# Textbook Help For Differential Geometry

#### caffeinemachine

##### Well-known member
MHB Math Scholar
Hello MHB.

I want to read differential geometry and I want to use J. M. Lee's 'Introduction to Smooth Manifolds' Introduction to Smooth Manifolds (Graduate Texts in Mathematics): John M. Lee: 9780387954486: Amazon.com: Books

In the preface the author says that this book is a natural sequel to his previous book 'Introduction to Topological Manifolds' Introduction to Topological Manifolds (Graduate Texts in Mathematics): John Lee: 9781441979391: Amazon.com: Books.

Is this book a necessary prerequisite for reading 'Introduction to Smooth Manifolds'?

Any other book recommendations for differential geometry are welcome.

#### Chris L T521

##### Well-known member
Staff member
Hello MHB.

I want to read differential geometry and I want to use J. M. Lee's 'Introduction to Smooth Manifolds' Introduction to Smooth Manifolds (Graduate Texts in Mathematics): John M. Lee: 9780387954486: Amazon.com: Books

In the preface the author says that this book is a natural sequel to his previous book 'Introduction to Topological Manifolds' Introduction to Topological Manifolds (Graduate Texts in Mathematics): John Lee: 9781441979391: Amazon.com: Books.

Is this book a necessary prerequisite for reading 'Introduction to Smooth Manifolds'?

Any other book recommendations for differential geometry are welcome.

For starters, Lee's Introduction to Smooth Manifolds (ISM) isn't really a differential geometry book; it's something that's definitely worth going through in preparation for differential geometry. I never went through Lee's Introduction to Topological Manifolds because it wasn't a prerequisite for the class that was offered where I did my masters degree, so I don't think you'll need it. I should say though that you should have a solid background in advanced linear algebra (i.e should know things about vector spaces, dual spaces, tensors, exterior algebras, etc) and topology. When I took manifolds courses as a grad student, we took 2 quarters to get through ISM, so I wouldn't rush through it if I were you. XD

If you want something more differential geometry focused, I would go with Do Carmo's book on differential geometry (typically used at the undergraduate level; this would be a very good book to introduce yourself to differential geometry without making things too complicated), and then his book on Riemannian geometry (which is typically used at the graduate level). In addition to this, there's also Spivak's five volume series A Comprehensive Introduction to Differential Geometry. In my opinion, to get a good taste of differential geometry, you only need to go through book 1 and book 2 (book 3 is also worth going through as well since it's pretty much all about surface theory, geodesics, and more curvature). Book 1, in essence, covers a good portion of what you'd learn in Lee's ISM. Book 2 focuses more on curvature, connections, moving frames, and principal bundles. You can actually read the table of contents for each volume at this link. The exercises in Spivak's books are very interesting, but at the same time, they can be pretty challenging. Lee, on the other hand, isn't as challenging as Spivak, although his problems aren't necessarily a walk in the park either.

I hope this helps!

#### caffeinemachine

##### Well-known member
MHB Math Scholar
For starters, Lee's Introduction to Smooth Manifolds (ISM) isn't really a differential geometry book; it's something that's definitely worth going through in preparation for differential geometry. I never went through Lee's Introduction to Topological Manifolds because it wasn't a prerequisite for the class that was offered where I did my masters degree, so I don't think you'll need it. I should say though that you should have a solid background in advanced linear algebra (i.e should know things about vector spaces, dual spaces, tensors, exterior algebras, etc) and topology. When I took manifolds courses as a grad student, we took 2 quarters to get through ISM, so I wouldn't rush through it if I were you. XD

If you want something more differential geometry focused, I would go with Do Carmo's book on differential geometry (typically used at the undergraduate level; this would be a very good book to introduce yourself to differential geometry without making things too complicated), and then his book on Riemannian geometry (which is typically used at the graduate level). In addition to this, there's also Spivak's five volume series A Comprehensive Introduction to Differential Geometry. In my opinion, to get a good taste of differential geometry, you only need to go through book 1 and book 2 (book 3 is also worth going through as well since it's pretty much all about surface theory, geodesics, and more curvature). Book 1, in essence, covers a good portion of what you'd learn in Lee's ISM. Book 2 focuses more on curvature, connections, moving frames, and principal bundles. You can actually read the table of contents for each volume at this link. The exercises in Spivak's books are very interesting, but at the same time, they can be pretty challenging. Lee, on the other hand, isn't as challenging as Spivak, although his problems aren't necessarily a walk in the park either.

I hope this helps!
I know linear algebra at the level of Axler's Linear Algebra Done Right. I know a few other things in Linear Algebra not given in Axler's book. I haven't read anything on tensors or exterior algebra though.

I am comfortable with point-set topology. More precisely, I have read point-set topology at the level of Part I in Simmon's 'Introduction to Topology and Modern Analysis.'

I am inclined to study ISM since it deals with things in an abstract way.
I, for some unknown reason, dislike DoCarmo's book.

#### Evgeny.Makarov

##### Well-known member
MHB Math Scholar
If you want to be a little exotic, you can try a standard Soviet-issued AK-47 course of differential geometry and topology used in Moscow State University. The textbooks are

Basic Elements of Differential Geometry and Topology by S.P. Novikov and A.T. Fomenko

and

A course of differential geometry and topology by A.S. Mishchenko and A.T. Fomenko

(both are on Amazon).

Fomenko, who is a member of the Russian Academy of Sciences, was the lecturer for the course I took. I have fond memories because he was usually quite precise: his lectures were sequences of definition-theorem-proof's, but he also provided plenty of geometric intuition. I usually tend more to algebra than geometry because it is easier to explain it to a computer; geometry may have hand-waving moments that rely on spacial perception and that are difficult to formalize. In this course, I had no such difficulties.

#### caffeinemachine

##### Well-known member
MHB Math Scholar
If you want to be a little exotic, you can try a standard Soviet-issued AK-47 course of differential geometry and topology used in Moscow State University. The textbooks are

Basic Elements of Differential Geometry and Topology by S.P. Novikov and A.T. Fomenko

and

A course of differential geometry and topology by A.S. Mishchenko and A.T. Fomenko

(both are on Amazon).

Fomenko, who is a member of the Russian Academy of Sciences, was the lecturer for the course I took. I have fond memories because he was usually quite precise: his lectures were sequences of definition-theorem-proof's, but he also provided plenty of geometric intuition. I usually tend more to algebra than geometry because it is easier to explain it to a computer; geometry may have hand-waving moments that rely on spacial perception and that are difficult to formalize. In this course, I had no such difficulties.
Thank you Evgeny for this suggestion.
Can you elaborate on what exactly you meant by 'exotic' or by an 'AK 47' course?
Is it very difficult?

#### Deveno

##### Well-known member
MHB Math Scholar
Spivak has also published a "reader's digest version" called "Calculus on Manifolds" whose small size belies the amount of material it covers. It suppresses a lot of the topological detail, focusing mainly on the metric topology induced by the Euclidean inner product in $\Bbb R^n$. It's a personal favorite of mine (probably because I still recall my shock upon learning how much this small book cost upon my first purchase!).

Essentially the entire book is a proof that:

$$\int_M d\omega = \int_{\partial M} \omega$$

with an undertone of subtle humor throughout.

@Evgeny: AK-47!!! hah!

#### caffeinemachine

##### Well-known member
MHB Math Scholar
Spivak has also published a "reader's digest version" called "Calculus on Manifolds" whose small size belies the amount of material it covers. It suppresses a lot of the topological detail, focusing mainly on the metric topology induced by the Euclidean inner product in $\Bbb R^n$. It's a personal favorite of mine (probably because I still recall my shock upon learning how much this small book cost upon my first purchase!).

Essentially the entire book is a proof that:

$$\int_M d\omega = \int_{\partial M} \omega$$

with an undertone of subtle humor throughout.

@Evgeny: AK-47!!! hah!
I have read the first two chapters from Spivak's book. Then I was pointed towards Mukres' 'Analysis on Manifolds' and I found it more detailed and user friendly and also more modern than Spivak's. Haven't finished it but all this reading made me interested in advanced geometry.

#### Evgeny.Makarov

##### Well-known member
MHB Math Scholar
Can you elaborate on what exactly you meant by 'exotic' or by an 'AK 47' course?
I mean that these books are as standard in Moscow University as AK-47 (or rather AK-74) is in the Russian army, but they would probably be exotic in an American university.

Is it very difficult?
I did not think so at the time. You could look at the table of contents on Amazon, or I could send you an excerpt (PM me).

Edit: It is an undergraduate course for third-year students, though it would probably correspond to senior-first year graduate course in the West.

#### Turgul

##### Member

Certainly it would be advisable to have some familiarity with algebraic topology. But if you know what the fundamental group and simplicial (or singular) homology groups of a topological space are (roughly), there is no reason not to try Lee's Smooth Manifolds book. Even then, this is not strictly necessary.

But then, what you want out of Lee's book? Are you trying to get basic familiarity with smooth manifolds? Are you trying to do computations with connections? Is this project for general learning or as a supplement to some other goal?

In particular, there is a lot of nice literature about differential topology which may actually be what you want, depending on what that is.

#### caffeinemachine

##### Well-known member
MHB Math Scholar

Certainly it would be advisable to have some familiarity with algebraic topology. But if you know what the fundamental group and simplicial (or singular) homology groups of a topological space are (roughly), there is no reason not to try Lee's Smooth Manifolds book. Even then, this is not strictly necessary.

But then, what you want out of Lee's book? Are you trying to get basic familiarity with smooth manifolds? Are you trying to do computations with connections? Is this project for general learning or as a supplement to some other goal?

In particular, there is a lot of nice literature about differential topology which may actually be what you want, depending on what that is.
I disagree with you.
I don't see how my question is funny.
When I asked if 'Introduction to Topological Manifolds' is a prerequisite for reading 'Introduction to Smooth Manifolds', I am not asking if I need it.
I am merely asking if it is a prerequisite. Many authors tell about the prerequisites in the preface. They are not about anybody in particular, they of course cannot be.

- - - Updated - - -

I mean that these books are as standard in Moscow University as AK-47 (or rather AK-74) is in the Russian army, but they would probably be exotic in an American university.

I did not think so at the time. You could look at the table of contents on Amazon, or I could send you an excerpt (PM me).

Edit: It is an undergraduate course for third-year students, though it would probably correspond to senior-first year graduate course in the West.
I saw the book. I think I can give it a try. Thanks.

- - - Updated - - -

Edit: It is an undergraduate course for third-year students, though it would probably correspond to senior-first year graduate course in the West.
I guess that's Russia for me!

#### Fantini

MHB Math Helper
For elementary differential geometry I recommend Barrett O'Neill's book. Far better than Do Carmo. For the middle term, this book I found recently seems to serve as a good treatment for what you want to delve deeper into. I agree with you: Do Carmo is not a good reference for me.

@Deveno: are you serious about Calculus on Manifolds? It's one of the worst books I've ever seen.

#### caffeinemachine

##### Well-known member
MHB Math Scholar
For elementary differential geometry I recommend Barrett O'Neill's book. Far better than Do Carmo. For the middle term, this book I found recently seems to serve as a good treatment for what you want to delve deeper into. I agree with you: Do Carmo is not a good reference for me.

@Deveno: are you serious about Calculus on Manifolds? It's one of the worst books I've ever seen.
Your suggestion of reading 'Analysis on Manifolds' was a really good one. I sure will check out the books you have suggested above.

#### Turgul

##### Member
To answer then what I think is your original query more precisely, Lee's topological manifolds book is more of a motivation prerequisite than it is a logical one to the smooth manifold book. The topological book develops basic algebraic topology of manifolds, so if you've worked through it, you would have spent reasonable time thinking about manifolds and understanding how they behave topologically (and if something works for a smooth structure, it must make sense for the underlying topological space, so you should have at least a coarse understanding of what may or may not work in the smooth world). But I don't recall the smooth manifolds book requiring many (if any) actual results from topology without at least some discussion to remind you what is going on.

I think Lee tried fairly hard to make the smooth manifolds book self contained. Most people I know who have worked through it have not worked through his book on topological manifolds beforehand. As far as I can tell, the smooth manifolds book is far more popular than the topological manifolds book. On the other hand, I don't know anyone who has tried this book without at least some sort of introduction to topology under their belts.

Hopefully that sheds some insight to your question.

As to other books, my recommendation about differential topology is worth keeping in mind. Books like Guillemin and Pollack's book, the book From Calculus to Cohomology, and Warner's book were the first books about manifolds that I studied as an undergraduate, but they do have a bit more of a topological flavor than most of the other recommendations you've been given here.

#### caffeinemachine

##### Well-known member
MHB Math Scholar
To answer then what I think is your original query more precisely, Lee's topological manifolds book is more of a motivation prerequisite than it is a logical one to the smooth manifold book. The topological book develops basic algebraic topology of manifolds, so if you've worked through it, you would have spent reasonable time thinking about manifolds and understanding how they behave topologically (and if something works for a smooth structure, it must make sense for the underlying topological space, so you should have at least a coarse understanding of what may or may not work in the smooth world). But I don't recall the smooth manifolds book requiring many (if any) actual results from topology without at least some discussion to remind you what is going on.

I think Lee tried fairly hard to make the smooth manifolds book self contained. Most people I know who have worked through it have not worked through his book on topological manifolds beforehand. As far as I can tell, the smooth manifolds book is far more popular than the topological manifolds book. On the other hand, I don't know anyone who has tried this book without at least some sort of introduction to topology under their belts.

Hopefully that sheds some insight to your question.

As to other books, my recommendation about differential topology is worth keeping in mind. Books like Guillemin and Pollack's book, the book From Calculus to Cohomology, and Warner's book were the first books about manifolds that I studied as an undergraduate, but they do have a bit more of a topological flavor than most of the other recommendations you've been given here.
Thanks. I will check out the books for sure.