affine connection

Affine connections are ways of taking “derivatives” of vector fields with other ones. It is a different notion than the Lie derivative, though.

Definition

Let $M$ be a smooth manifold with vector fields $\mathfrak{X}(M)$. An affine connection is a map

$$\begin{array}{rl}\nabla :\mathfrak{X}(M)\times \mathfrak{X}(M)& ⟶\mathfrak{X}(M)\\ (X,Y)& ⟼{\nabla }_{X}Y\end{array}$$

that satisfies the following properties:

1. Tensorial in $X$.

$${\nabla }_{fX+g\tilde{X}}Y=f{\nabla }_{X}Y+g{\nabla }_{\tilde{X}}Yf,g\in C^{\infty }(M)$$

2. $\mathbb{R}$- linear in $Y$

$${\nabla }_X(Y+\tilde{Y})={\nabla }_{X}Y+{\nabla }_X\tilde{Y}$$

3. Derivation in $Y$

$${\nabla }_X(fY)=(X\cdot f)Y+f{\nabla }_{X}Yf\in C^{\infty }(M)$$

Nonexample

The Lie bracket is NOT tensorial in X or Y.

Relationship with covariant derivative

Many authors consider covariant derivative to be just an alternate name for an affine connection.

In @docarmo1992Riemannian there is a distinction made.

Let $M$ be a manifold with an affine connection $\nabla$. Let $c:I\to M$ be a curve on the manifold $M$ and consider vector fields along that curve and denote these by $\mathfrak{X}(c)$. There is a unique mapping called the covariant derivative

$$\begin{array}{rl}\frac{D}{dt}:\mathfrak{X}(c)& ⟶\mathfrak{X}(c)\\ V& ⟼\frac{DV}{dt}\end{array}$$

that has the following properties:

1. Linear in $V$: $\frac{D}{dt}(V+W)=\frac{DV}{dt}+\frac{DW}{dt}$

2. Liebniz rule: $\frac{D}{dt}(fV)=\frac{df}{dt}V+f\frac{DV}{dt}$ for $f\in C^{\infty }(M)$

3. If $V$ is induced by a vector field $Y\in \mathfrak{X}(M)$ (that is, $V(t)=Y(c(t))$ ) then $\frac{DV}{dt}={\nabla }_{dc/dt}Y$.

This means a couple of things. First, an affine connection at a point really only depends on values of the vector field on a curve running through that point. Next, there is an intrinsic idea of differentiation for a vector field that has familiar looking properties that is equivalent to an affine connection. Thus, we should think of an affine connection as the “derivative” of a vector field along curves.

The proof is in @docarmo1992Riemannian page 51.

Local Property

Affine connections are local. That means that if for vector fields $X,Y\in \mathfrak{X}(M)$ and either $X$ or $Y$ vanish identically on an open set $U\subset M$ then ${\nabla }_{X}Y$ vanishes on $U$.

Proof

todo

Corollary

If $X,Y\in \mathfrak{X}(M)$ coincide with $\tilde{X},\tilde{Y}\in \mathfrak{X}(M)$ on some open set $W\subset M$ then ${\nabla }_{X}Y={\nabla }_{\tilde{X}}\tilde{Y}$.

In coordinates

An affine connection can be written in coordinates using the Christoffel symbols. Consider vector fields

$$X=X^i\frac{\partial }{\partial x^i}\text{and}Y=Y^i\frac{\partial }{\partial x^i}$$

on $W$ we have

$${\nabla }_{X}Y=\sum _{i=1}^n\left(X\cdot Y^i+\sum _{j,k=1}^n{\Gamma }_{jk}^i{X}^j{Y}^k\right)\frac{\partial }{\partial x^i}$$