Connection on a vector bundle

This lives as an element of: the space of $\mathbb {R}$-bilinear maps $\Gamma (TM)\times \Gamma (E)\to \Gamma (E)$ for a vector bundle $E$ over a manifold $M$

This article defines a basic construct that makes sense on any differential manifold
View a complete list of basic constructs on differential manifolds

Definition

Given data

• A connected differential manifold $M$ with tangent bundle denoted by $TM$
• A vector bundle $E$ over $M$

Definition part (pointwise form)

A connection is a smooth choice $\nabla$ of the following: at each point $p\in M$, there is a map ${}^{p}\nabla :T_{p}(M)\times \Gamma (E)\to E(p)$, satisfying some conditions. The map is written as ${}^{p}\nabla _{X}(v)$ where $X\in T_{p}(M)$ and $v\in \Gamma (E)$.

• It is $\mathbb {R}$-linear in $X$ (i.e., in the $T_{p}(M)$ coordinate).
• It is $\mathbb {R}$-linear in $\Gamma (E)$ (viz., the space of sections on $E$).
• It satisfies the following relation called the Leibniz rule:

${}^{p}\nabla _{X}(fv)=(Xf)(p)(v)+f(p)^{p}\nabla _{X}(v)$

Definition part (global form)

A connection is a map $\nabla :\Gamma (TM)\times \Gamma (E)\to \Gamma (E)$, satisfying the following:

• It is $C^{\infty }$-linear in $\Gamma (TM)$ (in other words, it is tensorial, or pointwise, in the $\Gamma (TM)$-coordinate)
• it is $\mathbb {R}$-linear in $\Gamma (E)$
• It satisfies the following relation called the Leibniz rule:

$\nabla _{X}(fv)=(Xf)(v)+f\nabla _{X}(v)$

where $f$ is a scalar function on the manifold and $fv$ denotes scalar multiplication of $v$ by $f$.

Alternative definitions

A connection is equivalent to the following:

• A choice of splitting of the first-order symbol sequence of a vector bundle: Further information: connection is splitting of first-order symbol sequence
• A module structure of the vector bundle, over the connection algebra: Further information: connection is module structure over connection algebra
• A connection on the corresponding bundle over the principal bundle over the general linear group. Further information: connection on vector bundle equals connection on principal GL-bundle

Particular cases

When $E=M\times \mathbb {R}$ is the trivial one-dimensional bundle, then sections of $E$ are the same as infinitely differentiable functions on $M$. For this bundle, there is a unique connection: the usual action of a vector field on a function.

When $E$ is itself the tangent bundle, we call the connection a linear connection.

Terminology

Covariant derivative of a section

Further information: covariant derivative of a section

Given a connection $\nabla$ on a vector bundle $E$ over a differential manifold $M$, the covariant derivative of a section $s\in \Gamma (E)$ with respect to a vector field $X$ is defined as the value:

$\nabla _{X}(s)$

The term covariant derivative can thus be used only if we already have a connection in mind.

Absolute derivative of a section

Further information: absolute derivative of a section

Given a connection $\nabla$, the absolute derivative of a section $s\in \Gamma (E)$, denoted $d_{\nabla }(s)$, is defined as the operator that sends a vector field $X$ to $\nabla _{X}(s)$. In the particular case where $E=M\times \mathbb {R}$ is the trivial one-dimensional bundle, this reduces to the de Rham derivative of a function, yielding a 1-form.

Connection, transport along a curve

Further information: connection along a curve, transport along a curve

Given a connection on the manifold, we can obtain a connection along any curve on the manifold, using the pullback connection. A connection along the curve gives a transport: a rule for transporting a basis for the fiber at one point, to a basis for the fiber at the other point. Thus, a connection is often thought of as a global transport rule.

Importance

Consider a vector field $X$ on $M$. We know that we can define a notion of directional derivatives for functions along this vector field: this differentiates the function at each point, along the vector at that point. The derivative of $f$ along the direction of $X$ is a new function, denoted as $Xf$.

Note that at any point $p$, the value of $(Xf)(p)$ depends on the local behavior of $f$ but only on the pointwise behavior of $X$, that is, it only depends on the tangent vector $X(p)$ and not on the behavior of $X$ in the neighborhood.

The idea behind a connection is to extend this differentiation rule, not just to functions, but also to other kinds of objects. In particular, we want to be able to have a differentiation rule for sections of the tangent and cotangent bundles, along vector fields. In this definition, what we would like is:

• The derivative with respect to a vector field at a point should just depend on the value of the vector field at the point -- it should not depend on the behavior in the neighborhood. We say it is a tensorial map with respect to $X$.

• A Leibniz rule is satisfied with respect to scalar multiplication by functions, which connects differentiation for this connection with the differentiation of scalar functions along vector fields

Note that the usual differentiation along vector fields is thus the canonical connection on the trivial one-dimensional bundle, and we would like that any other connections we define should be compatible with this via the Leibniz rule.

Existence

Further information: Connections exist

Given any vector bundle over a differential manifold, there exists a connection for that vector bundle.

Constructions

Connection on a direct sum

Further information: Direct sum of connections

Suppose we have connections $\nabla ,\nabla '$ on vector bundles $E,E'$ over a differential manifold $M$. Then, we can obtain a connection, that we'll denote $\nabla \oplus \nabla '$, on the direct sum $E\oplus E'$. This is defined by:

$(\nabla \oplus \nabla ')(s,s')=\nabla (s)\oplus \nabla '(s')$.

Connection on a tensor product

Further information: Tensor product of connections

Suppose we have connections $\nabla ,\nabla '$ on vector bundles $E,E'$ over a differential manifold $M$. Then, we can obtain a connection, that we'll denote $\nabla \otimes \nabla '$, on the tensor product $E\otimes E'$. On pure tensors, it is given by the formula:

$(\nabla \otimes \nabla ')_{X}(s\otimes s')=\nabla _{X}(s)\otimes s'+s\otimes \nabla '_{X}(s')$

In other words, the formula is chosen so that a Leibniz-like rule is satisfied for tensor products.

Connection on the dual

Further information: Dual connection

Given a connection $\nabla$ on a vector bundle $E$ over a differential manifold $M$, we can obtain a connection $\nabla ^{*}$on the dual bundle $E^{*}$ as follows:

$\nabla _{X}^{*}(l)=s\mapsto X(l(s))-l(\nabla _{X}(s))$

Particular kinds of connections

Metric connection

Further information: metric connection

The notion of a metric connection makes sense when we have a metric bundle: a vector bundle with an inner product on every fiber that varies compatibly. A metric connection is a connection with the property that it satisfes a Leibniz-like rule with respect to the inner product of sections:

$X\left\langle s_{1},s_{2}\right\rangle =\left\langle \nabla _{X}s_{1},s_{2}\right\rangle +\left\langle s_{1},\nabla _{X}s_{2}\right\rangle$

A case of particular interest is a metric linear connection: this is a metric connection on the tangent bundle, for a Riemannian manifold.

The set of all connections

As an affine space

Further information: Affine space of all connections

Given a manifold $M$ and a vector bundle $E$ over $M$, consider the set of all connections for $E$. Clearly, the connections live inside the space of $\mathbb {R}$-bilinear maps $\Gamma (TM)\times \Gamma (E)\to \Gamma (E)$. Hence, we can talk of linear combinations of connections. In general, a linear combination of connections need not be a connection. The problem arises from the Leibniz rule, which has a term $Xf$ that does not scale with the connection.

It is true that the set of differences of connections (if nonempty) forms a vector subspace of the vector space of all bilinear maps. Since there is a fundamental theorem that connections exist, we conclude that the set of connections is in fact an affine space, viz a translate of a subspace, and thus any affine linear combination of connections is again a connection.

The vector space in question is the space of $C^{\infty }$-bilinear maps $\Gamma (TM)\times \Gamma (E)\to \Gamma (E)$.

As the collection of module structures

Further information: Connection is module structure over connection algebra

Given a vector bundle $E$, a connection on $E$ makes $\Gamma (TM)$ act on $\Gamma (E)$. Thus, we could view $\Gamma (E)$ as a module over the free algebra generated by $\Gamma (TM)$. This action actually satisfies some extra conditions, and these conditions help us descend to an action of the connection algebra on $\Gamma (E)$.

Thus, a connection on a vector bundle $E$ is equivalent to equipping $\Gamma (E)$ with a module structure over the connection algebra.

Local description

Connections localize

Further information: Connections localize

Given a connection on the whole differential manifold $M$, we can get a connection on any open subset $U$ of $M$. Note that this is not a completely trivial statement, because not every vector field on an open subset extends to a vector field on the whole manifold. However, we can express any vector field on an open subset, as the product of a function and a vector field that can be extended to the whole manifold, and we can then use the Leibniz rule.

It is also true that connections piece together. In other words, to know $\nabla _{X}s$ at a point $p$, it suffices to know the germ of $s$ at $p$.

Describing connections using coordinate charts

Further information: Christoffel symbols of a connection, matrix of connection forms

A connection is a bilinear map, and because connections localize and piece together, it suffices to describe what happens to the connection inside coordinate charts. However, we need to remember that while $\nabla _{X}(s)$ depends only pointwise on $X$ (so it depends only on the value of $X$ at a point), it depends locally on $s$ (so it depends on the germ of $s$ at the point). So, to describe a connection at a point $p$, it is not enough to take a basis for $T_{p}(M)$ and a basis for $E(p)$ and describe what happens on that basis. Rather, we take a basis for $T_{p}(M)$, and pick a coordinate chart around $p$, and take constant vector fields corresponding to a choice of basis for that coordinate chart.