MATH 453: Differentiable Manifolds

Note: If this course is being taught this semester, more information can be found at the course home page.

Cross Listed

(none)

Prerequisites

MTH235, MTH265 or MTH240; desirable courses: MTH255, MTH256, MTH266

This course is a prerequisite or co-requisite for

MTH443 and 500 level geometry courses

Description

Basic definitions of manifolds, submanifolds, vector fields, vector bundles and differential forms. The undergraduate subjects of calculus/point set topology/linear algebra; specifically the inverse function theorem, Stokes theorem, the fundamental theorem of ODE, defined on Euclidean space at the undergraduate level are generalized to work on manifolds.

This course introduces the tools used in graduate level analysis, Riemannian geometry, algebraic topology to deal with high dimensional manifolds.

Topics covered

Topics include manifolds, vector fields and vector bundles, differential forms, partitions of unity, homogeneous spaces, inverse function theorem and submanifolds. Other topics may be chosen from tensors, Lie derivatives, deRham cohomology, integrable flows (fundamental theorem of ODE for manifolds), and Frobenius theorem.

MTH255 (“Curves and surfaces differential geometry”) and MTH256(“Riemann geometry”). While MTH453 develops the modern tools for high dimensional manifolds, MTH255 and MTH256 concentrate on what can be accomplished in 2 and 3 dimensions using mostly the tools of calculus. We recommend that undergraduates take those courses first to see what can be done.

One description of modern Riemannian geometry research: The attempt to discover for higher dimensions analogs to the results proven in MTH255 & MTH256 using the tools from MTH453. Since the answers in higher dimensions appear to be significantly more complicated than in 2 and 3 dimensions this is an on-going project.