Is differential geometry useful for algebra?

Question

Is differential geometry useful for algebra? Or are they not very related? I don't need a complicated answer; I just want to know if it would be useful for learning algebra.

Thanks in advance

Answer 1

Yes, differential geometry can be useful for algebra, although it is of course a different subject. Geometric structures on Lie groups often relate directly to algebraic structures on Lie algebras, by the work of Felix Klein, and later Elie Cartan. A torsion-free connection of curvature zero on a manifold can be expressed by algebraic identities, see Bianchi identities, etc. In general, Lie groups and Lie algebras are very much related, thus combining differential geometry and algebra.

Answer 2

In a general way what makes mathematics exciting as a subject is that as one learns more about its different parts, one sees how these parts illuminate each other. The connections become richer and more subtle as one reaches higher levels of knowledge in mathematics. Vastly oversimplifying, I think of differential geometry as being concerned with the properties of curves and surfaces. In this sense it is grows out of concepts and ideas one learns in Calculus, and involves lots of matters concerning continuity. By comparison, algebra lies more in the domain of discrete mathematics. It often distinguishes different kinds of mathematical structures, integers, rational numbers, groups, rings, fields, etc. by showing what axioms these different structures obey.