Differential Geometry

Catalog description

This course is a transition from vector calculus to differential geometry, the study of curved spaces. The course plan is to move from a study of extrinsic geometry of curves and surfaces in space, familiar from multivariable calculus, to the intrinsic geometry of manifolds. This includes the study of tangent spaces and vector fields and the concept of Riemannian manifolds and leading to explicit characterizations of metrics, connections, and curvatures. Computational tools will include tensor algebra and differential forms. Using these, derivatives on manifolds and integration on manifolds will generalize the corresponding notions from multivariable calculus. Further topics also include the Euler characteristic, The Gauss-Bonnet theorem, symmetry, homogeneous spaces, and applications such as Electromagnetism and General Relativity.