Tangent and Cotangent Spaces

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The tangent space $T_pM$ is the vector space of possible first-order motions through a point $p$ , defined without embedding the manifold in a larger space. Curve velocities and derivations of smooth functions give equivalent constructions; coordinates merely supply bases and components. The cotangent space $T_p^*M$ is the algebraic dual, whose elements measure tangent vectors and provide the natural home for differentials.

Required background. Calculus and Linear Algebra for Relativity — use the chain rule, directional derivatives, bases, dual spaces, and linear maps.

Tangent vectors from curves

Let $M$ be a smooth $n$ -manifold and $p\in M$ . Two smooth curves $\gamma_1,\gamma_2:(-\epsilon,\epsilon)\to M$ through $p$ are tangent at $p$ when, in one—and hence every—chart $x=(x^1,\ldots,x^n)$ around $p$ ,

\left.\frac{d(x^i\circ\gamma_1)}{dt}\right|_{0} = \left.\frac{d(x^i\circ\gamma_2)}{dt}\right|_{0}

for all $i$ . An equivalence class $[\gamma]$ is a tangent vector. Its coordinate components are $v^i=d(x^i\circ\gamma)/dt|_0$ , and

v=v^i\left.\frac{\partial}{\partial x^i}\right|_p.

Under new coordinates $y^a=y^a(x)$ , the chain rule gives $v'^a=(\partial y^a/\partial x^i)_p v^i$ . Components change, whereas the velocity class does not. A rigorous development of this construction and the coordinate transformation is given in Gualtieri 2017, §§2.1–2.5, pp. 14–23, PDF.

Derivations and equivalence

A derivation at $p$ is a linear map $D:C^\infty(M)\to\mathbb R$ satisfying the Leibniz rule

D(fg)=f(p)Dg+g(p)Df.

A curve velocity defines $D_{[\gamma]}f=d(f\circ\gamma)/dt|_0$ . To go the other way, first note that a derivation is local: if $h$ vanishes near $p$ , choose a smooth cutoff $\chi$ supported in that neighborhood with $\chi=1$ near $p$ . Since $h=(1-\chi)h$ , the Leibniz rule gives $Dh=0$ . Thus functions that agree near $p$ have the same derivative. A coordinate function $x^i$ , although defined only on a chart, may therefore be extended smoothly after multiplying by a cutoff; $Dx^i$ means the result of any such extension.

Choose the chart centered at $p$ and put $v^i=Dx^i$ . In a sufficiently small coordinate ball, the fundamental theorem of calculus gives the exact decomposition

f(x)-f(0)=x^i h_i(x), \qquad h_i(x)=\int_0^1\partial_i f(tx)\,dt.

Here $h_i(0)=\partial_i f(p)$ . A derivation annihilates constants because $D1=D(1\cdot1)=2D1$ . Applying the Leibniz rule to the decomposition therefore gives

Df=v^i\partial_i f(p).

Thus every derivation is the directional derivative of a curve such as $x^i(\gamma(t))=x^i(p)+tv^i$ for sufficiently small $t$ . The result is independent of the selected chart because both sides act identically on every smooth function. Curve velocities express motion; derivations express how that motion probes observables. The localization and product-rule argument is developed in Gualtieri 2017, §4.2, Theorem 4.15, pp. 45–47, PDF.

For an embedded surface $S=F^{-1}(c)\subset\mathbb R^m$ at a regular point, this intrinsic construction agrees with the familiar constraint calculation:

T_pS=\ker(dF_p).

For $S^2=\{X^2+Y^2+Z^2=1\}$ , $dF_p(v)=2(Xv_X+Yv_Y+Zv_Z)$ , so $T_pS^2$ consists of ambient vectors satisfying $p\mathbin{\cdot}v=0$ . The dot product here belongs to the ambient Euclidean example; it is not part of the abstract tangent-space definition.

Covectors and dual bases

The cotangent space is $T_p^*M=\operatorname{Hom}(T_pM,\mathbb R)$ . The coordinate covectors $dx^i|_p$ form the basis dual to $\partial_i|_p$ :

dx^i(\partial_j)=\delta^i{}_j.

If $f$ is smooth, its differential at $p$ is the covector $df_p(v)=v(f)$ . In coordinates, $df=(\partial_i f)dx^i$ . Coordinate covariance follows immediately:

dy^a=\frac{\partial y^a}{\partial x^i}dx^i, \qquad \frac{\partial}{\partial y^a} =\frac{\partial x^i}{\partial y^a}\frac{\partial}{\partial x^i}.

The inverse Jacobians ensure that the pairing $\alpha(v)=\alpha_i v^i$ is unchanged. This dual transformation is why gradients written as component arrays should not be confused with tangent vectors: $df$ is naturally a covector. Turning it into a vector later requires a nondegenerate metric.

What is and is not canonical

Each $T_pM$ has vector-space operations, but there is generally no canonical way to compare $v\in T_pM$ with $w\in T_qM$ when $p\ne q$ . Coordinates can make their components look comparable, yet a coordinate change can alter that comparison independently at the two points. A connection will later supply parallel transport, but it is extra geometric structure.

Similarly, no canonical identification $T_pM\cong T_p^*M$ exists. Finite dimensionality identifies each space with its double dual, not with its dual. The distinction prevents silent raising and lowering of indices before a metric is introduced. Tangent and cotangent bundles collect the spaces pointwise, $TM=\bigsqcup_pT_pM$ and $T^*M=\bigsqcup_pT_p^*M$ ; smooth sections of these bundles are vector fields and one-forms.

A coordinate-invariance check

Consider the real line with coordinates related by $y=x^3$ near a point with nonzero $x$ . At the point $x=1$ , let $v=2\partial_x$ and $\alpha=5\,dx$ . Since $dy=3\,dx$ and $\partial_y=\tfrac13\partial_x$ there,

v=6\partial_y, \qquad \alpha=\frac53\,dy, \qquad \alpha(v)=5\cdot2=\frac53\cdot6=10.

The components change in opposite ways, while the pairing stays fixed. The factor $3$ is evaluated at $x=1$ ; at a different point the Jacobian is $3x^2$ .

This elementary cancellation is the practical meaning of coordinate invariance. It also explains why a component list without its basis is incomplete: the list alone changes under reparametrization. At $x=0$ , the cubic map is still a homeomorphism, but its inverse is not smooth. It is therefore not a smooth chart compatible with the usual smooth structure there. That failure belongs to the coordinate choice, not to the tangent space.

For a smooth map between manifolds, the differential acts on these intrinsic velocities and obeys the chain rule. Its dual acts on covectors in the reverse direction. This pair of operations, rather than an unmotivated component rule, will explain pushforwards and pullbacks on the next page.

The distinction between tangent spaces at different points is equally important. A curve provides its own velocity at each point, but it does not canonically identify those velocities with one another. Differentiating a vector field along the curve will require a connection; differentiating a scalar along it already makes sense because scalars can be compared directly.

Exercises

At the north pole $N=(0,0,1)$ of $S^2$ , find the tangent vector of $\gamma(t)=(\sin t,0,\cos t)$ and evaluate $dZ_N$ on it. Then explain why this result does not require a metric on $S^2$ .

Solution

Differentiating in the ambient coordinates gives $\dot\gamma(0)=(1,0,0)$ , which satisfies $N\mathbin{\cdot}\dot\gamma(0)=0$ and therefore lies in $T_NS^2$ . Since $Z\circ\gamma=\cos t$ ,

dZ_N(\dot\gamma)=\left.\frac{d}{dt}\cos t\right|_0=0.

The calculation uses the differential of the restricted coordinate function $Z|_{S^2}$ acting on a curve velocity. The ambient dot product was only a convenient check of tangency; neither the velocity class nor $dZ_N(\dot\gamma)$ requires an intrinsic metric.

References

Gualtieri, Marco. Geometry and Topology I. University of Toronto course notes, 2017, §§2.1–2.5, pp. 14–23; §4.2, pp. 45–47, PDF.