Dear fellow Mathematicians and Physicists, As the fall term closes and spring term starting next year, I am deciding on which math class to take. Simply put, I'm a math major (possibly engineering if there are enough slots in my schedule), more applied, seeking to apply math concepts to science and engineering. I have taken a substantial amount of classes in control systems and fluid dynamics to know that area of engineering well enough. I have already covered the fundamental math classes - Analysis, Algebra, Topology, elementary ODE and PDE. Then when looking at the classes which caught my eye, one of them stood out: DIFFERENTIAL GEOMETRY. I talked to the professor and he said given my background, I should find the material accessible. In short, we will be studying Riemann Geometry and develop the theory using rigorous real analysis. So far, differential geometry gives me ideas of arc length, tangent, surface integral - all the multivariable calculus stuff. Yet, I am well aware this graduate level class is NOT about that. A quick look at the synopsis, I see stuff like 2-forms, implicit function theorem, manifolds, imbedding, (general case) of Stoke's theorem. Thus my question: Is this class, Differential Geometry (graduate level), be useful in an applied field like physics or engineering? I heard the phrase that General Relativity is written in the language of Differential Geometry so I feel somewhat glad that there is some applicability in this subject. Well, certainly I don't mind learning the theory if it is solely in the realm of Pure math, but it'll be good if there are some real world applications, at least from my point of view. Oh, and what the heck is the difference between Differential Geometry and Riemann Geometry? Thanks.