Introduction to Differential Geometry

[JasonRox]

So, where do I begin?

I have a current interests in going into Algebraic Topology and I am currently learning about Point-Set Topology and Abstract Algebra, which are pre-requisites to Algebraic Topology.

After talking to a professor, he recommended learning some Differential Geometry, but I think this requires some knowledge of Differential Forms first (not 100% sure).

So can anyone recommend me in a good direction towards Differential Geometry?

What should I know before pursuing this topic? I'm guessing some vector calculus like Green's/Stoke's Theorem and what not.

Any suggestions or advice is greatly appreciated.

Note: I noticed that a lot of Algebraic Topologists also know a lot of Differential Geometry too, which I know is not a coincidence.

 

[Cexy]

It's certainly possible to learn about differential geometry without knowing about differential forms in any detail - I did exactly that for a course on general relativity last term! I'm doing a more advanced course on differential geometry this term, which does involve a lot of differential forms, and it's much nicer.

There are certain things that you should probably be familiar with - obviously proving Stokes' Theorem for differential forms will look pretty weird if you don't know Stokes' Theorem in Euclidean space!

Then there are your basic bread-and-butter preliminaries - functions, vector spaces, linear algebra, groups, tensors, topological spaces, homeomorphisms... I'd say that all of those would be useful.

The book I learned from was 'Geometry, Topology and Physics' by Nakahara. I thoroughly recommend it - its explanations are clear and it is never vague or hand-wavey. It also has a good introductory section to get you up to speed. It has a fair number of applications to physics, but never too many - and you never know, you might find them interesting!

 

[JasonRox]

It's certainly possible to learn about differential geometry without knowing about differential forms in any detail - I did exactly that for a course on general relativity last term! I'm doing a more advanced course on differential geometry this term, which does involve a lot of differential forms, and it's much nicer.

There are certain things that you should probably be familiar with - obviously proving Stokes' Theorem for differential forms will look pretty weird if you don't know Stokes' Theorem in Euclidean space!

Then there are your basic bread-and-butter preliminaries - functions, vector spaces, linear algebra, groups, tensors, topological spaces, homeomorphisms... I'd say that all of those would be useful.

The book I learned from was 'Geometry, Topology and Physics' by Nakahara. I thoroughly recommend it - its explanations are clear and it is never vague or hand-wavey. It also has a good introductory section to get you up to speed. It has a fair number of applications to physics, but never too many - and you never know, you might find them interesting!

You need to know about Groups and Tensors for Differential Geometry/Form?

That seems a little odd.

I might want to add that I obviously want to learn this in rigorous detail. Knowing how to prove Stoke's Theorem is one thing, understanding the proof is another thing.

I had the option to take a General Relativity class with Differential Geometry, and I know that it wasn't going to contain rigorous mathematics. Of course, it was an introduction and I assume yours was too because otherwise Differential Geometry would have been a pre-requisite.

 

[selfAdjoint]

You need to know about Groups and Tensors for Differential Geometry/Form?

That seems a little odd.

Well, no, as a look into Nakahra will show you. Groups come into principal bundles, which are important in differential geometry, and tensors are an alternative to differential forms - you will eventually have to learn both as they are somewhat complementary. Nakahara is a text for physicists not mathematicians but it should be sufficiently rigorous for your purposes. Or you could ask that professor what text he recommends.

 

[JasonRox]

Well, no, as a look into Nakahra will show you. Groups come into principal bundles, which are important in differential geometry, and tensors are an alternative to differential forms - you will eventually have to learn both as they are somewhat complementary. Nakahara is a text for physicists not mathematicians but it should be sufficiently rigorous for your purposes. Or you could ask that professor what text he recommends.

My professor is actually looking into that I believe.

The problem is that I can't learn something just because it is complementary. Differential Geometry is a complementary to Algebraic Topology, but I don't plan on going into as deep as I'm planning for Algebraic Topology. I just don't have the time to do that. There is no doubt I will know lots of it, but having one topic as broad as Algebraic Topology is already more than a handful.

So, I can't go and learn Tensors just because it is a complementary to one of the topics I learned because that will lead to another complementary and then another and then another.

There is no doubt I will learn some basics about Tensors in some time to come, but I'm concerned about what I need to know that will prepare myself as much as possible for Algebraic Topology.

I'm covering the pre-requisites to Algebraic Topology, but I think doing some Differential Geometry would be beneficial since it will clarify a lot of things with regards to manifolds and what not.

So, I'm just wondering, what are the things that I should know for sure before reading into Differential Geometry?

Try to keep the things reasonable, like topics in Calculus, Linear Algebra, Abstract Algebra and what not. You can take a course in Differential Geometry with certain pre-requisites, what might they be specifically? Certainly it's not something you learn in Graduate School because they offer the course as an Undergraduate Course.

 

[mathwonk]

differential geometry is about curvature, so learn that.

 

[JasonRox]

differential geometry is about curvature, so learn that.

Can you be more specific?

 

[mathwonk]

well tangent lines, tangent planes, coordinate changes, gradients are all first derivative phenomena.

from a certain point of view, this is all that makes sense on an abstract manifold.

if you add to a manifold the concept of a distance, i.e. a "metric", you can measure angles and curvature, in terms of second derivatives. This subject is what is called differential geometry, i.e. geometry on a manifold that has a metric.

Curvature of curves is measured by comparing with a circle, just as slope is measured by comparing with a line. Gaussian curvature of a surface is calculated by comparing areas of the surface with areas on a sphere.

although defining curvature involves a metric, there is a strong link between intrinsic topology and global properties of curvature. The Gauss Bonnet theorem says the average curvature of a compact surface is governed by the topology of the surface. e.g. neither a sphere nor a torus can have a metric which is everywhere negatively curved.

This link generalizes to the concept of "chern classes" of bundles, intrinsic cohomology classes that can also be computed using curvature, e.g. via "connections".


Or maybe that discussion is less specific. Probably better to just look up "curvature" in the index of various books.

 

[mathwonk]

actually maybe you are more interested in differential, topology than differential geometry. A good place to begin is Spivak's Calculus on manifolds, to learn about manifolds and differential forms. Once you have that language, you might read volume I of his differential geometry book, which really does not have any differential geometry in it, but does have some algebraic and differential topology. Other good sources are Milnor's topology from the differentiable viewpoint (great!) and the differential topology textbook it inspired by Guillemin and Pollack.

 

[Doodle Bob]

Wow, this conversation keeps popping up on this list. Whatever you do, do *not* start with any text written by a Russian or published by Dover or written for physicists. So many people start down that path and are never seen again.

I would recommend that you start with the basics: low-dimensional differential geometry. Millman and Parker's Elements of Differential Geometry and Do Carmo's Differential Geometry of Curves and Surfaces and Oprea's Differential Geometry are all excellent introductions to the field and develop the proper intuition for the subject.

From there, Spivak makes much more sense.

Plus, don't worry about the tensor nonsense. Tensor theory is nothing but a mathematical hobgoblin. Learn about the curvature and the tensors will come on their own.

 

[JasonRox]

Wow, this conversation keeps popping up on this list. Whatever you do, do *not* start with any text written by a Russian or published by Dover or written for physicists. So many people start down that path and are never seen again.

I would recommend that you start with the basics: low-dimensional differential geometry. Millman and Parker's Elements of Differential Geometry and Do Carmo's Differential Geometry of Curves and Surfaces and Oprea's Differential Geometry are all excellent introductions to the field and develop the proper intuition for the subject.

From there, Spivak makes much more sense.

Plus, don't worry about the tensor nonsense. Tensor theory is nothing but a mathematical hobgoblin. Learn about the curvature and the tensors will come on their own.

Great advice.

I noticed that the Dover editions aren't always straightforward.

I'll take a look into those books. Hopefully our school has them.

 

[matt grime]

Dovers are reprints of Classics. Everyone should look at them but bear in mind that cheap as they are they may be somewhat out of date. Any old book on differential geometry tends to make me want to slit my wrists at the cumbersome notation and presentation. But that is just my recollection from learning it at university, so I apologise if it is warped with the mists of time and distaste for the subject I picked up then

 

[Doodle Bob]

Dovers are reprints of Classics. Everyone should look at them but bear in mind that cheap as they are they may be somewhat out of date.

Actually, this isn't quite true. Dovers are reprints of books whose copyrights are cheap enough that they are cheap to print. Thus, a great many of them are old texts that no one quite cares about any more or translated texts whose copyright ownership is problematic at best. Now, there are quite a few good ones out there, but it is just not worth the time and effort to find them, when there are so many much, much, much better and clearer and more contemporary introductions to diffl. geometry such as those I listed.

 

[DeadWolfe]

There are certain things that you should probably be familiar with - obviously proving Stokes' Theorem for differential forms will look pretty weird if you don't know Stokes' Theorem in Euclidean space!

I did it. It was good. Well, actually, I had seen the version in Euclidian space, but never worked with it and couldn't really remember it. Mostly because things like "Curl" seemed arbitrary and silly. Learning the more general case helped me appreciate the Euclidian version more. In general actually, learning Calculus on Manifolds helped me appreciate the standard multivariable calculus more.

 

[JasonRox]

Dovers are reprints of Classics. Everyone should look at them but bear in mind that cheap as they are they may be somewhat out of date. Any old book on differential geometry tends to make me want to slit my wrists at the cumbersome notation and presentation. But that is just my recollection from learning it at university, so I apologise if it is warped with the mists of time and distaste for the subject I picked up then

Yeah, Dovers are fairly old. They are sometimes prints from the 50's.

 

[mathwonk]

i was trying to make the distinction between the language of differential forms and tangent bundles that is widely used in algebraic and differential topology, as opposed to the more specialized concept of curvature which is really the concept peculiar to differential geometry per se.

I.e. I am not convinced anyone interested in algebraic topology as Jason says he is, needs either an elementary or an advanced book on differential geometry, excellent though the ones mentioned by Doodle Bob are.

I assumed he resally wants to learn the language of manifolds, forms, and bundles, as used by algebraic topologists, and that is a different body of material.

For instance it is contained in spivak's "calculus on manifolds" in a very basic undergraduate and elementary form.

then guillemin and pollack use those basic ideas to do intersection theory and transversality.

then spivak's vol 1, treats it more abstractly, but still falls short of using any differential geometry, until volume 2.

Indeed vol2 of spivak is a good introduction to curvature and differential geometry which uses very little of the abstratc machinery of volume 1.

but I think it is vol 1 that Jason wants rather than vol 2. Thus millman and oparker, while a nice easy intro to diff geom, will not give him whart an algebraic topologist needs in my opinion.

I could be wrong of course.

differential geometry is a more specialized subject, and really does need tensors to do it right.if you want to see the kind of differential tools used in algebraic topology, consult the book of Bott and Tu, Differential Forms in Algebraic Topology (Graduate Texts in Mathematics) (Hardcover)
by Raoul Bott, Loring W. Tu

 

[JasonRox]

i was trying to make the distinction between the language of differential forms and tangent bundles that is widely used in algebraic and differential topology, as opposed to the more specialized concept of curvature which is really the concept peculiar to differential geometry per se.

I.e. I am not convinced anyone interested in algebraic topology as Jason says he is, needs either an elementary or an advanced book on differential geometry, excellent though the ones mentioned by Doodle Bob are.

I assumed he resally wants to learn the language of manifolds, forms, and bundles, as used by algebraic topologists, and that is a different body of material.

For instance it is contained in spivak's "calculus on manifolds" in a very basic undergraduate and elementary form.

then guillemin and pollack use those basic ideas to do intersection theory and transversality.

then spivak's vol 1, treats it more abstractly, but still falls short of using any differential geometry, until volume 2.

Indeed vol2 of spivak is a good introduction to curvature and differential geometry which uses very little of the abstratc machinery of volume 1.

but I think it is vol 1 that Jason wants rather than vol 2. Thus millman and oparker, while a nice easy intro to diff geom, will not give him whart an algebraic topologist needs in my opinion.

I could be wrong of course.

differential geometry is a more specialized subject, and really does need tensors to do it right.


if you want to see the kind of differential tools used in algebraic topology, consult the book of Bott and Tu, Differential Forms in Algebraic Topology (Graduate Texts in Mathematics) (Hardcover)
by Raoul Bott, Loring W. Tu

Again, great advice.

That is precisely what I am looking for.

 

[Doodle Bob]

MW,

you might be right on this score. If he's already doing algebraic topology, then he's most likely already dipped into cohomology, which is after all a way to get at forms on manifolds. If this is where Jason wants to go, I would then recommend as well Glen Bredon's fine text, Topology and Geometry.

But what can I say: I'm a Riemannian geometer, so I tend towards the geometry rather than the topology. And I'm getting tired of these endless questions about tensors and forms.

Another plug for yet another fine geometry text: This may be a bit advanced but if anyone wants to get a truly broad spectrum of what differential geometry is all about and where it is going, I highly recommend Marcel Berger's A Panoramic View of Riemannian Geometry. Every time I open this tome, I learn something new. Ans, whether you know much about the topic or not, the first few chapters are quite readable.

 

[mathwonk]

doodlebob, for the intersection between algebraic topology and differential geometry, do you (know or) recommend goldberg's curvature and homology? I hear it is excellent but have never read it.

 

[Doodle Bob]

doodlebob, for the intersection between algebraic topology and differential geometry, do you (know or) recommend goldberg's curvature and homology? I hear it is excellent but have never read it.

OK, for my academic grandfather, I'll break my rule: this is a fine text and, yes, it is a Dover book. It motivates quite well the connection between the Riem. Geometry and Algebraic Topology. While I'm on a FISA-level of bad faith, I'll also recommend Bishop and Goldberg's Tensor Analysis on Manifolds.

 

[mathwonk]

so is your daddy david blair?

and of course this book is known to me as a fine Academic text.

dover occasionally catches a prize.

 

[Doodle Bob]

so is your daddy david blair?

Very good, he is indeed my academic father.

 

[George Jones]

Whatever you do, do *not* start with any text ...
or written for physicists.

Just for that, I may resurrect the thread "tensor questions"!

I understand your point, and I agree that there seems to be an uncountable number of bad books written for physicists, but, as with the Dovers, there are a few exceptions.

Regards,
George

 

[DeadWolfe]

...I like Dover books.

 

[CMoore]

I too like Dover books as a general rule; I'm not quite sure why Dover is getting knocked so much on here. Of course Dover, like any other publisher, has its share of stinkers but it also provides many excellent volumes at a reasonable price. For example, here's three good ones:

1. Advanced Calculus by C.H. Edwards is a readable and coherent treatment of calculus in $$R^{n}$$. This text costs about $15; compare that to current texts on advanced calculus that run over $100.

2. Introduction to Topology by B. Mendelson is an excellent elementary introduction to point set topology. This text covers all the basics including connectness, compactness, metric spaces, etc.

And finally,

3. Vector and Tensor Analysis with Applications by no fewer than 2 Russians: A. I. Borisenko and I. E. Tarapov. Yes, the notation is not as modern as you will find in other texts but what you will find is a very clear introduction to (classical) tensor analysis.

 

[mathwonk]

I have done some research since last night on connections between differential geometry and algebraic topology and wish to revise my advice.

Doodle Bob is the expert here but I will make some remarks subject to his review.

The concept of second derivatives which I aligned mostly with diff geom, do indeed have significant applications to algebraic topology, via morse theory.

Morse theory considers the second derivative matrix of a real valued function at a critical point, especially when that matrix of second derivatives is non singular. The subsequent identification of critical points as saddle points or maxs or mins is actually central to understanding the global topology of the manifold.

E.g. a torus is distinguished among compoact oriented 2 manifolds by having a function with a max a min and 2 saddle points.

Morse theory allows one to construct a CW complex reflecting the homotopy of the a manifold, just from knowing the critical poinmts of one non degenerate function, with its second derivatives at those points.

Moving on to Riemannian structures, it is also useful to introduce a metric to measure non topological entities like length, but which turn out to have topological implications.

E.g. in the study of homotopy groups, i.e. loop groups, it is useful to introduce a length for these loops, in order to discern "shortest" length loops or geodesics.


These give critical points for the length function on the space of loops and allow one to determine the homotopy of the loop space on the manifold.

Some sphere e.g. can be realized as loops spaces of smaller spheres, via the suspension construction of freudenthal, and thus riemannian geometry yeil;ds results on the homotopy groups of spheres, a purely topological question.

The deep periodicity theorems of bott on the stable homotopy of classical groups is another consequence of these differential geometry methods.


If one tries to read about this say in milnor's book on morse theory, he/she may well wish he had followed doodle bob's advice and learned an intuitive version of curvature and differential geometry ideas from an elementary book first, like millman parker, or do carmo.


so my original comments about alg top not using diff geom, did not go far enough into alg top it seems. just my ignorance. it is always better to know something than not to.

 

[mathwonk]

hardy was my acad great great grandfather, (but i still pick on his calculus book, pure mathematics, for tiny things).

that makes my great^5 grand dad = arthur cayley, the man who apparently first defined abstract groups (1854).

ours is a small family.

 

[mathwonk]

i think it was not meant that one cannot find gems among dover books, much appreciated ones at the price too, including some great classics.

I have owned dover versions of einstein's papers, the principia of Newton, electricity and magnetism by maxwell, riemann's works, transfinite numbers by cantor, and more recently a nice modern work on systems of ode's by paul waltman.

but there is a caution that dover books often are books whose terminology or point of view is out of date, and hence may lead beginners down a path of isolation, all too common among physicists wrt mathematics.

another unfortunate observation, dover books are no longer physically of the high quality they once were. Bindings in sewn signatures that lasted essentially forever, have been replaced by cheap glued pages, or "perfect" bindings that can easily fall apart. this is a sad change. I would willingly pay a few dollars more for a decent binding.

i wish dover would publish the all too scarce and costly book: foundations of modern analysis, by dieudonne, but i agree henry edwards' advanced calculus is a modern, well written book.

anything by richard silverman is also recommended.

 

[Doodle Bob]

I have done some research since last night on connections between differential geometry and algebraic topology and wish to revise my advice.

Doodle Bob is the expert here but I will make some remarks subject to his review.

ha! I'm still embarassed over that "the determinant is an open map" fiasco.

I very much agree with what you've written here. In a rather surprising synchronicity, this discussion of Morse theory ties in nicely to some papers that I've been reading in the past few days regarding polytopes in R^n in which an index is defined on the vertices of the polytopes and summing all of the indices up of course results in the Euler characteristic.

Anyway, I have repeatedly found that many mysterious results in differential geometry -- particularly, those which start with questions of the sort "Why the hell would anyone construct a tensor that looks like that?", e.g. the Schwartzian derivative -- usually have a fairly solid low-dimensional analogue that clears the whole thing up.