Levi-Civita connection

This lives as an element of: the space of all linear connections, which in turn sits inside the space of all $\mathbb {R}$-bilinear maps $\Gamma (TM)\times \Gamma (TM)\to \Gamma (TM)$

Definition

Given data

A Riemmanian manifold $(M,g)$ (here, $M$ is a differential manifold and $g$ is the additional structure of a Riemannian metric on it).

More generally, we can also look at a pseudo-Riemannian manifold, or a manifold with a smoothly varying nondegenerate (not necessarily positive definite) symmetric positive definite bilinear form in each tangent space.

Definition part

A Levi-Civita connection on $(M,g)$ is a linear connection $\nabla$ on $M$ satisfying the following two conditions:

• The connection is metric, viz $Xg(Y,Z)=g(\nabla _{X}Y,Z)+g(Y,\nabla _{X}Z)$
• The connection is torsion-free, viz $\nabla _{X}Y-\nabla _{Y}X=[X,Y]$

Facts

The Levi-Civita connection is unique

The proof of the uniqueness of the Levi-Civita connection is as follows.

• Take three vector fields $X,Y,Z$. Now, consider the three equations obtained by cycling $X,Y,Z$ in the first condition. Solving this system of linear equations, we can express $g(\nabla _{X}Y,Z)$ in terms of $X,Y,Z$.

Explicitly:

$g(\nabla _{X}Y,Z)+g(Y,\nabla _{X}Z)=Xg(Y,Z)$

$g(\nabla _{Y}X,Z)+g(X,\nabla _{Y}Z)=Yg(Z,X)$

$g(\nabla _{Z}X,Y)+g(X,\nabla _{Z}Y)=Zg(X,Y)$

Now let's choose to focus only on the clockwise cyclic expressions, that is, the three expressions $p=g(\nabla _{X}Y,Z),q=g(\nabla _{Y}Z,X),q=g(\nabla _{Z}X,Y)$.

Writing everything in terms of these three (we here make use of the torsion tensor vanishing):

$p+q=Xg(Y,Z)-g(Y,[X,Z])$

and similarly for the other three variables.

We thus get expressions for $g(\nabla _{X}Y,Z)$ in terms of $g$ and the Lie bracket.

• Now use the fact that $g$ is nondegenerate to conclude that knowledge of the function $g(\nabla _{X}Y,Z)$ for all $Z$ helps us fix $\nabla _{X}Y$ uniquely.
• This means that we have a unique definition for $\nabla$.

Note that the nondegeneracy of $g$ is very important, otherwise knowing the value of the inner product for any three vectors may not necessarily help us in computing the value of $nabla_{X}Y$

To show that the Levi-Civita connection exists, it suffices to check that the map sending $Z$ to what we propose for $g(\nabla _{X}Y,Z)$ is actually a linear map.

Christoffel symbols

The Levi-Civita connection on a manifold $M$ is a map $\nabla :\Gamma (TM)\times \Gamma (TM)\to \Gamma (TM)$. This means that at any point $p\in M$, it gives a map $T_{p}(M)\times T_{p}(M)\to T_{p}(M)$, which roughly differentiates one tangent vector along another.

Let $\partial _{1},\partial _{2},\ldots ,\partial _{n}$ form a basis for the tangent space $TM$. Then, the Christoffel symbol $\Gamma _{ij}^{k}$ is the component along $e_{k}$ of the vector $\nabla _{\partial _{i}}\partial _{j}$.

The Christoffel symbols thus give an explicit description of the Levi-Civita connection. Namely, the Levi-Civita connection can be expressed using the Christoffel symbols.