I want to start studying differential geometry but I can't seem to find a proper starting path. Whenever I try to search for differential geometry books/articles I get a huge list. I know that it is a broad topic, but I want some advice for you regarding the books and articles. I want to learn differential geometry and especially manifolds. I have some background in abstract algebra, linear algebra, topology, real/complex analysis.
[Math] Differential geometry study materials
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Stillwell's Naive Lie theory was essentially written as an answer to this question. I quote from the introduction:
It seems to have been decided that undergraduate mathematics today rests on two foundations: calculus and linear algebra. These may not be the best foundations for, say, number theory or combinatorics, but they serve quite well for undergraduate analysis and several varieties of undergraduate algebra and geometry. The really perfect sequel to calculus and linear algebra, however, would be a blend of the two —a subject in which calculus throws light on linear algebra and vice versa. Look no further! This perfect blend of calculus and linear algebra is Lie theory (named to honor the Norwegian mathematician Sophus Lie—pronounced “Lee ”).
I think, one of the big problem is that aside from theoretical physics (string theory, general relativity), most mathematicians aren't terribly aware of what engineers and scientists use differential geometry for. This certainly makes it difficult to write/plan a course in that regard.
It was only recently when I heard a talk by Alain Goriely did I find out that biologists care about differential geometry too! But during the talk there were quite a few theorems about curves in three dimensional space that I've never heard of, and I do geometry PDEs and general relativity for a living. This at least provides an isolated data point to illustrate the above, on how mathematicians typically don't know what is or is not important for applications to other fields.
Ideally such a course/textbook should be prepared by someone with great interdisciplinary familiarity.
In terms of differential geometry "as a natural extension of calculus", I think you may have better luck going to older textbooks, where instead of calling it differential geometry, the subject is just called "advanced calculus". Quite a few books are written back then with an eye toward the applied mathematician (but of course, I am incapable of giving recommendations).
Let me add that I am currently supervising a third-year undergraduate course in University of Cambridge on differential geometry. It fits half of your bill: it does not assume more than basic calculus and linear algebra (partly due to the funny way the Cambridge maths curriculum is rather scant on analysis); the current set of lecture notes is written by Gabriel Paternain (if you are interested you can try asking him for a copy). Unfortunately the way the degree program works, the course won't attract much non-pure-mathematicians other than the future-theoretical-physicists. So I can't really comment on how well it works for engineers and other scientists.
The course is divided in essentially four parts:
- Definition of manifolds as submanifolds in Euclidean space, diffeomorphisms and smooth maps, Sard's theorem and degree mod 2.
- Curves and surfaces in space. Frenet frame, curvature, torsion of curves; isoperimetric inequality. First and second fundamental form, mean and Gaussian curvature.
- Calculus of variations, geodesics, minimal surfaces.
- More about curvature, leading up to Gauss-Bonnet.
One more note: I just remembered that Gary Gibbons is teaching a course titled "Applications to Differential Geometry to Physics". It is not necessarily elementary, but certainly has a lot of applications. Being taught from the point of view of a polymath, the examples given in the notes do cover some more ground than is typical.
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I would recommend Lee's book "Introduction to Smooth Manifolds." It's a long book but is comprehensive, has complete proofs, and has lots of exercises.