Intuition behind Levi-Civita connection

Question

Can you tell me what the intuition behind the Levi Civita connection is? Is there any good picture to have in mind, when dealing with it?

Answer 1

The Levi-Civita connection, aka the Riemannian connection, is the preferred connection in the tangent bundle of a Riemannian manifold. Connections in the tangent bundle are also called linear, or affine, depending on personal preferences.

To appreciate its value please notice that there are a lot of connections that one can define in the tangent bundle (or any other vector bundle), and using any of them a notion of a covariant derivative of tensor fields may be developed. However, if an arbitrary connection is used, many things would not look natural, for instance we would not be able to manipulate with indices as we like.

Fortunately, in the case of Riemannian manifolds, there exist a connection that is compatible with the Riemannian structure (that is, the metric tensor $g$). More precisely, if such a connection is torsion-free, then it is unique. This one is called the Levi-Civita connection of the Riemmannian metric $g$, and sometimes we use special notation for it $\nabla^g$.

"Compatible" means that $\nabla g =0$, that is the metric tensor is parallel with respect to this connection. Intuitively, this means that parallel transport preserves the inner product. In calculations, this fact is responsible for our ability to raise and lower indices under the covariant differentiation.

The best picture in mind to have is the Koszul formula for the Levi-Civita connection, coupled with the fact that in $\mathbb{R}^n$ equipped with the standard Euclidean metric the Levi-Civita connection is exactly the partial differentiation of the components $\nabla_i = \partial_i$ (all the Christoffel symbols $\Gamma^k_{ij}$ vanish), so really it is a very familiar object for those who studied multivariable calculus.