I am teaching a introductory course on differentiable manifolds next term. The course is aimed at fourth year US undergraduate students and first year US graduate students who have done basic coursework in
point-set topology and multivariable calculus, but may not know the definition of differentiable manifold. I am following the textbook [Differential Topology](http://rads.stackoverflow.com/amzn/click/0132126052) by 
Guillemin and Pollack, supplemented by Milnor's [book](http://rads.stackoverflow.com/amzn/click/0691048339).


My question is: **What are good topics to cover that are not in assigned textbooks?** 



What should be taught in a 1st course on smooth manifolds?
I nominate Ehresmann's theorem according to which a proper submersion between manifolds is automatically a locally trivial bundle. It is incredibly useful, in deformation theory for example, but is sadly neglected in introductory courses and books on manifolds. It is completely elementary: witness [these lecture notes](http://www.math.ucla.edu/~petersen/manifolds.pdf) by Peter Petersen, where it is proved in a few lines on page 9, the prerequisites being about two pages long.


[Bjørn Ian Dundas](http://www.uib.no/People/nmabd/dt/080627dt.pdf) and our friend  [Andrew Stacey](http://www.math.ntnu.no/~stacey/documents/tma4190.lectures.handout.2009-04-24.pdf)  also have online documents proving this theorem.


