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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
I disagree that the manifolds articles on wikipedia are good at giving the intuition. for connections in general : the intuition is with the "directional derivative", and for the Levi-civita connection in particular : the idea is that in $\mathbb{R}^n$ there are many connections which are not torsion-free – user1952009 Feb 27 at 22:29

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.

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In a manifold with a metric, in the neighborhood of any point $p$ you can always choose coordinates so that the partial derivatives of the metric components vanish at $p$. Let's call that a "locally flat coordinate system at $p$". (For a Riemannian metric you can also choose the components of the metric at $p$ to be $\delta_{ij}$.) In a locally flat coordinate system at $p$, the components at $p$ of the Levi-Civita covariant derivative of any tensor are simply the first partial derivatives of the tensor components. The intuition for the connection itself is that it defines "parallel transport" of a tensor by "constant components" at $p$, where "constant" means the first partial derivatives vanish in locally flat coordinates.

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My marginal note to [The Absolute Differential Calculus, Levi-Civita,pp.101-104] says, "Pick a direction in the plane at the contact point at the start of roll. That direction agrees with the same direction in the tangent plane at the start point. Now, roll along the curve in the surface. That direction at the end of the roll picks out a direction in the tangent plane at the end of the roll." That is the intuition for parallel transport explained soooo much better in those pages.

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