Linear connection

This lives as an element of: the space of

$\mathbb{R}$

-bilinear maps

$\mathrm{\Gamma }(TM)\times \mathrm{\Gamma }(TM)\to \mathrm{\Gamma }(TM)$

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 $\mathrm{\nabla }$ of the following: at each point $p\in M$, there is a map $p^{}\mathrm{\nabla }:T_p(M)\times \mathrm{\Gamma }(TM)\to T_p(M)$, satisfying some conditions. The map is written as $p^{}{\mathrm{\nabla }}_X(v)$ where $X\in T_p(M)$ and $v\in \mathrm{\Gamma }(E)$.

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

$p^{}{\mathrm{\nabla }}_X(fv)=(Xf)(p)(v)+f(p{)}^p{\mathrm{\nabla }}_X(v)$

Definition part (global form)

A linear connection is a map $\mathrm{\nabla }:\mathrm{\Gamma }(TM)\times \mathrm{\Gamma }(TM)\to \mathrm{\Gamma }(TM)$, satisfying the following:

• It is $C^{\mathrm{\infty }}$-linear in the first $\mathrm{\Gamma }(TM)$
• it is $\mathbb{R}$-linear in the second $\mathrm{\Gamma }(TM)$
• It satisfies the following relation called the Leibniz rule:

${\mathrm{\nabla }}_X(fv)=(Xf)(v)f{\mathrm{\nabla }}_X(v)$

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, where the second $TM$ is replaced by an arbitrary vector bundle $E$ over $M$.

Operations on a linear connection

Torsion of a linear connection

Further information: torsion of a linear connection

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

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

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

Note that torsion makes sense only for linear connections.

Curvature of a linear connection

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

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

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.