AS/SC/AK/ MATH 4250 6.0

Differential Geometry

      Curves and surfaces in 3-space, tangent vectors, normal vectors, curvature, introduction to topology, manifolds, tangent spaces, multilinear algebra and tensors. Normally offered in alternate years.
      Differential geometry uses the methods of multivariable calculus and linear algebra to study curves, surfaces, and higher-dimensional "manifolds". The subject was initiated by Gauss, further developed by Riemann, and has seen important advances in this century due to Cartan and Lie. The concept of a manifold and the geometric structures associated to it play central roles in such fields as dynamical systems, topology, harmonic analysis and differential equations. Many basic ideas are involved in differential geometry: how can space be curved, what is the relationship between local and global information, why don't you always end up where you started when going around a loop? Differential geometry is now a central tool in theoretical physics; this century has seen the "geometrization of physics" with significant applications to the fields of general relativity , classical mechanics and elementary particle theory.
      The first part of the course will study surfaces, both in three- and higher-dimensional spaces, using as its main tool vector fields on these surfaces. We will treat geodesics, parallel transport and curvature. We then discuss differential forms and a general form of Stokes's Theorem. This will lead to a significant and beautiful result, the Gauss-Bonnet Theorem, relating local geometric to global topological information. It is a prototype of many important modern results. The later part of the course will be an introduction to differentiable manifolds.
      The course will be accessible to students with a background in vector calculus and linear algebra. Some knowledge of differential equations will also be useful. In fact, the course may be seen as putting the finishing touches on these courses, unifying them and showing their significant applications.
      The grade will be based on term tests and assignments (70%) and a final project (30%). The text has not been chosen.