Linear connection

This lives as an element of: the space of $\mathbb {R}$-bilinear maps $\Gamma (TM)\times \Gamma (TM)\to \Gamma (TM)$

Definition

Given data

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

Definition part (pointwise form)

A linear connection is a smooth choice $\nabla$ of the following: at each point $p\in M$, there is a map ${}^{p}\nabla :T_{p}(M)\times \Gamma (TM)\to T_{p}(M)$, 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 $\Gamma (TM)$ (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 linear connection is a map $\nabla :\Gamma (TM)\times \Gamma (TM)\to \Gamma (TM)$, satisfying the following:

• It is $C^{\infty }$-linear in the first $\Gamma (TM)$
• it is $\mathbb {R}$-linear in the second $\Gamma (TM)$
• 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 $f$.

Generalizations

The notion of linear connection can be generalized to the more general notion of a connection.

Operations on a linear connection

Torsion of a linear connection

The torsion of a linear connection $\nabla$ is denoted as $\tau (\nabla )$. It is a $(1,2)$-tensor defined as:

$\tau (\nabla )(X,Y)=\nabla _{X}Y-\nabla _{Y}X-[X,Y]$.

A connection whose torsion is zero is termed a torsion-free linear connection.

Curvature of a linear connection

The curvature of a linear connection $\nabla$ is denoted as $R_{\nabla }$. It is defined as:

$R_{\nabla }(X,Y)=\nabla _{X}\nabla _{Y}-\nabla _{Y}\nabla _{X}-\nabla _{[X,Y]}$

The notion of curvature actually makes sense for any connection. In the case of a linear connection, it is tensorial, with lots of nice properties. It is in fact a (1,3)-tensor called the Riemann curvature tensor.