Tangent and Cotangent Bundles - Core Definitions

The tangent bundle provides the natural framework for studying velocities and derivatives on manifolds. It collects all tangent spaces into a single manifold structure, enabling global analysis of vector fields and differential equations.

DefinitionTangent Vector

A tangent vector at $p \in M$ is a linear map $v: C^\infty(M) \to \mathbb{R}$ satisfying the Leibniz rule:

v(fg) = v(f) \cdot g(p) + f(p) \cdot v(g)

for all smooth functions $f, g \in C^\infty(M)$ . The set of all tangent vectors at $p$ forms a vector space $T_pM$ called the tangent space.

This abstract definition captures the essence of differentiation: tangent vectors are directional derivatives. The Leibniz rule ensures they behave like derivatives under multiplication of functions.

DefinitionTangent Bundle

The tangent bundle of $M$ is

TM = \bigsqcup_{p \in M} T_pM = \{(p, v) : p \in M, v \in T_pM\}

with projection $\pi: TM \to M$ given by $\pi(p, v) = p$ . This carries a natural smooth manifold structure making $\pi$ a smooth map.

ExampleTangent Bundle of $\mathbb{R}^n$

For $M = \mathbb{R}^n$ , we have $T_p\mathbb{R}^n \cong \mathbb{R}^n$ for each $p$ , so $T\mathbb{R}^n \cong \mathbb{R}^n \times \mathbb{R}^n = \mathbb{R}^{2n}$ . A tangent vector at $p$ is a pair $(p, v)$ where $v \in \mathbb{R}^n$ represents the velocity direction.

The tangent bundle is a $2n$ -dimensional manifold when $M$ has dimension $n$ . Its charts are constructed from charts on $M$ via coordinate representations of tangent vectors.

DefinitionCoordinate Basis

Given a chart $(U, \varphi)$ with coordinates $(x^1, \ldots, x^n)$ , the coordinate vector fields $\frac{\partial}{\partial x^i}$ form a basis for each tangent space $T_pM$ with $p \in U$ . In this basis, a tangent vector is written

v = v^i \frac{\partial}{\partial x^i}

where $v^i = v(x^i)$ are the components.

Remark

The notation $\frac{\partial}{\partial x^i}$ is justified because this vector acts on functions by partial differentiation: $\frac{\partial}{\partial x^i}(f) = \frac{\partial(f \circ \varphi^{-1})}{\partial x^i}$ evaluated at $\varphi(p)$ .

DefinitionCotangent Space

The cotangent space $T_p^*M$ is the dual space of $T_pM$ , consisting of linear maps $\omega: T_pM \to \mathbb{R}$ . Elements are called covectors or cotangent vectors. The cotangent bundle is

T^*M = \bigsqcup_{p \in M} T_p^*M

ExampleDifferential of a Function

For $f \in C^\infty(M)$ , the differential $df_p: T_pM \to \mathbb{R}$ is a covector defined by $df_p(v) = v(f)$ . This generalizes the gradient from multivariable calculus. In coordinates, $df = \frac{\partial f}{\partial x^i} dx^i$ where $dx^i$ are the dual basis covectors.

The cotangent bundle provides the natural setting for differential forms, which are antisymmetric tensor products of covectors. Together, the tangent and cotangent bundles form the foundation for tensor analysis on manifolds.