What do the curvature and torsion measure? Consider a smooth surface for simplicity. What does its curvature measure? What does its Gaussian/Riemannian curvature measure? What does its torsion measure?
What does the Ricci curvature measure?

If you search the questions on MSE and/or MO, I think you will,find some pretty good insights into these topics; if I recall correctly, these topics have come up more than once; many times, in fact.  For example, this one might be a good place to start:
Geometric interpretation of connection forms, torsion forms, curvature forms, etc
Good luck in you searches!
