Levi-Civita Connection

On a Riemannian manifold, there is a natural choice for which connection to use. Choosing a connection is choosing a sense of acceleration on our manifold. For a Riemannian manifold \(M\), a natural choice is to agree that geodesics have \(0\) accelertaion. Indeed, geodesics are paths that go in a “straight line” without changing velocity. Thus we would like a connection \(\nabla\) such that for any geodesic \(\gamma(t)\) we have \(\nabla_{\dot{\gamma}(t)}\dot{\gamma}(t) = 0\). If we have \(\gamma(t) = (x^1(t), \cdots, x^n(t))\) in local coordinates, this requirement is equivalent to \begin{align} 0 &= \nabla_{\dot{\gamma}(t)} \dot{\gamma}(t)\\
&= \nabla_{\dot{\gamma}(t)} (\dot{x}^j(t)\partial_j)\\
&= \dot{\gamma}(t)(\dot{x}^j(t))\partial_j + \dot{x}^j(t)\nabla_{\dot{\gamma}(t)} \partial_j\\
&= \ddot{x}^j(t)\partial_j + \dot{x}^j(t)\nabla_{\dot{x}^i\partial_i} \partial_j\\
&= (\ddot{x}^k(t) + \dot{x}^i\dot{x}^j(t)A_{ij}^k) \partial_k. \end{align} That is, this is equivalent to \[ \ddot{x}^k(t) + \dot{x}^i\dot{x}^j(t)A_{ij}^k = 0 \quad \text{for all } k. \] Thus we want to choose our functions \(A_{ij}^k\) so that all geodesics satisfy the above equation. However, comparing the above equation to the geodesic equation, we realize that this will always be true if we choose \(A_{ij}^k = \Gamma_{ij}^k\) to be the Christoffel symbols! Therefore, on a Riemannian manifold we should define our connection in coordinates by setting \[ \nabla_{\partial_i}\partial_j = \Gamma_{ij}^k \partial_k. \tag{C} \label{eq:lc-coords} \] This definition, however, opens one important question: does this choice of connection rely on the choice of coordinates? The answer is no, which we will prove by later defining this natural connection in a coordinate independent way. The resulting connection is called the “Levi-Civita connection”.

Recall the connection for Euclidean space \(\mathbb{R}^n\) is given by \[ \nabla_{X}Y = XY^i\partial_i. \] This Euclidean connection satisfies the product rule \[ \nabla_{Z}\langle X , Y\rangle = \nabla_{X} \left(\sum_{i}X^iY^i\right) = \sum_{i}(XZ^i)Y^i + X^i(ZY^i) = \langle \nabla_Z X , Y\rangle + \langle X , \nabla_ZY\rangle. \] Additionally, observe that for any vector fields \(X,Y\) over \(\mathbb{R}^n\), we can compute the difference of the Euclidean connection \(\nabla_X Y - \nabla_Y X\) to be \[ \nabla_X Y - \nabla_Y X = XY^i\partial_i - YX^i\partial_i = (XY^i - YX^i) \partial_i. \] But this is juts the Lie bracket \([X,Y]\). That is, the Euclidean connection satisfies the commutator relation \[ \nabla_X Y - \nabla_Y X = [X,Y]. \tag{S} \label{eq:symmetry} \]

It turns out that for a Riemannian manifold \(M\), the Levi-Civita connection defined in local coordinates by (\ref{eq:lc-coords}) also satisfies the commutator relation (\ref{eq:symmetry}) as well as the following product rule with respect to the metric \(g\). \[ \nabla_Z\langle X , Y\rangle_g = \langle \nabla_Z X , Y\rangle_g + \langle X , \nabla_Z Y\rangle_g. \tag{M} \label{eq:compatibility} \] A connection satisfying (\ref{eq:symmetry}) is called symmetric and a connection satisfying the product rule (\ref{eq:compatibility}) is said to be compatible with the metric. We begin by showing the symmetry which is simply a coordinate computation.

Prop. The Levi-Civita connection as defined in coordinates by (\ref{eq:lc-coords}) is symmetric.

Proof. Compute \begin{align} \nabla_X Y - \nabla_Y X &= \nabla_{X^i \partial_i}(Y^j \partial_j) - \nabla_{Y^j\partial_j}(X^i \partial_i)\\
&= X^i((\partial_iY^j)\partial_j + Y^j \nabla_{\partial_j}\partial_i) - Y^j((\partial_jX^i)\partial_i + X^i \nabla_{\partial_i}\partial_j)\\
&= (XY^j\partial_j - YX^i\partial_i) + X^iY^j(\nabla_{\partial_j}\partial_i - \nabla_{\partial_i}\partial_j)\partial_k\\
&= [X,Y] + X^iY^j(\Gamma_{ij}^k - \Gamma_{ji}^k)\partial_k. \end{align} Then the result follows from \(\Gamma_{ij}^k = \Gamma_{ji}^k\) which we can see from the definition of Christoffel symbols: \[ \Gamma_{ij}^k = \frac{1}{2}g^{kl}(\partial_ig_{jl} + \partial_jg_{li} - \partial_lg_{ij}) \]

Next we show the Levi-Civita connection as defined in coordinates is compatible with the metric, which follows from a substantially longer coordinate computation.

Prop. The Levi-Civita connection as defined in coordinates by (\ref{eq:lc-coords}) is compatible with the metric.

Proof. First we expand out the right side of (\ref{eq:compatibility}). \begin{align} \langle \nabla_Z X , Y\rangle + \langle X , \nabla_Z Y\rangle &= \langle \nabla_{Z^k \partial_k}(X^i \partial_i) , Y^j\partial_j \rangle + \langle X^i \partial_i , \nabla_{Z^k \partial_k}(Y^j \partial_j)\rangle\\
&= Z^k(Y^j\langle \partial_k X^i\partial_i + X^i \nabla_{\partial_k}\partial _i , \partial_j\rangle + X^i\langle \partial_i , \partial_kY^j \partial_j + Y^j \nabla_{\partial_k}\partial_j\rangle)\\
&= Z^k(Y^j\langle (\partial_kX^l + X^i\Gamma_{ik}^l)\partial_l , \partial_j\rangle + X^i \langle \partial_i , (\partial_kY^l + Y^j\Gamma_{kj}^l)\partial_l\rangle)\\
&= Z^kY^j(\partial_kX^l + X^i\Gamma_{ik}^l)g_{lj} + Z^kX^i(\partial_kY^l + Y^j\Gamma_{kj}^l)g_{il}\\
&= Z^k(Y^j\partial_kX^lg_{lj} + X^i\partial_kY^l g_{il}) + Z^kX^iY^j(\Gamma_{ik}^lg_{lj} + \Gamma_{kj}^lg_{il}). \end{align} Next we expand out the left size of (\ref{eq:compatibility}). \[ \nabla_Z\langle X , Y\rangle = Z^k \partial_k\langle X^i \partial_i , Y^j \partial_j\rangle = Z^k \partial_k(X^i Y^j g_{ij}) = Z^k(Y^j\partial_kX^ig_{ij} + X^i\partial_kY^jg_{ij}) + Z^kX^iY^j\partial_kg_{ij}. \] Note these expansions are quite similar, and we see that in fact the right and left sides of (\ref{eq:compatibility}) are equal so long as we can show \[ \partial_kg_{ij} = \Gamma_{ik}^lg_{lj} + \Gamma_{kj}^lg_{il}. \] Indeed, to show this we use the definition of the Christoffel symbols \[ \Gamma_{ij}^k = \frac{1}{2}g^{kl}(\partial_ig_{jl} + \partial_jg_{li} - \partial_lg_{ij}) \] and apply the matrix \(g_{km}\) to both sides to conclude \[ \Gamma_{ij}^k g_{km} = \frac{1}{2}(\partial_ig_{jm} + \partial_jg_{mi} - \partial_mg_{ij}). \] Thus using the above expression twice we can compute \[ \Gamma_{ik}^lg_{lj} + \Gamma_{kj}^lg_{il} = \frac{1}{2}(\partial_ig_{kj} + \partial_kg_{ji} - \partial_jg_{ik}) + \frac{1}{2}(\partial_kg_{ji} + \partial_jg_{ik} - \partial_ig_{kj}) = \partial_kg_{ij} \] as needed.

It turns out that these two properties – symmetry and metric compatibility – are quite special. In fact, on a Riemannian manifold there will only be one connection that satisfies both properties.

Prop. (Fundamental Theorem of Riemannian Geometry). For any Riemannian manifold \(M\), there exists a unique connection \(\nabla\) that is both symmetric and compatible with the metric. This connection is called the Levi-Civita connection.

Proof in coordinates. We have already demonstrated existence, for the Levi-Civita connection is symmetric and metric-compatible. To see why an arbitrary symmetric and metric-compatible connection \(\nabla\) must be the Levi-Civita connection, we work locally in coordinates \((x^i)\) and write \(\nabla_{\partial_i}\partial_j = A_{ij}^k\). By the same computation we performed to show symmetry of the Levi-Civita connection, we see the symmetry of \(\nabla\) is equivalent to \(A_{ij}^k = A_{ji}^k\). Similarly, we see \(\nabla\) is compatible with the metric exactly when \[ \partial_kg_{ij} = A_{ik}^lg_{lj} + A_{kj}^lg_{il}. \] by the corresponding computation for the Levi-Civita connection; this expression is often called the first Christoffel identity. These two requirements give a linear system of \(\frac{1}{2}n^2(n+1)\) equations with the same amount of unknowns. The trick to solve this system is to permute the first Christoffel identity to get cancellation and solve for the sum \begin{align} \partial_i g_{jl} + \partial_j g_{il} - \partial_l g_{ij} = (A_{ij}^p g_{pl} + A_{il}^p g_{jp}) + (A_{ji}^p g_{pl} + A_{jl}^p g_{ip}) - (A^p_{li} g_{pj} + A_{lj}^p g_{ip}) = 2A_{ij}^p g_{pl}. \end{align} Then applying the inverse matrix \(g^{kl}\) we recover the definition of the Christoffel symbols: \[ A_{ij}^k = \frac{1}{2}g^{kl}(\partial_i g_{jl} + \partial_j g_{il} - \partial_l g_{ij}). \]

Proof without coordinates. Existence follows from the Levi-Civita connection. For uniqueness, suppose \(\nabla\) is a symmetric and metric-compatible connection and use both properties to write \begin{align} X\langle Y , Z\rangle_g = \langle \nabla_X Y , Z \rangle_g + \langle Y , \nabla_X Z \rangle_g = \langle \nabla_X Y , Z \rangle_g + \langle Y , \nabla_Z X \rangle_g + \langle Y , [X, Z]\rangle_g. \end{align} We will use a similar trick as the proof in coordinates to find an expression for \(\nabla\). By cyclically permuting the above, we get two more identities: \begin{align} Y\langle Z , X\rangle_g = \langle \nabla_Y Z , X \rangle_g + \langle Z , \nabla_Y X \rangle_g = \langle \nabla_Y Z , X \rangle_g + \langle Z , \nabla_X Y \rangle_g + \langle Z , [Y, X]\rangle_g\\
Z\langle X , Y\rangle_g = \langle \nabla_Z X , Y \rangle_g + \langle X , \nabla_Z Y \rangle_g = \langle \nabla_Z X , Y \rangle_g + \langle X , \nabla_Y Z \rangle_g + \langle X , [Z, Y]\rangle_g. \end{align} Now adding the first two equations and subtracting the third gives the cancellation \begin{align} X\langle Y , Z\rangle_g + Y\langle Z , X\rangle_g - Z\langle X , Y\rangle_g = 2 \langle \nabla_X Y , Z \rangle_g + \langle Y , [X, Z]\rangle_g + \langle Z , [Y, X]\rangle_g - \langle X , [Z, Y]\rangle_g. \end{align} Thus we can solve for \(\langle \nabla_X Y , Z \rangle_g\) to find \begin{align} \langle \nabla_X Y , Z \rangle_g = \frac{1}{2}(X\langle Y , Z\rangle_g + Y\langle Z , X\rangle_g - Z\langle X , Y\rangle_g - \langle Y , [X, Z]\rangle_g - \langle Z , [Y, X]\rangle_g + \langle X , [Z, Y]\rangle_g). \end{align} which uniquely determines the connection \(\nabla\). The above is thus a coordinate-invariant expression for the Levi-Civita connection and is called Koszul’s formula.