I am a physics undergrad, and need to study differential geometry ASAP to supplement my studies on solitons and instantons. How much topology do I need to know? I know some basic concepts reading from the internet on topological spaces, connectedness, compactness, metrics and quotient hausdorff spaces. Do I need to go deeper? Also, could you suggest me some chapters from topology textbooks to brush up this knowledge? Could you please also suggest a good differential geometry books that covers the topics in differential geometry that are needed in physics in sufficient detail (without too much emphasis on mathematical rigour)? I have heard of the following textbook authors: Nakhara, Fecko, Spivak. Would you recommend these?
[Math] Topology needed for differential geometry
differential-geometrygeneral-topologyreference-requestsoft-question
Related Solutions
By ``basic topology of $\mathbb{R}^n$'' I assume that you are familiar with the notions of openness, closedness, connectedness, and compactness. If you are unclear on these notions (I found compactness hard to get used to), you should remedy that before attempting to learn differential geometry.
If you understand these, then you're probably already prepared to read an introductory book on differential geometry, such as do Carmo's Differential Geometry of Curves and Surfaces or O'Neill's Elementary Differential Geometry. Apart from the concepts I mentioned above, all the necessary topology is developed alongside the geometry in these books (e.g. homeomorphism, homotopy, Euler characteristic, and so on).
If you want to learn quickly about the topology of smooth manifolds without having to learn about general topological spaces, there is probably no better place to look than Milnor's Topology from the Differentiable Viewpoint. A more in-depth treatment along the same lines is Guillemin and Pollack's Differential Topology.
I offer that differential geometry may be a much broader field than algebraic topology, and so it is impossible to have textbooks analogous to Switzer or Whitehead.** So, although it isn't precisely an answer to your question, these are the most widely cited differential geometry textbooks according to MathSciNet. I've roughly grouped them by subject area:
- Bridson and Haefliger "Metric spaces of non-positive curvature"
- Burago, Burago, and Ivanov "A course in metric geometry"
- Gromov "Metric structures for Riemannian and non-Riemannian structures"
- Kobayashi and Nomizu "Foundations of differential geometry"
- Lawson and Michelsohn "Spin geometry"
- Besse "Einstein manifolds"
- Abraham and Marsden "Foundations of mechanics"
- Arnold "Mathematical methods of classical mechanics"
- O'Neill "Semi-Riemannian geometry with applications to relativity"
- Wald "General relativity"
- Hawking and Ellis "The large scale structure of spacetime"
- Helgason "Differential geometry, Lie groups, and symmetric spaces"
- Olver "Applications of Lie groups to differential equations"
- Rabinowitz "Minimax methods in critical point theory with applications to differential equations"
- Willem "Minimax theorems"
- Mawhin and Willem "Critical point theory and Hamiltonian systems"
- Katok and Hasselblatt "Introduction to the modern theory of dynamical systems"
- Temam "Infinite-dimensional dynamical systems in mechanics and physics"
- Guckenheimer and Holmes "Nonlinear oscillations, dynamical systems, and bifurcations of vector fields"
- Hale "Asymptotic behavior of dissipative systems"
- Hirsch, Pugh, and Shub "Invariant manifolds"
- Giusti "Minimal surfaces and functions of bounded variation"
Of the metric geometry books (#1), BBI's book is good for self-study, while Gromov's book is nice to have around and open to random pages.
Kobayashi and Nomizu is a hard book, but it is extremely rewarding, and I don't know of any comparable modern book - I would disagree in the extreme with whoever told you to skip it. It is only aged in superficial ways, such as some notations. Lawson and Michelsohn's book is quite advanced, and K-N vol. 1 (at least) would be a prerequisite. It includes a chapter on the Atiyah-Singer index theorem.
Besse's book covers "special Riemannian metrics", including a review of Riemannian, Kahler, and pseudo-Riemannian geometry. It is more of a reference book, good to look through sometimes.
For classical mechanics, Abraham and Marsden is quite sophisticated, and it is necessary to have a solid geometrical footing (roughly K-N vol 1) before going into it; Arnold's book is more introductory and would probably be very nice for self-study.
The general relativity books in #5 are all introductory and pretty approachable.
I'm not so familiar with the books #6-9. Guckenheimer and Holmes seems very friendly.
Personally, I'd also recommend Chow, Lu, and Ni's "Hamilton's Ricci flow," the content of which is necessary to understand the proofs of the Poincare and geometrization conjectures. The first chapter is an excellent mini-textbook on "classical" Riemannian geometry, reaching just beyond introductory books like Do Carmo's.
** just to underline the point in the first sentence - there are only five general or algebraic topology textbooks (Hatcher, Spanier, Rolfsen, Engelking, and Kelley), four differential topology textbooks (Bredon, Hirsch, Milnor "Morse Theory", and Milnor-Stasheff) and two convex geometry textbooks (Schneider and Ziegler) as widely cited as the above differential geometry textbooks
Best Answer
You shouldn't need much. Almost all you need to know about topology (especially of the point-set variety) should have been covered in a course in advanced calculus. That is to say, you really need to know about "stuff" in $\mathbb{R}^n$. (The one main exception is when you study instantons and some existence results are topological in nature; for that you will need to know a little bit about fundamental groups and homotopy.) The reason is that differential topology and differential geometry study objects which locally look like Euclidean spaces. This dramatically rules out lots of the more esoteric examples that point-set topologists and functional analysts like to consider. So most introductory books in differential geometry will quickly sketch some of the basic topological facts you will need to get going.
In terms of topology needed for differential geometry, one of the texts I highly recommend would be
It is quite mathematical and quite advanced, and covers large chunks of what you will call differential geometry also. One can complement that with his Riemannian Manifolds to get some Riemannian geometry also.
But since you are asking from the point of view of a Physics Undergrad, perhaps better for you would be to start with either (or both of)
and follow-up with