Grad student Geometry and Topology learning seminar

Robert Garbary, Department of Pure Mathematics, University of Waterloo

“Linearizing a smooth map”

I’m going to continue talking about the tangent spaces of a manifold. Given a smooth map f :M →N andapointp∈M,we’regoingtoconstructalinearmapfp :TpM →Tf(p)N -it’s this crazy thing you may have heard about before called the ’chain rule’. I will go through an explicit example or two in painstaking detail, and talk about why we care about points where this map has rank 0.