Linear connection

This lives as an element of: the space of $\R$-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 $v \in \Gamma(TM)$.

• It is $\R$-linear in $X$ (that is, in the $T_p(M)$ coordinate).
• It is $\R$-linear in $\Gamma(TM)$ (viz the space of sections on $TM$).
• 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, 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 $\nabla$ is denoted as $\tau(\nabla)$. It is a $(1,2)$-tensor defined as:

$\tau(\nabla)(X,Y)= \nabla_XY - \nabla_YX - [X,Y]$.

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

Further information: Riemann curvature tensor

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.