1.6 Vector fields and flows

Vector fields are fundamental in differential geometry, assigning tangent vectors to each point on a manifold. They describe infinitesimal behavior of curves and flows, crucial for analyzing geometry and topology. Understanding vector fields unlocks insights into manifold structure and dynamics.

Smooth vector fields form a vector space, with properties like linearity and scalar multiplication. They can be represented in local coordinates, enabling explicit calculations. Vector fields interact with differential forms through operations like interior product and Lie derivative, revealing deep connections in differential geometry.

Definition of vector fields

• Vector fields are a fundamental object of study in differential geometry, assigning a tangent vector to each point on a manifold
• They provide a way to describe the infinitesimal behavior of curves and flows on the manifold
• Understanding vector fields is crucial for analyzing the geometry and topology of manifolds

Tangent vectors on manifolds

• At each point $p$ on a manifold $M$, the tangent space $T_pM$ is a vector space consisting of tangent vectors to curves passing through $p$
• Tangent vectors can be thought of as velocity vectors of curves at the point $p$
• The collection of all tangent spaces forms the tangent bundle $TM$, a vector bundle over the manifold $M$

Smooth vector fields

• A vector field $X$ on a manifold $M$ is a smooth assignment of a tangent vector $X_p \in T_pM$ to each point $p \in M$
• Smoothness means that the components of $X$ in any local coordinate chart vary smoothly as functions of the coordinates
• The set of all smooth vector fields on $M$ is denoted by $\mathfrak{X}(M)$ and forms a vector space over $\mathbb{R}$

Local coordinate representations

• In a local coordinate chart $(U, (x^1, \ldots, x^n))$, a vector field $X$ can be expressed as a linear combination of the coordinate basis vectors: $X = \sum_{i=1}^n X^i \frac{\partial}{\partial x^i}$
• The coefficients $X^i$ are smooth functions on $U$ called the components of $X$ in the given coordinate chart
• The local coordinate representation allows for explicit calculations and analysis of vector fields

Properties of vector fields

• Vector fields possess various algebraic and analytical properties that make them a rich and versatile tool in differential geometry
• These properties enable the study of the geometry and topology of manifolds through the behavior of vector fields
• Understanding these properties is essential for working with vector fields in applications

Linearity and scalar multiplication

• The set of smooth vector fields $\mathfrak{X}(M)$ forms a vector space over $\mathbb{R}$
• For vector fields $X, Y \in \mathfrak{X}(M)$ and scalars $a, b \in \mathbb{R}$, the linear combination $aX + bY$ is defined pointwise: $(aX + bY)_p = aX_p + bY_p$
• Scalar multiplication of a vector field $X$ by a smooth function $f \in C^\infty(M)$ is defined pointwise: $(fX)_p = f(p)X_p$

Support and compact support

• The support of a vector field $X$ is the closure of the set of points where $X$ is non-zero: $\operatorname{supp}(X) = \overline{\{p \in M : X_p \neq 0\}}$
• A vector field $X$ is said to have compact support if its support is a compact subset of $M$
• Vector fields with compact support are particularly useful in the study of local properties and in the construction of partitions of unity

Divergence and curl

• The divergence of a vector field $X$ on a Riemannian manifold $(M, g)$ is a scalar function that measures the infinitesimal rate of change of volume under the flow of $X$: $\operatorname{div} X = \frac{1}{\sqrt{|g|}} \frac{\partial}{\partial x^i} (\sqrt{|g|} X^i)$
• The curl of a vector field $X$ on a 3-dimensional Riemannian manifold is a vector field that measures the infinitesimal rotation of $X$: $\operatorname{curl} X = (*dX^\flat)^\sharp$, where $\flat$ and $\sharp$ denote the musical isomorphisms and $*$ is the Hodge star operator
• Divergence and curl provide important information about the local behavior of vector fields and are related to conservation laws and the topology of the manifold

Vector fields and differential forms

• Vector fields and differential forms are dual objects on a manifold, with each providing complementary information about the geometry and topology
• The interplay between vector fields and differential forms is a fundamental aspect of modern differential geometry
• This duality is expressed through various operations, such as the interior product, contraction, and Lie derivative

Dual relationship with 1-forms

• The dual space to the tangent space $T_pM$ at each point $p \in M$ is the cotangent space $T_p^*M$, consisting of linear functionals on $T_pM$
• A differential 1-form $\omega$ on $M$ is a smooth assignment of a cotangent vector $\omega_p \in T_p^*M$ to each point $p \in M$
• The pairing between a vector field $X$ and a 1-form $\omega$ is given by the evaluation: $\omega(X) = \omega_p(X_p)$ at each point $p \in M$

Interior product and contraction

• The interior product (or contraction) of a vector field $X$ with a differential $k$-form $\omega$ is a $(k-1)$-form denoted by $i_X\omega$ or $X \lrcorner \omega$
• In local coordinates, $(i_X\omega)_{i_1 \ldots i_{k-1}} = X^j \omega_{j i_1 \ldots i_{k-1}}$
• The interior product satisfies various properties, such as linearity, the Leibniz rule, and compatibility with the exterior derivative

Lie derivative of differential forms

• The Lie derivative $\mathcal{L}_X\omega$ of a differential $k$-form $\omega$ along a vector field $X$ measures the change of $\omega$ under the flow of $X$
• It is defined as $\mathcal{L}_X\omega = \lim_{t \to 0} \frac{1}{t} (\varphi_t^*\omega - \omega)$, where $\varphi_t$ is the flow of $X$
• The Lie derivative satisfies the Leibniz rule, commutes with the exterior derivative, and is related to the interior product by Cartan's magic formula: $\mathcal{L}_X = d \circ i_X + i_X \circ d$

Integral curves and flows

• Integral curves and flows are fundamental concepts in the study of vector fields, describing the motion of points under the influence of a vector field
• They provide a way to visualize and analyze the global behavior of vector fields on a manifold
• The existence and uniqueness of integral curves and flows are guaranteed by the fundamental theorem of ordinary differential equations

Definition of integral curves

• An integral curve of a vector field $X$ on a manifold $M$ is a smooth curve $\gamma: I \to M$ (where $I \subseteq \mathbb{R}$ is an interval) such that $\gamma'(t) = X_{\gamma(t)}$ for all $t \in I$
• Integral curves are parametrized curves whose velocity vector at each point coincides with the value of the vector field at that point
• They can be thought of as the trajectories of particles moving under the influence of the vector field

Existence and uniqueness theorem

• The fundamental theorem of ordinary differential equations guarantees the existence and uniqueness of integral curves for a given vector field and initial condition
• For a smooth vector field $X$ on $M$ and a point $p \in M$, there exists a unique maximal integral curve $\gamma: I \to M$ such that $\gamma(0) = p$
• The theorem ensures that the flow of a vector field is well-defined and determines the global behavior of the vector field

Local and global flows

• The flow of a vector field $X$ on $M$ is a smooth map $\varphi: D \to M$, where $D \subseteq \mathbb{R} \times M$ is an open subset containing $\{0\} \times M$, such that for each $p \in M$, the curve $t \mapsto \varphi(t, p)$ is the unique maximal integral curve of $X$ starting at $p$
• The flow is defined locally around each point, and the domain $D$ may not be the entire $\mathbb{R} \times M$ due to the possibility of incomplete integral curves
• When the vector field is complete (i.e., all integral curves are defined for all $t \in \mathbb{R}$), the flow is global and defined on the entire $\mathbb{R} \times M$

One-parameter groups of diffeomorphisms

• For each $t \in \mathbb{R}$, the map $\varphi_t: M \to M$ defined by $\varphi_t(p) = \varphi(t, p)$ is a diffeomorphism of $M$
• The collection $\{\varphi_t\}_{t \in \mathbb{R}}$ forms a one-parameter group of diffeomorphisms, satisfying $\varphi_0 = \operatorname{id}_M$ and $\varphi_{t+s} = \varphi_t \circ \varphi_s$ for all $t, s \in \mathbb{R}$
• The one-parameter group of diffeomorphisms encodes the symmetries and dynamics of the vector field

Lie brackets and Lie derivatives

• Lie brackets and Lie derivatives are fundamental operations on vector fields that capture their infinitesimal behavior and relationships
• They play a crucial role in the study of symmetries, integrability, and the geometry of manifolds
• Understanding Lie brackets and Lie derivatives is essential for working with vector fields and their applications

Lie bracket of vector fields

• The Lie bracket of two vector fields $X, Y \in \mathfrak{X}(M)$ is another vector field $[X, Y] \in \mathfrak{X}(M)$ that measures the failure of $X$ and $Y$ to commute
• In local coordinates, $[X, Y]^i = X^j \frac{\partial Y^i}{\partial x^j} - Y^j \frac{\partial X^i}{\partial x^j}$
• The Lie bracket satisfies antisymmetry ($[X, Y] = -[Y, X]$) and the Jacobi identity ($[X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0$)

Jacobi identity and Lie algebra

• The Jacobi identity is a crucial property of the Lie bracket, ensuring that the space of vector fields $\mathfrak{X}(M)$ forms a Lie algebra
• A Lie algebra is a vector space equipped with a bilinear operation (the Lie bracket) satisfying antisymmetry and the Jacobi identity
• The Lie algebra structure of vector fields encodes the infinitesimal symmetries and deformations of the manifold

Lie derivative of vector fields

• The Lie derivative $\mathcal{L}_XY$ of a vector field $Y$ along another vector field $X$ is a vector field that measures the change of $Y$ under the flow of $X$
• It is defined as $\mathcal{L}_XY = \lim_{t \to 0} \frac{1}{t} (\varphi_{-t}^*Y - Y)$, where $\varphi_t$ is the flow of $X$
• The Lie derivative satisfies various properties, such as linearity, the Leibniz rule, and compatibility with the Lie bracket

Relation to Lie brackets and flows

• The Lie bracket of two vector fields $X$ and $Y$ can be expressed in terms of their Lie derivatives: $[X, Y] = \mathcal{L}_XY - \mathcal{L}_YX$
• The flow of a vector field $X$ is related to its Lie derivative by the formula $\frac{d}{dt} \varphi_t^*Y = \varphi_t^*(\mathcal{L}_XY)$, where $\varphi_t$ is the flow of $X$
• These relations highlight the deep connections between Lie brackets, Lie derivatives, and the geometry of vector fields and flows

Invariant submanifolds and foliations

• Invariant submanifolds and foliations are geometric structures that capture the symmetries and decompositions of a manifold with respect to a given vector field or distribution
• They provide a way to study the global behavior of vector fields and their integral curves
• The Frobenius theorem is a fundamental result that characterizes the integrability of distributions and the existence of invariant submanifolds and foliations

Invariant submanifolds under flows

• A submanifold $N \subseteq M$ is said to be invariant under the flow of a vector field $X$ if for every $p \in N$ and $t \in \mathbb{R}$ such that $\varphi_t(p)$ is defined, we have $\varphi_t(p) \in N$
• Invariant submanifolds are preserved by the flow of the vector field and contain entire integral curves
• They play a crucial role in the study of dynamical systems and the qualitative behavior of vector fields

Frobenius theorem and integrability

• A distribution $\Delta$ on a manifold $M$ is a smooth assignment of a subspace $\Delta_p \subseteq T_pM$ to each point $p \in M$
• A distribution is called involutive if for any two vector fields $X, Y$ tangent to $\Delta$, their Lie bracket $[X, Y]$ is also tangent to $\Delta$
• The Frobenius theorem states that a distribution $\Delta$ is integrable (i.e., there exists a foliation of $M$ tangent to $\Delta$) if and only if it is involutive

Codimension-one foliations

• A codimension-one foliation of a manifold $M$ is a decomposition of $M$ into disjoint connected submanifolds (called leaves) of codimension one
• Codimension-one foliations are particularly important in the study of contact structures and Reeb vector fields
• They can be defined by integrable codimension-one distributions or by the kernels of non-vanishing 1-forms

Reeb vector fields and contact structures

• A contact structure on a manifold $M$ of odd dimension $2n+1$ is a maximally non-integrable codimension-one distribution $\xi$
• A contact form $\alpha$ is a 1-form such that $\alpha \wedge (d\alpha)^n \neq 0$ everywhere, and its kernel defines a contact structure
• The Reeb vector field $R_\alpha$ associated with a contact form $\alpha$ is the unique vector field satisfying $\alpha(R_\alpha) = 1$ and $i_{R_\alpha} d\alpha = 0$
• Reeb vector fields and contact structures play a central role in contact geometry and have applications in mechanics and thermodynamics

Applications and examples

• Vector fields and their associated concepts have numerous applications in various areas of mathematics and physics
• They provide a powerful framework for studying the geometry, topology, and dynamics of manifolds
• Some notable applications include Hamiltonian mechanics, Riemannian geometry, and the study of symmetries and conservation laws

Hamiltonian vector fields in symplectic geometry

• A symplectic manifold $(M, \omega)$ is an even-dimensional manifold $M$ equipped with a closed, non-degenerate 2-form $\omega$
• For a smooth function $H \in C^\infty(M)$, the Hamiltonian vector field $X_H$ is defined by the equation $i_{X_H}\omega = dH$
• Hamiltonian vector fields describe the dynamics of conservative mechanical systems and are a fundamental object of study in symplectic geometry

Geodesic flows on Riemannian manifolds

• On a Riemannian manifold $(M, g)$, the geodesic flow is a vector field on the tangent bundle $TM$ that describes the motion of particles along geodesics
• The geodesic flow is defined by the Hamiltonian vector field of the energy function $E(v) = \frac{1}{2} g(v, v)$ on $TM$ with respect to the canonical symplectic structure
• Geodesic flows encode the geometry of the Riemannian manifold and have applications in mechanics and optics

Killing vector fields and isometries

• A Killing vector field on a Riemannian manifold $(M, g)$ is a vector field $X$ that preserves the metric, i.e., $\mathcal{L}_Xg = 0$
• Killing vector fields are the infinit