Which Book Is Best for Learning Differential Geometry?

[AbhilashaEha]

HI, am a newbie to differential geometry..Can anyone please suggest me a book suitable for Maths hons student...

Before posting read this out...

required topics-
one parameter family of surfaces, developables associated with a curve : polar and rectifying & osculating developables ,two parameter family of surfaces, curvillinear coordinates, curves on a surface, eulers theorem, dupin's theorem, surface of revolution, conjugate directions, conjugate systems, asymptotic lines,curvature and torsion, geodesics !

 

[Studiot]

Hello Abhilasha, I hope you Diff Geom course turns out well.

What you have listed looks like a list of traditional topics.

I say this because there are two approcahes to Diff Geom - the classical and the modern.

The Oxford University Book

an Introduction to Differential Geometry

by Willmore

Offers a good basis for the subject with lots of useful examples in what can be a rather dry academic subject.

He uses traditional differentials for the presentation.


*****************************************************

The modern approach is exemplified by

Elementary Differential Geometry

by ONeill

He launches straight into forms, the connection to topology and manifolds.

It is quite important that you find out which approach your course will be following before choosing a text as the wrong one will be more of a hinderance than a help.

I have underlined the two appropriate sentences to ask.

go well

 

[AbhilashaEha]

Thanks even i hope the same but my professor and his book are useless and leave me helpless...My course in mainly theory centered ...I think its the classic one...that is why it appears too boring...

BTW my course mainly deals with use of derivatives(total and partial) merged along with some high school 3-D geometry concepts (like equation of plane, normal plane, tangent planes, dot and cross products etc)

I need a book with relevant elementary theory in the subject and a lots to examples to practice...That's it !


So Wilmore would be a perfect book ? What about Schaum's ? Should i take that up also ?

 

[yenchin]

I think classical (traditional) differential geometry of surfaces in 3-dimensional space is a very beautiful subject; it is also a great arena for you to train your intuition in preparation for more abstract modern differential geometry of manifolds. The text I used was do Carmo's "https://www.amazon.com/dp/0132125897/?tag=pfamazon01-20".

 

[AbhilashaEha]

OnE thing please don't get me wrong ! I don't want to go into pretty much details...the topics i have mentioned comprise 98% of the syllabus ...these are the only topics that i will deal at undergraduate level and want to learn that much only which can earn me good marks ( but the college has recommended Erwin kreyszig's book as well...Do u think that's readable...another thing i have spivak's comprehensive intro into DG as well but hardly find that useful... seems like written mainly for engineering students...)

So please help me decide which one would let me conquer the topics without much pre-requistives of a decent chapter-wise study...i just want to study it topic wise

intuition in preparation for more abstract modern differential geometry of manifolds. The text I used was do Carmo's "https://www.amazon.com/dp/0132125897/?tag=pfamazon01-20".

Hello Abhilasha, I hope you Diff Geom course turns out well.

What you have listed looks like a list of traditional topics.

The Oxford University Book

an Introduction to Differential Geometry

by Willmore

 

[Studiot]

You did say honours mathematics.

Differential Geometry is generally a third level subject here.

As for Kreysig, he is an author with two faces.

He has written some highly accessible textbooks for engineers and scientists. The section on diff geom in 'Advanced Engineering Mathematics' is an excellent introduction.

But he is also a respected professor of Mathematics, author of some full blooded mathematics textbooks where he takes no prisoners.

His textbook 'Differential Geometry' is one such.
He makes extensive use of Tensor calculus and probably goes way beyond your requirements.

The smaller and simpler book by Struik

'Lectures on Classical Differential Geometry'

is probably about the level you seem to require.

One other book worth mentioning because she incorporates all approaches, classical modern differential forms and groups and the bridge of tensor calculus, is

Prakash

'Differential Geometry an Integrated approach'.

go well

 

[Anonymous217]

I'm not exactly sure which is best, but I'm taking a grad-level diff geometry course at Berkeley right now and we're using Spivak's Comprehensive Intro to Diff Geometry. I've only read the first chapter, but it seems pretty straightforward.

 

[AbhilashaEha]

Thnks Studiot "lectures on DG " by Struik is a gem indeed...This made all my searches fruitful...it has all the topics i needed in required length and depth...:P

Is the book enough or should i do the practice stuff along with Schaum's Outline- differential geometry ?
I mean should it be read simultaneously with some other book or it deserves an independent study ? :-)

 

[AbhilashaEha]

Thnks Studiot "lectures on DG " by Struik is a gem indeed...This made all my searches fruitful...it has all the topics i needed in required length and depth...:P

Is the book enough or should i do the practice stuff along with Schaum's Outline- differential geometry ?
I mean should it be read simultaneously with some other book or it deserves an independent study ? :-)

 

[Studiot]

Is the book enough or should i do the practice stuff along with Schaum's Outline- differential geometry ?
I mean should it be read simultaneously with some other book or it deserves an independent study ? :-)

How long is a piece of string?

If Struik covers your course then fine. But it is too small to have many practice examples ( the exercises are more further theory teaching eg "find the geodesics of the plane by integrating 2.2 in polar coordinates") than real uses.
Examples are what Schaum's series does well but their explanation of theory is rather short.