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.
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