On General Relativity the Levi Civita connection is quite important. Indeed, General Relativity is all about connecting the curvature of spacetime with the distribution of matter and energy, at least that is the intuition I've always read about.
Now, given a smooth manifold $M$ which is supposed to be spacetime, there is not a direct way to talk about "curvature" of $M$. The meaningful thing is to talk about the curvature of a connection defined on some bundle over $M$.
In General Relativity, the curvature appearing in Einstein's equations, is curvature of a connection on the bundle $TM$ introduced by means of a covariant derivative operation.
More than that, one picks one specific connection: the Levi Civita connection, which is the unique torsion free connection for which the covariant derivative of the metric tensor vanishes.
So in summary: the curvature of spacetime which is dealt in General Relativity comes from a connection, the connection is introduced by a coviarant derivative and finally the covariant derivative chosen (hence the connection chosen) is the Levi Civita connection.
Why is that? I mean, this is not the only existing connection. Why in General Relativity, the relevant connection from the Physics point of view is the Levi Civita connection?
What is the Physics motivation for the need of the Levi Civita connection? Reasoning with Physics, and remembering that what we want to achieve is a description of spacetime and gravity where matter influences the curvature of spacetime, what would be the Physics motivation for the Levi Civita connection?
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