7.2 Vector bundles

Vector bundles are a key concept in differential geometry, generalizing vector spaces over topological spaces. They bridge local and global properties of manifolds, providing a framework for studying geometric structures and invariants.

This topic explores various aspects of vector bundles, including their definition, construction methods, and operations. It also delves into connections, curvature, and characteristic classes, highlighting their importance in mathematics and physics applications.

Definition of vector bundles

• Vector bundles are a fundamental concept in differential geometry and topology that generalize the notion of a vector space varying continuously over a topological space
• They provide a framework for studying geometric structures and invariants associated with manifolds and serve as a bridge between local and global properties

Fiber bundles with vector spaces

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• A vector bundle is a specific type of fiber bundle where the fibers are vector spaces of a fixed dimension $k$
• The total space $E$ of a vector bundle is a topological space that locally looks like a product of the base space $B$ and a vector space $V$
• The projection map $\pi: E \to B$ assigns to each point in the total space its corresponding point in the base space
• The fibers $\pi^{-1}(b)$ over each point $b \in B$ are isomorphic to the typical fiber $V$

Trivial vs non-trivial bundles

• A vector bundle is called trivial if it is isomorphic to the product bundle $B \times V$, where the fibers are simply copies of the same vector space $V$ at each point
• Non-trivial bundles, on the other hand, have a more complex structure and cannot be globally trivialized
• The Möbius band is a classic example of a non-trivial vector bundle over the circle $S^1$, where the fibers twist as one moves around the base space

Transition functions

• To describe a vector bundle, one often uses local trivializations and transition functions between them
• A local trivialization is a homeomorphism $\phi_i: \pi^{-1}(U_i) \to U_i \times V$ over an open set $U_i \subset B$, which identifies the fibers over $U_i$ with the typical fiber $V$
• Transition functions $g_{ij}: U_i \cap U_j \to GL(V)$ are defined on the overlaps of local trivializations and describe how the fibers are glued together
• They satisfy the compatibility condition $\phi_j \circ \phi_i^{-1}(b, v) = (b, g_{ij}(b)v)$ for $b \in U_i \cap U_j$ and $v \in V$

Cocycle condition

• The transition functions of a vector bundle must satisfy the cocycle condition, which ensures the consistency of the bundle structure
• For any triple overlap $U_i \cap U_j \cap U_k$, the cocycle condition requires that $g_{ij}g_{jk} = g_{ik}$
• This condition guarantees that the composition of transition functions around any closed loop in the base space is the identity, preventing any global inconsistencies

Constructions of vector bundles

• There are several ways to construct new vector bundles from existing ones, which allows for a rich variety of examples and applications
• These constructions often preserve or reflect important properties of the original bundles and provide tools for studying their geometry and topology

Pullback bundles

• Given a vector bundle $\pi: E \to B$ and a continuous map $f: B' \to B$, the pullback bundle $f^*E$ is a vector bundle over $B'$ obtained by "pulling back" the fibers of $E$ along $f$
• The total space of $f^*E$ is defined as the set of pairs $(b', e)$ such that $f(b') = \pi(e)$, and the projection map sends $(b', e)$ to $b'$
• Pullback bundles allow for the transfer of geometric structures from one base space to another and play a crucial role in the functoriality of vector bundles

Whitney sum

• The Whitney sum of two vector bundles $\pi_1: E_1 \to B$ and $\pi_2: E_2 \to B$ over the same base space is a new vector bundle $E_1 \oplus E_2$ over $B$
• The fiber over each point $b \in B$ is the direct sum of the fibers $(\pi_1^{-1}(b), \pi_2^{-1}(b))$
• The Whitney sum corresponds to the fiberwise direct sum of vector spaces and is a fundamental operation in K-theory

Tensor product

• The tensor product of two vector bundles $\pi_1: E_1 \to B$ and $\pi_2: E_2 \to B$ is a vector bundle $E_1 \otimes E_2$ over $B$ whose fibers are the tensor products of the corresponding fibers of $E_1$ and $E_2$
• The tensor product of vector bundles is compatible with the tensor product of vector spaces and provides a way to combine local data into global geometric objects

Dual bundle

• The dual bundle of a vector bundle $\pi: E \to B$ is a vector bundle $E^*$ over $B$ whose fibers are the dual vector spaces of the fibers of $E$
• For each $b \in B$, the fiber of $E^*$ over $b$ is the dual space $(\pi^{-1}(b))^*$, consisting of linear functionals on the fiber of $E$ over $b$
• The dual bundle is a contravariant functor and is related to the cotangent bundle in differential geometry

Determinant bundle

• The determinant bundle of a rank $k$ vector bundle $\pi: E \to B$ is a line bundle (rank 1 vector bundle) $\det(E)$ over $B$ whose fibers are the top exterior powers of the fibers of $E$
• For each $b \in B$, the fiber of $\det(E)$ over $b$ is the one-dimensional vector space $\Lambda^k(\pi^{-1}(b))$
• The determinant bundle encodes information about the orientation and volume of the fibers and is related to the Chern classes of the vector bundle

Sections of vector bundles

• Sections of a vector bundle are a fundamental object of study in the theory of vector bundles and provide a way to define and analyze geometric structures on manifolds
• They generalize the notion of vector fields and differential forms and play a crucial role in the formulation of differential equations and variational problems

Definition and properties

• A section of a vector bundle $\pi: E \to B$ is a continuous map $s: B \to E$ such that $\pi \circ s = \text{id}_B$
• Intuitively, a section assigns to each point $b \in B$ a vector in the fiber $\pi^{-1}(b)$ over $b$ in a continuous way
• The space of sections of a vector bundle is an infinite-dimensional vector space over the field of scalars (usually $\mathbb{R}$ or $\mathbb{C}$)
• Sections can be added pointwise and multiplied by scalar functions on the base space, making them a module over the ring of functions on $B$

Sheaf of sections

• The sections of a vector bundle form a sheaf over the base space $B$, which captures the local-to-global properties of the bundle
• For each open set $U \subset B$, the set of sections over $U$ is denoted by $\Gamma(U, E)$ and forms a vector space over the field of scalars
• The restriction maps between sections over different open sets satisfy the sheaf axioms, making the assignment $U \mapsto \Gamma(U, E)$ a sheaf of vector spaces
• The sheaf of sections is a fundamental object in the study of vector bundles and their cohomology

Frames and trivializations

• A frame of a vector bundle $\pi: E \to B$ is a set of sections $\{s_1, \ldots, s_k\}$ that form a basis for the fibers at each point $b \in B$
• The existence of a global frame is equivalent to the triviality of the vector bundle, as it provides a global isomorphism between $E$ and the product bundle $B \times \mathbb{R}^k$
• Local frames over open sets $U \subset B$ correspond to local trivializations of the bundle and are used to define transition functions between different local trivializations

Morphisms between vector bundles

• A morphism between two vector bundles $\pi_1: E_1 \to B$ and $\pi_2: E_2 \to B$ over the same base space is a continuous map $\varphi: E_1 \to E_2$ that is linear on each fiber and commutes with the projection maps
• In other words, for each $b \in B$, the restriction of $\varphi$ to the fiber $\pi_1^{-1}(b)$ is a linear map to the fiber $\pi_2^{-1}(b)$, and $\pi_2 \circ \varphi = \pi_1$
• Morphisms between vector bundles form a category, with composition given by the usual composition of maps and identity morphisms given by the identity maps on each bundle
• Isomorphisms in this category are vector bundle isomorphisms, which are morphisms with a two-sided inverse

Operations on vector bundles

• Operations on vector bundles allow for the construction of new bundles from existing ones and provide a rich algebraic structure to the category of vector bundles
• These operations are compatible with the corresponding operations on vector spaces and often have geometric interpretations in terms of the underlying manifolds

Subbundles and quotient bundles

• A subbundle of a vector bundle $\pi: E \to B$ is a vector bundle $\pi': E' \to B$ such that for each $b \in B$, the fiber $(\pi')^{-1}(b)$ is a linear subspace of the fiber $\pi^{-1}(b)$
• The quotient bundle $E/E'$ is a vector bundle over $B$ whose fibers are the quotient spaces of the fibers of $E$ by the corresponding fibers of $E'$
• Subbundles and quotient bundles are related by the short exact sequence of vector bundles $0 \to E' \to E \to E/E' \to 0$

Direct sums and Whitney formula

• The direct sum of two vector bundles $\pi_1: E_1 \to B$ and $\pi_2: E_2 \to B$ is a vector bundle $E_1 \oplus E_2$ over $B$ whose fibers are the direct sums of the corresponding fibers of $E_1$ and $E_2$
• The Whitney formula relates the Chern classes of the direct sum bundle to the Chern classes of the individual bundles: $c(E_1 \oplus E_2) = c(E_1) \smile c(E_2)$, where $\smile$ denotes the cup product in cohomology
• The direct sum operation is compatible with the Whitney sum construction and provides a group structure on the set of isomorphism classes of vector bundles over a fixed base space

Tensor powers and exterior powers

• The tensor power of a vector bundle $\pi: E \to B$ is a vector bundle $E^{\otimes k}$ over $B$ whose fibers are the $k$-fold tensor products of the fibers of $E$
• The exterior power of a vector bundle $\pi: E \to B$ is a vector bundle $\Lambda^k E$ over $B$ whose fibers are the $k$-th exterior powers of the fibers of $E$
• Tensor powers and exterior powers are related by the binomial formula: $\Lambda^k(E \oplus F) \cong \bigoplus_{i+j=k} \Lambda^i E \otimes \Lambda^j F$
• These operations provide a way to construct new vector bundles with interesting geometric and algebraic properties, such as the determinant bundle and the Grassmann bundles

Connections and curvature

• Connections and curvature are fundamental concepts in the differential geometry of vector bundles and provide a way to study the intrinsic geometry of manifolds
• They generalize the notion of derivatives and parallel transport from Euclidean spaces to vector bundles and are essential tools in the formulation of gauge theories and general relativity

Definition of connections

• A connection on a vector bundle $\pi: E \to B$ is a linear map $\nabla: \Gamma(E) \to \Gamma(T^*B \otimes E)$ satisfying the Leibniz rule $\nabla(fs) = df \otimes s + f\nabla s$ for any smooth function $f$ on $B$ and section $s$ of $E$
• Intuitively, a connection provides a way to differentiate sections of the vector bundle along tangent vectors on the base manifold
• Connections can be expressed locally in terms of Christoffel symbols with respect to a chosen frame, and different choices of frames lead to different expressions for the same connection

Parallel transport

• Given a connection $\nabla$ on a vector bundle $\pi: E \to B$ and a curve $\gamma: [0,1] \to B$, parallel transport along $\gamma$ is a linear isomorphism $P_\gamma: E_{\gamma(0)} \to E_{\gamma(1)}$ between the fibers over the endpoints of the curve
• Parallel transport is defined by solving the differential equation $\nabla_{\dot{\gamma}} s = 0$ for sections $s$ of $E$ along $\gamma$, with the initial condition $s(\gamma(0)) = v$ for some vector $v \in E_{\gamma(0)}$
• Parallel transport is path-dependent in general, and the failure of parallel transport to be path-independent is measured by the curvature of the connection

Curvature tensor

• The curvature of a connection $\nabla$ on a vector bundle $\pi: E \to B$ is a 2-form $R^\nabla \in \Omega^2(B, \text{End}(E))$ with values in the endomorphism bundle of $E$
• In local coordinates, the curvature tensor is given by $R^\nabla = dA + A \wedge A$, where $A$ is the connection 1-form (or gauge potential) associated to $\nabla$
• The curvature measures the non-commutativity of parallel transport around infinitesimal loops and provides an obstruction to the existence of global parallel frames

Flat bundles and monodromy

• A connection $\nabla$ on a vector bundle $\pi: E \to B$ is called flat if its curvature tensor vanishes identically, i.e., $R^\nabla = 0$
• Flat bundles are characterized by the property that parallel transport along any closed loop in the base space is the identity, i.e., the monodromy representation $\pi_1(B) \to GL(V)$ is trivial
• The monodromy of a flat connection provides a representation of the fundamental group of the base space into the general linear group of the typical fiber, which can be used to study the global topology of the bundle

Chern classes and characteristic classes

• Chern classes are cohomological invariants associated to complex vector bundles that measure the non-triviality of the bundle structure
• The $k$-th Chern class $c_k(E)$ of a complex vector bundle $\pi: E \to B$ is an element of the $2k$-th cohomology group $H^{2k}(B, \mathbb{Z})$
• Chern classes are invariant under vector bundle isomorphisms and satisfy certain axioms, such as naturality under pullbacks and the Whitney product formula
• More generally, characteristic classes are cohomological invariants associated to vector bundles that provide obstructions to the existence of certain geometric structures (e.g., Stiefel-Whitney classes for orientability, Pontryagin classes for spin structures)

Examples and applications

• Vector bundles arise naturally in many areas of mathematics and physics, and their study has led to numerous applications and deep connections between different fields
• Some of the most important examples and applications of vector bundles include:

Tangent and cotangent bundles

• The tangent bundle $TM$ of a smooth manifold $M$ is a vector bundle over $M$ whose fibers are the tangent spaces $T_pM$ at each point $p \in M$
• The cotangent bundle $T^*M$ is the dual bundle of the tangent bundle, whose fibers are the cotangent spaces (dual vector spaces) $T_p^*M$
• Tangent and cotangent bundles are fundamental objects in differential geometry and provide a natural setting for the study of differential forms, Riemannian metrics, and geodesics

Normal bundles and tubular neighborhoods

• The normal bundle $N_X$ of a submanifold $X \subset M$ is a vector bundle over $X$ whose fibers are the normal spaces $N_pX = T_pM / T_pX$ at each point $p \in X$
• The normal bundle describes the infinitesimal neighborhood of the submanifold and is used to construct tubular neighborhoods and define the Euler class
• Tubular neighborhoods provide a way to extend geometric structures (e.g., Riemannian metrics