Recommending books for introductory differential geometry
My favourite book on the subject is Introduction to Smooth Manifolds by John M. Lee. It is quite explicit, which is sometimes what you need when you first start to learn a topic, as you haven't developed any intuition yet. Also, Lee's writing style is excellent, in particular, he does a great job of motivating each topic. I like this book so much, my parents are buying me a copy for Christmas.
Added later: I just checked, and it seems that the book doesn't have that much about the Hodge star (only exercises 12-16 in Chapter 14). I'm not sure how much you need to know about the Hodge star, but the exercises cover the standard facts.
There's no arguing that John Lee's texts are excellent: the following are part of the series "Graduate Texts in Mathematics":
- Introduction to Smooth Manifolds
- Introduction to Topological Manifolds
- Riemannian Manifolds: An Introduction to Curvature
Each of the above links to Amazon, simply because you can preview the texts, e.g., the Table(s) of Contents, to see if any/all meet your needs. Each is also accompanied by credible "reviews", which may help you select the appropriate text(s) to meet your needs.
As you seem to be looking for a more elementary introduction to differential geometry:
You might want to check out the the course on Differential Geometry via MIT Open Course Ware, (Prof. Paul Seidel):
This course is an introduction to differential geometry. The course itself is mathematically rigorous, but still emphasizes concrete aspects of geometry, centered on the notion of curvature.
The text used for this course is:
Kuhnel, Wolfgang. Differential Geometry: Curves – Surfaces – Manifolds. Student mathematical library, vol. 16. Providence, RI: American Mathematical Society, 2002. ISBN: 9780821826560.
There are also lecture notes that accompany the course and text.
Other choices for Elementary Introductions:
- You might also want to look into O'Neill's Elementary Differential Geometry, perhaps a good choice to start off with.
"Written primarily for students who have completed the standard first courses in calculus and linear algebra, it provides an introduction to the geometry of curves and surfaces." - Also look into the book with the same title: Elementary Differential Geometry, 2nd Ed (2010), [Springer Undergraduate Mathematics Series], this one authored by Andrew Pressley.
"Elementary Differential Geometry presents the main results in the differential geometry of curves and surfaces suitable for a first course on the subject. Prerequisites are kept to an absolute minimum – nothing beyond first courses in linear algebra and multivariable calculus – and the most direct and straightforward approach is used throughout."
- Once you get a foundation in Differential Geometry, I'd recommend looking into J. M. Lee's: Manifolds and Differential Geometry.
In that you mention "intro," here is a link to an excellent book on differential geometry (free download) about 1/3 down Prof. Shifrin's homepage. It is quite highly regarded and clearly written for self-study:
http://math.uga.edu/~shifrin/