Tangent and Cotangent Bundles - Examples and Constructions

The pushforward and pullback operations transfer geometric data between manifolds via smooth maps. They provide the functorial properties that make differential geometry natural and coordinate-free.

DefinitionPushforward of Vectors

Let $f: M \to N$ be a smooth map. The pushforward or differential of $f$ at $p \in M$ is the linear map $f_*: T_pM \to T_{f(p)}N$ (also denoted $df_p$ ) defined by

(f_*v)(g) = v(g \circ f)

for $v \in T_pM$ and $g \in C^\infty(N)$ . In coordinates, if $v = v^i \frac{\partial}{\partial x^i}$ , then

f_*v = v^i \frac{\partial f^j}{\partial x^i} \frac{\partial}{\partial y^j}

where $(y^j)$ are coordinates on $N$ .

The pushforward encodes the linear approximation to $f$ at each point. It generalizes the Jacobian matrix from multivariable calculus to the manifold setting.

ExamplePushforward of Curves

If $\gamma: I \to M$ is a smooth curve, its velocity vector at $t$ is $\gamma'(t) \in T_{\gamma(t)}M$ . For a smooth map $f: M \to N$ , the chain rule gives

(f \circ \gamma)'(t) = f_*(\gamma'(t))

This shows that the pushforward correctly encodes the chain rule.

DefinitionPullback of Forms

Let $f: M \to N$ be smooth. The pullback of a covector $\omega \in T_{f(p)}^*N$ is the covector $f^*\omega \in T_p^*M$ defined by

(f^*\omega)(v) = \omega(f_*v)

for $v \in T_pM$ . This extends to differential forms by $f^*(\omega \wedge \eta) = f^*\omega \wedge f^*\eta$ and $f^*(d\omega) = d(f^*\omega)$ .

Remark

The pullback is contravariant: $(f \circ g)^* = g^* \circ f^*$ . While vectors push forward along maps, covectors and differential forms pull back. This duality reflects the functorial nature of tangent and cotangent bundles.

ExamplePullback of Exact Forms

If $f: M \to \mathbb{R}^n$ is smooth and $g$ is a function on $\mathbb{R}^n$ , then

f^*(dg) = d(g \circ f)

In coordinates, if $g = g(y^1, \ldots, y^n)$ , then

f^*(dg) = \frac{\partial g}{\partial y^j} \circ f \cdot d(f^j) = \frac{\partial g}{\partial y^j} \circ f \cdot \frac{\partial f^j}{\partial x^i} dx^i

DefinitionRelated Vector Fields

Two vector fields $X \in \mathfrak{X}(M)$ and $Y \in \mathfrak{X}(N)$ are $f$ -related for a smooth map $f: M \to N$ if

f_*(X_p) = Y_{f(p)}

for all $p \in M$ . Equivalently, $Y \circ f = f_* \circ X$ as maps $M \to TN$ .

TheoremProperties of Pushforward

The pushforward satisfies:

$(g \circ f)_* = g_* \circ f_*$ (functoriality)
$(\text{id}_M)_* = \text{id}_{TM}$ (identity)
If $X$ and $X'$ are $f$ -related to $Y$ and $Y'$ respectively, then $[X, X']$ is $f$ -related to $[Y, Y']$

These properties ensure that geometric constructions behave well under smooth maps, forming the foundation for naturality in differential geometry.