# Differential Geometry book that emphasizes on visualization

• Geometry
Hello!
I would like to know if anybody here knows if there's any good book on academic-level dfferential geometry(of curves and surfaces preferably) that emphasizes on geometrical intuition(visualization)?
For example, it would be great to have a technical textbook that explains the geometrical meaning of the covariant derivative in more detail or that gives further examples of geodesics and not just explain the geodesics on a sphere.
Thank you!

Here's a website on Discrete Differential Geometry:

http://ddg.cs.columbia.edu/

and this ebook by Keenan Crane has a lot of visualization in its presentation:

https://www.cs.cmu.edu/~kmcrane/Projects/DDG/
Nice, thanks.
Is there any book that you know of that deals with differential geometry as taught in the university(not discrete)?

jedishrfu
Mentor
I used some old books at college notably Lanczos so I don't know the latest and greatest ones. Perhaps @fresh_42 can provide a good reference.

fresh_42
Mentor
Perhaps @fresh_42 can provide a good reference.
To be honest with you, I entered this thread full of curiosity about:
I would like to know if anybody here knows if there's any good book on academic-level dfferential geometry(of curves and surfaces preferably) that emphasizes on geometrical intuition(visualization)?
The ones I have are rather theoretical, except this one:

https://www.amazon.com/dp/0387903577/?tag=pfamazon01-20&tag=pfamazon01-20

Maybe it's because it is meant to be read by undergraduates which makes it easier to read and understand. At least it is full of geometrical sketches and examples to underline the corresponding concepts. I like it, but in the end it's always a matter of taste.

Demystifier
To be honest with you, I entered this thread full of curiosity about:

The ones I have are rather theoretical, except this one:

https://www.amazon.com/dp/0387903577/?tag=pfamazon01-20&tag=pfamazon01-20

Maybe it's because it is meant to be read by undergraduates which makes it easier to read and understand. At least it is full of geometrical sketches and examples to underline the corresponding concepts. I like it, but in the end it's always a matter of taste.
Looks good, thanks!

mathwonk
Homework Helper
2020 Award
I suggest the book :Differential Geometry, a geometric introduction, apparently available free on the author David Henderson's website at Cornell. There is a clear elementary geometric discussion of covariant derivatives in chapter 8.1. The main idea is that of parallel transport. This enables you to make sense of the numerator of the definition of a derivative, i.e. it makes sense of the subtraction of two vectors which start out having different base points, by transporting one of them over to the base point of the other. So the main thing to learn first is parallel transport. This concept of parallel transport is discussed in an elementary hands on way also in his book "Experiencing Geometry". here is a link to the first book:

http://www.math.cornell.edu/~henderson/books/dg.html

The closest I've found is William Burke's Applied Differential Geometry. He puts a heavy emphasis on basic geometrical and visual intuition to back up applications of the formal mathematics. It doesn't reach the "average one figure per page" that I'd like in a book like the thread title suggests, but it does have over 230 figures(mostly very simple ones) in its 400-ish pages.

I think the field of differential geometry could benefit greatly from a few books like Needham's Visual Complex Analysis and Klaus Jänich's Topology. Books that don't try to completely teach the subject or cover every course topic, but focus instead on the areas where visual intuition can help codify the subject in the reader's mind.

Joker93
mathwonk
Homework Helper
2020 Award
FWIW, from what I can search on amazon, the book I linked by Henderson is MUCH more elementary than Burke's applied diff geom. I.e. parallel transport is just given as an elementary geometry concept in pictures rather than in conjunction with (to me) scary ideas like fiber bundles and connections. it is really written for the very naive student, like me. This is not to disparage Burke, which may be ideal for some, but merely to offer information to help those who may be more or less sophisticated choose for themselves. I myself like really really simple explanations, suitable for the man on the street. I have been known to sneak into lectures aimed at students with much less background than me hoping to understand something. It often works well. I benefited from reading Riemann e.g. who described curvature in terms of the ratio between the radius and the circumference of a small circle around a given point.

Last edited:
I suggest the book :Differential Geometry, a geometric introduction, apparently available free on the author David Henderson's website at Cornell. There is a clear elementary geometric discussion of covariant derivatives in chapter 8.1. The main idea is that of parallel transport. This enables you to make sense of the numerator of the definition of a derivative, i.e. it makes sense of the subtraction of two vectors which start out having different base points, by transporting one of them over to the base point of the other. So the main thing to learn first is parallel transport. This concept of parallel transport is discussed in an elementary hands on way also in his book "Experiencing Geometry". here is a link to the first book:

http://www.math.cornell.edu/~henderson/books/dg.html
I can't find the free online version of it(through your link or via google search). Were did you find it?

mathwonk