Lecture 07- Connections

#math

Let $X$ be a vector field. It can be used to provide a directional derivative.

Definition

$\nabla_{X} f := X f$

Since $X$ is a vector field then it 'eats' a $C^{\infty}$ function and gives you a $C^{\infty}$ function so this may seem to be notational overkill but we can extend $\nabla_{X}$ to act on $(p, q)$ tensor fields.

Directional Derivatives of Tensor Fields

Definition

A connection $\nabla$ (aka linear connection, covariant derivative, affine connection) on a smooth manifold $M$ is a map that takes a pair consisting of a vector (field) $X$ and a $(p, q)$ -tensor field $T$ and sends them to a $(p, q)$ -tensor (field) $\nabla_{X} T$ . Satisfying:

$\nabla_{X} f = X f, \forall f \in C^{\infty} (M)$
$\nabla_{X} (T + S) = \nabla_{X} T + \nabla_{X} S$
$\nabla_{X} (T (ω, ψ)) = \nabla_{X} (T) (ω, ψ) + T (\nabla_{X} ω, ψ) + T (ω, \nabla_{X} ψ)$ . This is for a $(1, 1)$ -tensor field but analogously for a $(p, q)$ -tensor field. (Leibniz Rule)
$\nabla_{f X + Z} T = f \cdot \nabla_{X} T + \nabla_{Z} T$ for functions $f \in C^{\infty} (M) .$

A manifold with connection is a quadruple of structures $(M, O, A, \nabla)$ .
Consider $X, Y$ vector fields:

\begin{aligned} \nabla_{X} Y & = \nabla_{X^{i} \frac{\partial}{\partial x^{i}}} (Y^{m} \frac{\partial}{\partial x^{m}}) \\ = X^{i} (\nabla_{\frac{\partial}{\partial x^{i}}} Y^{m}) \frac{\partial}{\partial x^{m}} + X^{i} Y^{m} \nabla_{\frac{\partial}{\partial x^{i}}} (\frac{\partial}{\partial x^{m}}) \\ = X^{i} (\frac{\partial}{\partial x^{i}} Y^{m}) \frac{\partial}{\partial x^{m}} + X^{i} Y^{m} Γ^{q}_{m i} (\frac{\partial}{\partial x^{q}}) \end{aligned}

Definition

Let $(M, O, A, \nabla)$ , with $(U, x) \in A$ , then the connection coefficient functions (referring to the $Γ$ 's) w.r.t to the chart $(U, x)$ are the dim( $M$ ) $^{3}$ many functions:

\begin{aligned} Γ^{i}_{j k} & : U \to R \\ p \to (d x^{i} (\nabla_{\frac{\partial}{\partial x^{k}}} \frac{\partial}{\partial x^{j}})) (p) \end{aligned}

Remark: On a chart domain $U$ its choice of the dim( $M$ ) $^{3}$ functions suffices to fix the action of $\nabla$ on a vector field. Fortunately, the same dim( $M$ ) $^{3}$ functions fix the action of $\nabla$ on any tensor field.

Commit to memory:
$(\nabla_{X} Y)^{i} = X (Y^{i}) + Γ^{i}_{j m} Y^{j} X^{m}$
$(\nabla_{X} ω)_{i} = X (ω_{i}) + Γ^{j}_{i m} ω_{j} X^{m}$

Definition

Let $X$ be a vector field. The divergence of $X$ is the function:

div (X) = (\nabla_{\frac{\partial}{\partial x^{i}}} X)^{i}

Change of $Γ$ 's under change of chart

Let $(U, x), (V, y) \in A$ and $U \cap V \neq \emptyset$ :

\begin{aligned} Γ_{(y)}^{i}_{j k} & = d y^{i} (\nabla_{\frac{\partial}{\partial y^{k}}} \frac{\partial}{\partial y^{j}}) \\ = \frac{\partial y^{i}}{\partial x^{q}} d x^{q} (\nabla_{\frac{\partial x^{p}}{(\partial y^{k)}} \frac{\partial}{\partial x^{p}}} \frac{\partial x^{s}}{\partial y^{j}} \frac{\partial}{\partial x^{s}}) \\ = \frac{\partial y^{i}}{\partial x^{q}} \frac{\partial x^{p}}{\partial y^{k}} \frac{\partial}{\partial x^{p}} (\frac{\partial x^{s}}{\partial y^{j}}) δ_{s}^{q} + \frac{\partial y^{i}}{\partial x^{q}} \frac{\partial x^{s}}{\partial y^{j}} \frac{\partial x^{p}}{\partial y^{k}} Γ_{(x)}^{q}_{s p} \\ = \frac{\partial y^{i}}{\partial x^{q}} \frac{\partial}{\partial y^{k}} (\frac{\partial x^{s}}{\partial y^{j}}) δ_{s}^{q} + \frac{\partial y^{i}}{\partial x^{q}} \frac{\partial x^{s}}{\partial y^{j}} \frac{\partial x^{p}}{\partial y^{k}} Γ_{(x)}^{q}_{s p} \end{aligned}

In summary:

Γ_{(y)}^{i}_{j k} = \frac{\partial y^{i}}{\partial x^{q}} \frac{\partial x^{s}}{\partial y^{j}} \frac{\partial x^{p}}{\partial y^{k}} Γ_{(x)}^{q}_{s p} + \frac{\partial y^{i}}{\partial x^{q}} \frac{\partial^{2} x^{q}}{\partial y^{k} y^{j}}

Commit this to memory. This is the change of connection coefficient function under change of chart $(U \cap V, x)$ to $(U \cap V, y)$ .

Normal Coordinates

Let $p \in M$ . Then one can construct a chart $(U, x)$ with $p \in U$ such that:

Γ_{(x)}^{i}_{(j k)} (p) = 0

at the point p. Not necessarily in any neighborhood.

Proof

Let $(V, q)$ be any chart $p \in V$ such that $y (p) = (0, . . ., 0)$ .
Thus in general: $Γ_{(y)}^{i}_{j k} \neq 0$ .
Then consider a new chart $(U, x)$ to which one transits by virtue of:

(x \circ y^{- 1})^{i} (α^{1}, . . ., α^{d}) := α^{i} - \frac{1}{2} Γ_{(y)}^{i}_{(j k)} α^{j} α^{k}

\frac{\partial x^{i}}{\partial y^{j}} = \partial_{j} (x^{i} \circ y^{- 1}) = δ_{j}^{i} - Γ_{(y)}^{i}_{(m j)} (p) α^{m}

\frac{\partial x^{i}}{\partial y^{j} \partial y^{k}} = - Γ_{(y)}^{i}_{(k j)} (p)

Thus: $Γ_{(x)}^{i}_{j k} (p) = Γ_{(y)}^{i}_{j k} (p) - Γ_{(y)}^{i}_{(k j)} (p) = Γ_{(y)}^{i}_{[j k]} (p)$

Terminology: $(U, x)$ is called a normal coordinate chart of $\nabla$ at $p \in M$ .

Directional Derivatives of Tensor Fields

Change of Γ's under change of chart

Normal Coordinates

Change of $Γ$ 's under change of chart