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I've done a masters taught module in GR and from what I've learnt these are two of some of the most important significance of needing a Riemannian Geometry:

1) If we consider the Lagrangian of a freely-falling particle given by $L= \int ds \sqrt{g_{uv}\dot{dx^u}\dot{dx^v}} $and find the equations of motion, by the principle of least action this is the shortest path and so must be the definition of a geodesic.

The alternate way to define a geodesic is that the tangent vector of is parallel transported along itself :

$V^u \nabla_u V^a =0 $

Then via the fundamental theorem of Riemannian geometry,( given a manifold equipped with a non-degenerate, symmetric, differentiable metric there exists a unique torsion-free connection such that $\nabla_a g_bc =0 $), we can show that these two definitions of a geodesic are important

2) Due to the fundamental theorem of Riemannian geometry, equipped with a metric on the space-time, we can express important objects such as the Christoffel symbol and Riemman tensor in terms of the metric, and so the metric effectively encodes all the information about the space-time

Are there other important roles played by Riemannian geometry?

I find the first one pretty interesting- is it Palatini formalism that looks at when the geometry is non-Riemmanian and so the geodesics would not be the same?

Thanks in advance.

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closed as too broad by user99914, Lord Shark the Unknown, ahulpke, Strants, Parcly Taxel Mar 6 '18 at 16:06

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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Even a "very simple" problem of a point particle moving freely on top of a curved surface in a framework of classical mechanics heavily involves Riemann geometry

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