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Geometry Classical and modern differential geometry

  1. Dec 2, 2018 #1
    Im planning on taking a course on classical differential geometry next term. This is the outline:
    The differential geometry of curves and surfaces in three-dimensional Euclidean space. Mean curvature and Gaussian curvature. Geodesics. Gauss's Theorema Egregium.
    The textbook is "differential geometry of curves and surfaces" by do carmo.

    I was wondering if anyone knows a good textbook or set of lecture notes that covers this material and modern differential geometry at the same time. Ideally i will work from this textbook for the class and learn the same ideas in modern differential geometry at the same time. Thanks!
  2. jcsd
  3. Dec 2, 2018 #2


    Staff: Mentor

    Have you tried a google search?

    I found a book by Isham for graduate physics students. It was pub in 1989 from a collection of course notes on coordinate free differential geometry.

    There must be more recent publications.

    I also found one on DDG for computer work.

    What topics of DG are you interested in?
  4. Dec 2, 2018 #3
    Millman & Parker, Elements of Differential Geometry


    The last chapter covers manifolds.

    Wolfgang Kühnel, Differential Geometry: Curves - Surfaces - Manifolds


    See my comments on Amazon.

    And somewhat more eccentric and with a different emphasis:

    R. W. R. Darling, Differential Forms and Connections

  5. Dec 3, 2018 #4
    I'm mostly interested in how it can be applied to QFT and electromagnetism. Ive seen the phrase fibre bundle come up a lot. But I also like DG just for the subject itself. Atleast I liked the section on curves and surfaces that I learned about in my calc course.
  6. Dec 3, 2018 #5
    Thanks for the reference, the one by Millman and Parker seems to be quite close to what I'm looking for.
  7. Dec 3, 2018 #6
    The Darling book and Frankel below would be the most relevant to gauge theory.

    A couple I forgot:

    Frankel, The Geometry of Physics

    This has everything you need worked out with pedagogical care, but it's not a quick read.

    Schutz, Geometrical Methods of Mathematical Physics

    Good on the basics and an easy read, but not much on fiber bundles.

  8. Dec 5, 2018 #7
    "Differential Geometry of Curves and Surfaces", by Manfredo do Carmo, is great as an introduction to DG because DG should start with the study of curves and surfaces in 3D before going into the geometry of n-dimensional differential manifolds.

    If you want to go further then you can use two books

    (1) "An Introduction to Manifolds" by Loring Tu. Chapters 1 through 6 make the transition to the differential geometry of manifolds up to Stokes's theorem on manifolds. In my opinion, it's a good idea to read the second book and learn some algebraic topology before reading chapter 7.

    (2) "Riemannian Geometry" by Manfredo do Carmo. This book builds upon the knowledge from the previous book and focuses on the study of distance and curvature on manifolds.

    A good alternative to these two books is "An Introduction to Differentiable Manifolds and Riemannian Geometry" by William Boothby, but it seems that it is more popular with mathematicians than it is with students.
  9. Dec 7, 2018 #8
    I must add that "at the same time" is not a good idea. DG of curves and surfaces works in R^3. DG of manifolds works on topological spaces that, locally, look like R^n. The mathematical techniques to deal with each situation are significantly different. And I think it's much better to learn DG of curves and surfaces before learning DG on manifolds because the latter can easily get away from our intuition without the former.
  10. Dec 8, 2018 #9
    Thanks for the book reference. As for learning both at the same time, i wasnt too worried about the techniques and abstraction in modern dg since I have already studied topology. But thanks for the suggestion, ill keep it in mind.
  11. Dec 8, 2018 #10


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    Science Advisor

    I like (end of each chapter summarizes the abstract notation and the useful coordinate-based expressions)
    Differential Geometry and Lie Groups for Physicists
    by Marián Fecko

    A standard book for physicists is
    Geometry, Topology and Physics (Graduate Student Series in Physics) 2nd Edition
    by Mikio Nakahara
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