Connection on a vector bundle

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 $C^{\infty }$-linear in $X$ (that is, 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 $X$. 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 denoted as $Xf$.

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

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 differentiate 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 behaviour in the neighbourhood. 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 bundle, and we would like that any other connections we define should be compatible with this via the Leibniz rule.

Constructions

Connection on a tensor product

Fill this in later

Finding canonical connections

Connection for a bilinear form

A nondegenerate bilinear form gives a canonical isomorphism between the tangent bundle and its dual bundle. We say that a connection is compatible with the bilinear form if the dual connection on the dual bundle gets identified with the original connection via this natural isomorphism.