# Intuition behind Levi-Civita connection

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?

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The wikipedia article on connections isn't bad. The Levi-Civita connection essentially tells you how the tangent spaces at each point are "glued" together -- "How should a tangent change as I move from point to point?" Answer: "Connection". –  Bill Cook Oct 18 '11 at 23:11

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.