Introduction to Differential Geometry

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mathwonk

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I have done some research since last night on connections between differential geometry and algebraic topology and wish to revise my advice.

Doodle Bob is the expert here but I will make some remarks subject to his review.

The concept of second derivatives which I aligned mostly with diff geom, do indeed have significant applications to algebraic topology, via morse theory.

Morse theory considers the second derivative matrix of a real valued function at a critical point, especially when that matrix of second derivatives is non singular. The subsequent identification of critical points as saddle points or maxs or mins is actually central to understanding the global topology of the manifold.

E.g. a torus is distinguished among compoact oriented 2 manifolds by having a function with a max a min and 2 saddle points.

Morse theory allows one to construct a CW complex reflecting the homotopy of the a manifold, just from knowing the critical poinmts of one non degenerate function, with its second derivatives at those points.

Moving on to Riemannian structures, it is also useful to introduce a metric to measure non topological entities like length, but which turn out to have topological implications.

E.g. in the study of homotopy groups, i.e. loop groups, it is useful to introduce a length for these loops, in order to discern "shortest" length loops or geodesics.


These give critical points for the length fucntion on the space of loops and allow one to determine the homotopy of the loop space on the manifold.

Some sphere e.g. can be realized as loops spaces of smaller spheres, via the suspension construction of freudenthal, and thus riemannian geometry yeil;ds results on the homotopy groups of spheres, a purely topological question.

The deep periodicity theorems of bott on the stable homotopy of classical groups is another consequence of these differential geometry methods.


If one tries to read about this say in milnor's book on morse theory, he/she may well wish he had followed doodle bob's advice and learned an intuitive version of curvature and differential geometry ideas from an elementary book first, like millman parker, or do carmo.


so my original comments about alg top not using diff geom, did not go far enough into alg top it seems. just my ignorance. it is always better to know something than not to.
 

mathwonk

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hardy was my acad great great grandfather, (but i still pick on his calculus book, pure mathematics, for tiny things).

that makes my great^5 grand dad = arthur cayley, the man who apparently first defined abstract groups (1854).

ours is a small family.
 

mathwonk

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i think it was not meant that one cannot find gems among dover books, much appreciated ones at the price too, including some great classics.

I have owned dover versions of einstein's papers, the principia of newton, electricity and magnetism by maxwell, riemann's works, transfinite numbers by cantor, and more recently a nice modern work on systems of ode's by paul waltman.

but there is a caution that dover books often are books whose terminology or point of view is out of date, and hence may lead beginners down a path of isolation, all too common among physicists wrt mathematics.

another unfortunate observation, dover books are no longer physically of the high quality they once were. Bindings in sewn signatures that lasted essentially forever, have been replaced by cheap glued pages, or "perfect" bindings that can easily fall apart. this is a sad change. I would willingly pay a few dollars more for a decent binding.

i wish dover would publish the all too scarce and costly book: foundations of modern analysis, by dieudonne, but i agree henry edwards' advanced calculus is a modern, well written book.

anything by richard silverman is also recommended.
 
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mathwonk said:
I have done some research since last night on connections between differential geometry and algebraic topology and wish to revise my advice.

Doodle Bob is the expert here but I will make some remarks subject to his review.
ha! I'm still embarassed over that "the determinant is an open map" fiasco.

I very much agree with what you've written here. In a rather surprising synchronicity, this discussion of Morse theory ties in nicely to some papers that I've been reading in the past few days regarding polytopes in R^n in which an index is defined on the vertices of the polytopes and summing all of the indices up of course results in the Euler characteristic.

Anyway, I have repeatedly found that many mysterious results in differential geometry -- particularly, those which start with questions of the sort "Why the hell would anyone construct a tensor that looks like that?", e.g. the Schwartzian derivative -- usually have a fairly solid low-dimensional analogue that clears the whole thing up.
 

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