Differential geometry vs Riemannian geometry

Question

I'd like to know the relation between differential geometry and Riemannian geometry.

exactly what makes them different?

Is one an special case of another? exactly how?

Answer 1

You can say that

• Riemannian geometry
• Lorentz geometry
• Symplectic geometry
• Poisson geometry
• Contact geometry
• Complex geometry
• Kähler geometry

and others are instances of differential geometry, which is the study of differentiable manifolds. You have several possibilities of geometric structures to equip your manifold with, and each one gives a "subarea" of differential geometry.