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?
Intuition behind Levi-Civita connection
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differential-geometry
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1The wikipedia article on [connections](http://en.wikipedia.org/wiki/Connection_%28vector_bundle%29) 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". – 2011-10-18
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1I 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 – 2016-02-27