[Tu Differential Geometry] 13. More on Affine Connections

 This article is one of Manifold, Differential Geometry, Fibre Bundle.

This chapter is a compilation of some properties of an affine connection that will prove useful later. First we discuss how an affine connection on a manifold $M$ induces a unique covariant derivative of vector field along a smooth curve in $M$. This generalizes the derivative $dV/dt$ of a vector field $V$ along a smooth curve in $\mathbb{R}^n$. Secondly, we present a way of describing a connection in local coordinates, using the so-called Christoffel symbols.

13.1 Covariant Differentiation Along a Curve

Let $c:[a,b]\rightarrow M$ be a smooth parametrized curve in a manifold $M$. Recall that a vector field along the curve $c$ in $M$ is a function $$V:[a,b]\rightarrow \coprod_{t\in [a,b]}T_{c(T)}M,$$ where $\coprod$ stands for the disjoint union, such that $V(t)\in T_{c(t)}M$ (Figure 13.1). Such a vector field $V(t)$ is $C^\infty$ if for any $C^\infty$ function $f$ on $M$, the function $V(t)f$ is $C^\infty$ as a function of $t$. We denote the vector space of all $C^\infty$ vector fields along the curve $c(t)$ by $\Gamma(TM|_{c(t)})$.

For a smooth vector field $V(t)=\sum v^i(t)\partial /\partial x^i$ along a smooth curve $c(t)$ in $\mathbb{R}^n$, we defined its derivative $dV/dt$ by differetiating the components $v^i(t)$ with respect to $t$. Then $dV/dt=\sum v^i\partial /\partial x^i$ satisfies the following properties:

(i) $dV/dt$ is $\mathbb{R}$-linear in $V$;

(ii) for any $C^\infty$ function $f$ on $[a,b]$, $$\frac{d(fV)}{dt}=\frac{df}{dt}V+f\frac{dV}{dt};$$

(iii) (Proposition 4.11) if $V$ is induced from a $C^\infty$ vector field $\tilde{V}$ on $\mathbb{R}^n$, in the sense that $V(t)=\tilde{V}_{c(t)}$ and $D$ is the directional derivative in $\mathbb{R}^n$, then $$\frac{dV}{dt}=D_{c'(t)}\tilde{V}.$$

It turns out that to every connection $\nabla$ on a manifold $M$ one can associate a way of differentiating vector fields along a curve satisfying the same properties as the derivative above.

Theorem 13.1 (Uniqueness of Covariant Derivative along Line). Let $M$ be a manifold with an affine connection $\nabla$, and $c:[a,b]\rightarrow M$ a smooth curve in $M$. Then there is a unique map $$\frac{D}{dt}:\Gamma(TM|_{c(t)})\rightarrow \Gamma(TM|_{c(t)})$$ such that for $V\in \Gamma(TM|_{c(t)})$, 

(i) ($\mathbb{R}$-linearity) $DV/dt$ is $\mathbb{R}$-linear in $V$;

(ii) (Leibnize rule) for any $C^\infty$ function $f$ on the interval $[a,b]$, $$\frac{D(fV)}{dt}=\frac{df}{dt}B+f\frac{DV}{dt};$$

(iii) (Compatibility with $\nabla$) if $V$ is induced from a $C^\infty$ vector field $\tilde{V}$ on $M$, in the sense that $V(t)=\tilde{V}_{c(t)}$, then $$\frac{DV}{dt}(t)=\nabla_{c'(t)}\tilde{V}.$$

We called $DV/dt$ the covariant derivative (associated to $\nabla$) of the vector field $V$ along the curve $c(t)$ in $M$.

Proof. Suppose such a covariant derivative $D/dt$ exists. On a framed open set $(U,e_1,\cdots,e_n)$, a vector field $V(t)$ along $c$ can be written as a linear combination $$V(t)=\sum v^i(t)e_{i,c(t)}.$$ Then $$\begin{align}\frac{DV}{dt}=\sum \frac{D}{dt}\left( v^i(t)e_{i,c(t)}\right) = \sum \frac{dv^i}{dt}e_i+v^i\frac{De_i}{dt}=\sum \frac{dv^i}{dt}e_i+v^i\nabla_{c'(t)}e_i\end{align}$$ (Use properties) where we abuse notation and write $e_i$ instead of $e_{i,ct(t)}$. This formula proves the uniquness of $D/dt$ if it exists.

As for existence, we define $DV/dt$ for a curve $c(t)$ in a framed open set $U$ by the formula (1). It is easily verified that $DV/dt$ satisfies the three properties (i), (ii), (iii). Hence, $D/dt$ exists for curves in $U$. If $\bar{e}_1,\cdots,\bar{e}_n$ is another frame on $U$, then $V(t)$ is a linear combination $\sum \bar{v}^i(t)\bar{e}_{i,c(t)}$ and the covariant derivative $\bar{D}V/dt$ defined by $$\frac{\bar{D}V}{dt}=\sum_i \frac{d\bar{v}^i}{dt}\bar{e}_i+\bar{v}^i\nabla_{c'(t)}\bar{e}_i$$ also satisfies the three properties of the theorem. By the uniqueness of the covariant derivative, $DV/dt=\bar{D}V/dt$. This proves that the covariant derivative $DV/dt$ is independent of the frame. By covering $M$ with framed open sets, (1) then defines a covariant derivative $DV/dt$ for the curve $c(t)$ in $M$.

Theorem 13.2 (Compatible with the metric). Let $M$ be a Riemannian manifold, $\nabla$ an affine connection on $M$, and $c[a,b]\rightarrow M$ a smooth curve in $M$. If $\nabla$ is compatible with the metric, then for any smooth vector field $V,W$ along $c$, $$\frac{d}{dt}\left\langle V,W\right\rangle = \left\langle \frac{DV}{dt},W\right\rangle + \left\langle V,\frac{DW}{dt}\right\rangle.$$

Proof. It suffices to check this equality locally, so ket $U$ be an open set on which an orthonormal frame $e_1,\cdots,e_n$ exists. With respect to this frame, $$V=\sum v^i(t)e_{i,c(t)},\quad W=\sum w^j(t)e_{j,c(t)}$$ for some $C^\infty$ functions $v^i$, $w^i$ on $[a,b]$. Then $$\frac{d}{dt}\left\langle V,W\right\rangle=\frac{d}{dt}\sum v^iw^i = \sum \frac{dv^i}{dt}=\sum \frac{dv^i}{dt} w^i+\sum v^i\frac{dw^i}{dt}.$$ By the defining properties of a covariant derivative, $$\frac{DV}{dt}=\sum_i \frac{dv^i}{dt}e_i+v^i\frac{De_i}{dt}=\sum_i \frac{dv^i}{dt}e_i+v^i\nabla_{c'(t)}e_i,$$ where we again abuse notation and write $e_i$ instead of $e_i\circ c$. Similarly, $$\frac{DW}{dt}=\frac{dw^j}{dt}e_j+w^j\nabla_{c'(t)}e_j.$$ Hence, $$\left\langle \frac{DV}{dt},W\right\rangle+\left\langle V,\frac{DW}{dt}\right\rangle = \sum_i \frac{dv^i}{dt}w^i+\sum_{i,j} v^iw^j\left\langle \nabla_{c'(t)}e_i,e_j\right\rangle + \sum_i v^i \frac{dw^i}{dt}+\sum_{i,j} v^iw^j\left\langle e_i,\nabla_{c'(t)}e_j\right\rangle .$$

Since $e_1,\cdots,e_n$ are orthonormal vector fields on $U$ and $\nabla$ is compatible with the metric, $$\left\langle \nabla_{c'(t)}e_i,e_j\right\rangle + \left\langle_i,\nabla_{c'(t)}e_j\right\rangle=c'(t)\left\langle e_i,e_j\right\rangle = c'(t)\delta_{ij}=0.$$ Therefore, $$\left\langle \frac{DV}{dt},W\right\rangle + \left\langle V,\frac{DW}{dt}\right\rangle = \sum \frac{dv^i}{dt}w^i+v^i\frac{dw^i}{dt}=\frac{d}{dt}\left\langle V,W\right\rangle.$$

Example 13.3. If $\nabla$ is the directional derivative on $\mathbb{R}^n$ and $V(t)=\sum v^i(t)\partial /\partial x^i$ is a vector field along a smooth curve $c(t)$ in $\mathbb{R}^n$, then the covariant derivative is $$\frac{DV}{dt}=\sum \frac{dv^i}{dt}\frac{\partial}{\partial x^i}+\sum v^iD_{c'(t)}\frac{\partial}{\partial x^i}=\sum \frac{dv^i}{dt}{\partial}{\partial x^i}=\frac{dV}{dt},$$ since $D_{c'(t)}\partial/\partial x^i=0$ by (4.2).

13.2 Connection-Preserving Diffeomorphisms

Although it is in general not possible to push forward a vector field except under a diffeomorphism, it is always possible to push forward a vector field along a curve under any $C^\infty$ map. Let $f:M\rightarrow \tilde{M}$ be a $C^\infty$ map (not necessarily a diffeomorphism) of manifold, and $c:[a,b]\rightarrow M$ a smooth curve in $M$. The pushforward of the vector field $V(t)$ along $c$ in $M$ is the vector field $(f_*V)(t)$ along the image curve $f\circ c$ in $\tilde{M}$ defined by $(f_*V)(t)=f_{*,c(t)}\left( V(t)\right).$$

Denote by $(M,\nabla)$ a $C^\infty$ manifold with an affine connection $\nabla$. We say that a $C^\infty$ diffeomorphism $f:(M,\nabla)\rightarrow (\tilde{M},\tilde{\nabla})$ preserves the connection or is connection-preserving if for all $X,Y\in \mathfrak{X}(M)$, $$f_*(\nabla_XY)=\tilde{\nabla}_{f_*X}f_*Y.$$ In this terminology, an isometry of Riemannian manifolds preserves the Riemannian connection. 

We now show that a connection-preserving differomorphism also preserves the covariant derivative along a curve.

Proposition 13.4. Suppose $f:(M,\nabla)\rightarrow (\tilde{M},\tilde{\nabla})$ is a connection-preserving differmorphism, $c(t)$ a smooth curve in $M$, and $D/dt$, $\tilde{D}/dt$ the covariant derivatives along $c$ in $M$ and $f\circ c$ in $\tilde{M}$, respectively. If $V(t)$ is a vector field along $c$ in $M$, then $$f_*\left( \frac{DV}{dt}\right) = \frac{\tilde{D}(f_*V)}{dt}.$$

Proof. Choose a neighborhood $U$ of $c(t)$ on which there is a frame $e_1,\cdots,e_n$ and write $V(t)=\sum v^i(t)e_{i,.c(t)}$. Then $\tilde{e}_1:=f_*e_1,\cdots,\tilde{e}_n:=f_*e_n$ is a frame on the neighborhood $f(U)$ of $f(c(t))$ in $\tilde{M}$ and $$(f_*V)(t)=\sum v^i(t)\tilde{e}_{i,(f\circ c)(t)}.$$ (Is $f_*$ is $\mathcal{F}$-linear???)

By the definition of the covariant derivative, $$\frac{DV}{dt}(t)=\sum \frac{dv^i}{dt}(t)e_{i,c(t)}+v^i(t)\nabla_{c'(t)}e_i.$$ Because $f$ preserves the connection, $$\left( f_*\frac{DV}{dt}\right) (t)=\sum \frac{dv^i}{dt}f_{*,c(t)}(e_{i,c(t)})+v^i(t)\tilde{\nabla}_{f_*c'(t)}f_*e_i\\ =\sum \frac{dv^i}{dt}\tilde{e}_{i,(f\circ c)(t)}+v^i(t)\tilde{\nabla}_{(f\circ c)'(t)}\tilde{e}_i =\left( \frac{\tilde{D}}{dt}f_*V\right) (t).$$

13.3 Christoffel Symbols

One way to describe a connection locally is by the connection forms relative to a frame. Another way, which we now discuss, is by a set of $n^3$ functions called the Chirstoffel symbols.

The Christoffel symbols are defined relative to a coordinate frame. Let $\nabla$ be an affine connection on a manifold $M$ and let $(U,x^1,\cdots,x^n)$ be a coordinate open set in $M$. Denote by $\partial_i$ the coordinate vector field $\partial/\partial x_i$. Then $\nabla_{\partial_i}\partial_j$ is a linear combination of $\partial_1,\cdots,\partial_n$, so there exist numbers $\Gamma^k_{ij}$ at each point such that $$\nabla_{\partial_i}\partial_j=\sum^n_{k=1}\Gamma^k_{ij}\partial_k.$$ These $n^3$ functions $\Gamma^k_{ij}$ are called the Chirstoffel symbols of the connection $\nabla$ on the coordinate open set $(U,x^1,\cdots,x^n)$. By the Leibniz rule and $\mathcal{F}$-linearity in the first argument of a connection, the Christoffel symbols completely describe a connection on $U$.

Proposition 13.5. An affine connection $\nabla$ on a manifold is torsion-free if and only if every coordinate chart $(U,x^1,\cdots,x^n)$ the Christoffel symbol $\Gamma^k_{ij}$ is symmetric in $i$ and $j$: $$\Gamma_{ij}^k=\Gamma_{ji}^k.$$

Proof. ($\Rightarrow$) Let $(U,x^1,\cdots,x^n)$ be a coordinate open set. Since partial differentiation is independent of the order of differentiation, $$[\partial_i,\partial_j]=\left[ \frac{\partial}{\partial x^i},\frac{\partial}{\partial x^j}\right] =0.$$ By torsion-freeness, $$\nabla_{\partial_i}\partial_j -\nabla_{\partial_j}\partial_i=[\partial_i,\partial_j]=0.$$ In terms of Christoffel symbols, $$\sum_k \Gamma^k_{ij}\partial_k-\sum_k\Gamma^k_{ji}\partial_k=0.$$ Since $\partial_1,\cdots,\partial_n$ are linearly independent at each point, $\Gamma^k_{ij}=\Gamma^k_{ji}.$

($\Leftarrow$) Conversely, suppose $\Gamma^k_{ij}=\Gamma^k_{ji}$ in the coordinate chart $(U,x^1,\cdots,x^n)$. Then $\nabla_{\partial_i}\partial_j=\nabla_{\partial_j}\partial_i$. Hence, $$T(\partial_i,\partial_j)=\nabla_{\partial_i}\partial_j-\nabla_{\partial_j}\partial_i=0.$$ Since $T(\ ,\ )$ is a bilinear function on $T_pM$, this proves that $T(X,Y)_p:=T(X_p,Y_p)=0$ for all $X_p,Y_p\in T_pM$. Thus, for all $X,Y\in \mathfrak{X}(M)$, we have $T(X,Y)=0$.