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

$\mathrm{\Gamma }(TM)\times \mathrm{\Gamma }(TM)\to \mathrm{\Gamma }(TM)$

Definition

Given data

A Riemannian 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 $\mathrm{\nabla }$ on $M$ satisfying the following two conditions:

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

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 $g({\mathrm{\nabla }}_{X}Y,Z)$ in terms of $X,Y,Z$.

Explicitly:

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

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

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

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

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 $g({\mathrm{\nabla }}_{X}Y,Z)$ in terms of $g$ and the Lie bracket.

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

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 $nabl{a}_{X}Y$

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

Christoffel symbols

The Levi-Civita connection on a manifold $M$ is a map $\mathrm{\nabla }:\mathrm{\Gamma }(TM)\times \mathrm{\Gamma }(TM)\to \mathrm{\Gamma }(TM)$. 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 ${\partial }_1,{\partial }_2,\dots ,{\partial }_n$ form a basis for the tangent space $TM$. Then, the Christoffel symbol $\underset{ij}{\overset{k}{\mathrm{\Gamma }}}$ is the component along $e_k$ of the vector ${\mathrm{\nabla }}_{{\partial }_i}{\partial }_j$.

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.