Roadmap for learning differential geometry

Question

I'm trying to learn differential geometry (or at least, some aspect of it). However, I've come across various textbooks on `this topic', each building up to one major result or the other: Gauss-Bonet (as in do Carmo), Gauss' theorem egregium, or Stoke's theorem (as in Spivak's Calculus on Manifolds). Moreover, while some textbooks seem to concern general manifolds, some are focused with Riemannian manifolds; and while some focus on general spaces, some focus on surfaces in $\mathbb{R}^n$.

As someone who is trying to learn differential geometry, this above variety is unclear to me in their connections with each other. What is a good trajectory if I want to learn about the fundamentals of diff. geo's major branches? I particularly wish to learn about manifolds in general, and about Riemann manifolds. I'd appreciate if someone could give me such a roadmap + an explanation of how things are connected in it.

As requested: my background is in analysis and probability. I have a graduate-level understanding of algebra and analysis, up to measure theory. I took a course on Calculus on Manifolds but we didn’t touch manifolds (it was a bit too slow). I’m trying to learn about general manifolds and then Riemannian manifolds, but I don’t know what the difference between the two is on a meaningful level.

Answer 1

What follows is specific to the late 1970s and my own experience, and because I'm not particularly knowledgeable about the subject, I have not tried to provide more recent references (which I'm sure others will).

This goes back a while, but when I was an undergraduate in the late 1970s the standard undergraduate texts in the U.S. seemed to be Elementary Differential Geometry by Barrett O'Neill (1966 edition) and Differential Geometry of Curves and Surfaces by Manfredo P. do Carmo (1976 edition). Where I was an undergraduate, the course was offered every Fall (taken by a mixture of about half strong upper level undergraduate students and half first year graduate students), with these two texts alternating during my undergraduate years in the late 1970s, depending on who taught the course (do Carmo was the text when I took the course).

There was also a slightly lower level text used at a less-prestigious university near where I lived, which I audited the summer after my 1st year -- Elements of Differential Geometry by Richard S. Millman and George D. Parker (1977 edition) which, for what it's worth, is more reasonable to cover in one semester than the other two books and I think for at least a few years was also a relatively popular choice of text in the U.S. FYI, in the U.S. during this time, undergraduate level differential geometry was much more widely offered and taken than was the case by the mid to late 1980s and afterwards, probably due to the growing popularity of discrete math topics and statistics and computer science taken by math majors.

The prerequisite for O'Neill's and do Carmo's texts were (at the time) a 2-semester sequence in advanced calculus (such as would use [2] Buck or [4] Kaplan or [6] in this MSE answer) and the prerequisites for Millman/Parker's book was elementary linear algebra (such as would use a "1st level book" in this MSE answer) and the 3rd semester (or 3rd & 4th semester, depending on the U.S. university) of the elementary calculus sequence (such as the multivariable calculus chapters in Stewart's Calculus).

The first semester graduate level differential geometry course where I was an undergraduate (actually, this course was primarily intended as a course providing machinery and background in manifolds for more advanced work in differential geometry, which was typically done in graduate student seminars), usually taken by 2nd or 3rd year graduate students because the 1st year was typically taken up with the following 6 courses the Ph.D. qualifying exams were based on -- real analysis using Smith's Primer of Modern Analysis or Hoffman's Analysis in Euclidean Space (this maybe only taken by half of the graduate students who needed such a "transition course", the other half taking perhaps the "undergraduate" differential geometry course or a qualitative ODE course using Hirsch/Smale or another such "elective course" not covered by the Ph.D. qualifying exams), measure theory using Royden, 2-semester algebra sequence using Lang or Hungerford, complex analysis using Ahlfors, topology using Munkres -- used Warner's Foundations of Differentiable Manifolds and Lie Groups or Spivak's A Comprehensive Introduction to Differential Geometry, Volume 1, with Warner's text used maybe twice as often as Spivak's book (Spivak was the text when I took the course). (Sorry about this approx. 180-word sentence!) This MSE question deals with prerequisites for Spivak's text.

Also, during this time and throughout at least the 1980s, a popular choice for a more geometrically focused first graduate course in differential geometry (than Warner or Spivak) was An Introduction to Differentiable Manifolds and Riemannian Geometry by William M. Boothby, which was the text used for a course I actually took (but did not put forth much effort in) as a graduate student.

Answer 2

do Carmo actually has two textbooks and not one. The more elementary one is Differential Geometry of curves and surfaces. This gives a good introduction to the idea of curvature (for both curves and surfaces), and is a good basis for pursuing more advanced topics in differential geometry. I have used parts of this textbook in teaching classical differential geometry to undergraduates. His second textbook is Riemannian geometry which is an introduction to classical differential geometry with an emphasis on the Riemann curvature tensor. This is also a useful and well-written source. An aspect of the second textbook that you should be aware of is that it hardly touches upon differential forms and the de Rham theorem, but there are many other books that cover that topic. Also, you could consult my course notes for undergraduate differential geometry (with focus on curvature, Theorema Egregium, and Gauss-Bonnet) and graduate differential geometry (the latter focuses on differential forms and the de Rham theorem rather than on curvature).