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 $M$

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 $\Gamma (TM)$
• 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 $f$.

Particular cases

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

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. This is called the pointwise property.

• 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 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:

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

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 Leibnixz-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

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.

As the collection of module structures

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.