends on the path you carry the vector along. That path-dependence is precisely curvature. So resist the urge to subtract X_p from X_q: without a connection, the expression is meaningless.