Can anyone recommend a good starting book for differential geometry? I want this book to be a book with geometric explanations and geometric interpretations.
Dear Temel Ermiş
I recommend
(1) M. P. Do Carmo; Differential Geometry of Curves and Surfaces, Pearson, 1st edition, 1976.
(2) B. Oneill, Elementary Differential Geometry, Academic Press, 2nd edition, 2006.
(3) M. M. Lipschutz, Theory and problems of differential geometry (Schaum’s outline series), McGraw-Hill, 1969.
Regard
https://books.google.gm/books?id=dbIAAQAAQBAJ&printsec=copyright&source=gbs_pub_info_r#v=onepage&q&f=false
There are tons of them from elementary geometry of curves and surfaces in Euclidean space to abstract formulations using ringed spaces. As an entry level I don't think one can beat Michael Spivak's books. However, they might be difficult to find these days. For a very good text that integrates the modern coordinate free aspects with while maintaining and using geometric intuition for a first course, the book by R. W. R. Darling is excellent.
https://www.amazon.com/Differential-Forms-Connections-R-Darling/dp/0521468000/ref=sr_1_1?Adv-Srch-Books-Submit.x=0&Adv-Srch-Books-Submit.y=0&keywords=differential+geometry&qid=1640799283&refinements=p_27%3ADarling&s=books&sr=1-1&unfiltered=1
I recommend these books
Book Elementary Differential Geometry
Chapter Geometric preliminaries
https://www.amazon.com/Manifolds-Monge-Amp%C3%A8re-Grundlehren-mathematischen-Wissenschaften/dp/0387907041
see
1-Elementary Differential Geometry
Springer
Andrew Pressley
Year:
2010
-------------------
2-Elementary Differential Geometry
Cambridge University Press
Christian Bär
Year:
2010
Depends who is "starting". For ages of 4 onwards try "Turtle geometry" by Harold Abelson and Andrea di Sessa using Logo software.
Pre-university might like "Geometries other than Euclid's" in Sawyer 's "Prelude to Mathematics" or "Crocheting Adventures with Hyperbolic Planes" by Daina Taimiņa.
Most approaches are for undergraduate/postgraduate level. As Truman Prevatt suggests above, Spivak is excellent but hard to get. One that I have found useful is "Elementary Topics in Differential Geometry" by J.A. Thorpe which is full of examples and diagrams to support the maths being developed. Ofcourse "DIfferential Geometry" by Erwin Kreyszig is great as well as the more recent books mentioned above. Unless a specific book is chosen for a course, it might be useful get learners to have a look for themselves and pick the one(s) they find clear in its presentation and explanations with lots of examples and exercises. Good luck. It's a great area of mathematics.
A Course in Differential Geometry
W. Klingenberg.
Foundations of Differential Geometry
Kobayashi And Nomizu.
Spectral Geometry of the Laplacian
Spectral Analysis and Differential Geometry of the Laplacian
Hajime Urakawa.
Introduction to Geometry and Topology
Werner Ballmann, Walker Stern.
1. Schaum's Outline of Differential Geometry by Martin M. M. Lipschutz;
2. Schaum's Outline of Differential Geometry by Seymour Lipschutz;
3. Fundamentals of Differential Geometry by Serge Lang.
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