is there a free book or publication with some applications of differential geometry (especially 2-manifolds) in graph theory. At that moment I know that graph which are not planar on plane, can be planar on manifolds. But that does not use differential geometry, only treats manifolds as subsets of R^3. I know that every class of graphs which is closed under taking minors, is defined by finite subset of "forbidden" minors (it is powerful Robertson-Seymour theorem).
I need to write an essay for difgeo course and I thought about something like that, but I can't see anything in my range. I mean something that uses curves, 2-manifolds, easy tensors (in particular metrics). E. g. where I can find a proof that every graph Kn is planar on manifold being sphere with big enough number of ears (torus is sphere with 1 ear) - maybe it would be cool part.
I know this answer is somewhat late, but have you viewed this book?
https://www.amazon.com/Emerging-Differential-Geometry-Mathematics-Developments/dp/1607410117
you might find sections of your interest.