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 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 pseudo-Riemannian metric: a smoothly varying nondegenerate (not necessarily positive definite) symmetric bilinear form $g$ 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 $\nabla _{X}Y-\nabla _{Y}X=[X,Y]$.

Definition by formula

The formula for the Levi-Civita connection requires us to use the fact that $g$ is nondegenerate. So, instead of directly specifying $\nabla _{X}Y$ for vector fields $X$ and $Y$, the formula specifies $g(\nabla _{X}Y,Z)$ for $X,Y,Z$ vector fields, as follows:

$g(\nabla _{X}Y,Z)={\frac {Xg(Y,Z)+Yg(Z,X)-Zg(X,Y)+g(Y,[Z,X])+g(Z,[X,Y])-g(X,[Y,Z])}{2}}$.

Facts

The Levi-Civita connection is unique

Further information: Levi-Civita connection exists and is unique

The proof combines the metric condition, the torsion-free condition, and the nondegeneracy of $g$.

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

Further information: Christoffel symbols of a connection

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.