Tangent and Cotangent Bundles - Key Properties

Vector fields and their flows provide the link between differential geometry and dynamical systems. They represent velocity fields on manifolds and generate one-parameter families of diffeomorphisms.

DefinitionVector Field

A vector field on $M$ is a smooth section $X: M \to TM$ of the tangent bundle, meaning $X$ assigns to each $p \in M$ a tangent vector $X_p \in T_pM$ smoothly. Equivalently, $X$ is a derivation on $C^\infty(M)$ : a linear map $X: C^\infty(M) \to C^\infty(M)$ satisfying

X(fg) = X(f) \cdot g + f \cdot X(g)

The space of vector fields, denoted $\mathfrak{X}(M)$ or $\Gamma(TM)$ , forms a module over $C^\infty(M)$ and a Lie algebra under the bracket operation.

DefinitionLie Bracket

The Lie bracket of two vector fields $X, Y$ is the vector field $[X, Y]$ defined by

[X, Y](f) = X(Y(f)) - Y(X(f))

for all $f \in C^\infty(M)$ . In local coordinates, if $X = X^i \frac{\partial}{\partial x^i}$ and $Y = Y^j \frac{\partial}{\partial x^j}$ , then

[X, Y] = \left(X^i \frac{\partial Y^j}{\partial x^i} - Y^i \frac{\partial X^j}{\partial x^i}\right) \frac{\partial}{\partial x^j}

Remark

The Lie bracket measures the failure of vector fields to commute. It satisfies $[X, Y] = -[Y, X]$ (antisymmetry) and the Jacobi identity $[X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0$ , making $\mathfrak{X}(M)$ a Lie algebra.

DefinitionIntegral Curve

An integral curve of a vector field $X$ is a smooth curve $\gamma: I \to M$ (where $I \subset \mathbb{R}$ is an interval) satisfying

\gamma'(t) = X_{\gamma(t)}

for all $t \in I$ . This is the manifold version of an ordinary differential equation.

ExampleIntegral Curves on $\mathbb{R}^2$

For the vector field $X = -y \frac{\partial}{\partial x} + x \frac{\partial}{\partial y}$ on $\mathbb{R}^2$ , the integral curves are circles centered at the origin. The differential equation is $\frac{dx}{dt} = -y$ , $\frac{dy}{dt} = x$ , with solutions $\gamma(t) = (r\cos(t + \theta_0), r\sin(t + \theta_0))$ .

DefinitionFlow of a Vector Field

The flow of a vector field $X$ is the collection of maps $\phi_t: M \to M$ defined by $\phi_t(p) = \gamma_p(t)$ , where $\gamma_p$ is the integral curve starting at $p$ with $\gamma_p(0) = p$ . When defined for all $t \in \mathbb{R}$ and all $p \in M$ , this forms a one-parameter group of diffeomorphisms.

TheoremExistence and Uniqueness of Flows

For any smooth vector field $X$ on $M$ and any point $p \in M$ , there exists a unique maximal integral curve $\gamma_p: I_p \to M$ with $\gamma_p(0) = p$ . If $M$ is compact, then $I_p = \mathbb{R}$ for all $p$ .

This fundamental result guarantees that vector fields generate local flows, which can be thought of as infinitesimal symmetries of the manifold. The compactness assumption ensures global existence.