In other math, what is the role of the Riemannian metric?

a$)$ The Riemannian metric determines the eigenvalues of a matrix.

b$)$ The Riemannian metric characterizes the elasticity of materials.

c$)$ The Riemannian metric provides a way to measure distances and angles on smooth surfaces.

d$)$ The Riemannian metric predicts the trajectories of particles in a fluid.