# Intuition for Levi-Civita connection?

Levi-Civita connection is usually defined as the unique connection which is torsion free and preserves metric.

Question Is there some intuitively transparent constructive way to define it (or corresponding parallel transport) ?

"intuitively transparent" is up to "good will" of the ones answering.

PS

I remember the following construction but it is not intrinsic and it is not clear for me how to derive the formula for Christoffel symbols from it in transparent way. Nevertheless let me mention it.

Assume we have a submanifold in some Riemann manifold. To define the transport along the curve on a submanifold we can do infinitesemal translation in bigger manifold - the resulting vector may not be tangent to submanifold - so we will make a projection on the tangent space of the manifold.

In this way starting from standard metric on R^n we can derive parallel transport and hence Levi-Civita connection on a submanifold. Expressing result in terms of submanifold's metric may be considered as way to answer the question - but it seems to be very indirect.

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– Steven Gubkin Nov 14 '12 at 20:27

Often it helps to look at any notion through its history. Parallel translation first appears in the 2-dimensional case (Minding, 1837).

The idea was to peel a small neighborhood of a curve in the surface and push it into the plane with minimum distorsion near the curve. You need to prove that under such map the image of the curve on the plane is uniquely defined up to congruence. The parallel fields on the plane can be lifted to the tangent bundle of the surface; they form parallel fields along the curve --- the parallel translation is defined.

Exactly the same idea works in higher dimensions.

This does not help to write formulas, but it is a nice way to get intuition.

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Thank you very much ! May I ask I do not quite understand "to peel ... of curve in the surface" and "push it into the plane". Surface and plane is it the same here ? Можно по-русски ? – Alexander Chervov Nov 15 '12 at 7:45
Look at the appendix of Arnold's Mathematical Methods of Classical Mechanics. He does a nice job at giving an intuitive picture. – alvarezpaiva Nov 15 '12 at 14:35

Here is a very visual way to think of what parallel transport is. It is Levi-Civita's original description for surfaces.

Take a surface embedded in Euclidean 3 space. draw a curve on it between 2 points. take the envelope of the tangent planes to the surface that are tangent planes to the curve between the two points. Note that that this envelope is a developable surface (of course, not the same surface), developable surfaces are isometric to a region in the plane, so that provides a unique way along the curve to move the tangent plane and so to say whether vectors are "parallel"

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