Which Book on Riemannian Geometry Balances Intuition and Minimal Prerequisites?

[Ragnar]

I'm looking for a good book on riemannian geometry, with a minimum of prerequistes and that takes a more intutive rather than formal approach.

I know a bit of calculus of variations, multivariable calculus, vector calculus, and a bit of linear algebra.

 

[quasar987]

Do you want to learn it in the context of general relativity?

If so, I recommand, "Relativity on curved manifolds" by de Felice and Clarke. It is a math book, not a physics book and it meets your requirement of "informality" while still being completely rigourous.

I actually haven't read any other "maths of relativity" books, so it's not like out of a selection of tens of books, I think this one is the best, but I'm using it and find it totally adequate.

 

[Doodle Bob]

I'm looking for a good book on riemannian geometry, with a minimum of prerequistes and that takes a more intutive rather than formal approach.

I know a bit of calculus of variations, multivariable calculus, vector calculus, and a bit of linear algebra.

This is what I wrote about this topic about a year ago:

"...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."

All three of the above books, though often technical, require not much more than what you claim to have covered already.

 

[Ragnar]

This is what I wrote about this topic about a year ago:

"...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. .

Why would it be wrong to chose those?

 

[Doodle Bob]

Why would it be wrong to chose those?

...because they will not provide you with what you requested.

 

[Ragnar]

Is riemannian geometry by sylvestre gallot good?

 

[DeadWolfe]

Maybe it's just me, but I would start with the low dimensional case, ie:

Differential Geometry of Curves and Surfaces by Do Carmo

 

[mathwonk]

everyone agrees docarmo is outstanding.

 

[Daverz]

Not having seen do Carmo, my recommendation for a text would be https://www.amazon.com/dp/0132641437/?tag=pfamazon01-20. They cover the differential geometry of curves and surfaces and then in the final chapter generalize to R^n.

I do have do Carmo's follow on Riemannian Geometry book, which is very good, but does ideally require a previous course in differential geometry (he refers to his own differential geometry book quite often).

By the way, there is nothing wrong with the books that Dover has put out. They are older books that cover classical differential geometry, and there's nothing wrong with that. I think Struik's book is pretty good, actually.

 

[yenchin]

Do Carmo's books are great. It is good to have a familiarity with 3-dimensional differential geometry first (i.e. geometry of curves and surfaces embedded in R^3). However you need to sort of re-learn from an intrinsic point of view when you go to Riemannian geometry or GR for that matter. So just be prepared.

 

[hrc969]

Is riemannian geometry by sylvestre gallot good?

I like this book. I'd say it is a pretty good book. A plus for you is that it only assumes knowledge of calculus in euclidean space. However, that's not to say that you'll be able to go thorugh it with no problems. Familiarity with geometry on curves and surface is a lot of help. Familiarity with differentiable manifolds also helps.

But I think that if you devote enough time trying to understand everything you might be able to go through it just fine.

I'm not so sure what you mean by intuitive rather than formal. The presentation and format of the bool seems a bit formal to me.

 

[Daverz]

I forgot to mention this book: https://www.amazon.com/dp/1568810733/?tag=pfamazon01-20

 

[Ragnar]

I like this book. I'd say it is a pretty good book. A plus for you is that it only assumes knowledge of calculus in euclidean space. However, that's not to say that you'll be able to go thorugh it with no problems. Familiarity with geometry on curves and surface is a lot of help. Familiarity with differentiable manifolds also helps.

But I think that if you devote enough time trying to understand everything you might be able to go through it just fine.

I'm not so sure what you mean by intuitive rather than formal. The presentation and format of the bool seems a bit formal to me.

What i meant was, does it present the intuition behind every theorem or a t least the major ones.

 

[yenchin]

I also like "Riemannian Manifolds: An Introduction to Curvature" by John M. Lee.

 

[mathwonk]

the reason to prefer the newer books over the older books is the observatiion that people who learn the old way first tend not to be able or willing to learn the new way afterwards, whereas those who learn the modern approach can easily read the old books as well.

are you a counterexample to this belief daverz?

 

[hrc969]

What i meant was, does it present the intuition behind every theorem or a t least the major ones.

Like I said, Gallot's book is pretty formal and if you have not had experience with the geometry of curves and surfaces then it might not be as intuitive. You can always try to look at a theorem and try to make an example for a 2D surface or something but it won't be as strightforwardas it would be if you've studied the geometry of curves and surfaces.

A book that I think takes a more intuitive approach is https://www.amazon.com/dp/0821839888/?tag=pfamazon01-20. This book will present you with the geometry of curves and surfaces and to some of the concepts of riemannian geometry in the context of curves and surfaces and then generalize later. This will make it easier to understand the more general approach (which is the kind that is done in Gallot).

Some people rather just deal with the general case from the beginning (I'm not sure whether that's a good idea, though) but as you want an intuitive approach Kuhnel's book seems right.

 

[gvk]

This is what I wrote about this topic about a year ago:

"...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.
...

"A foolish consistency is the hobgoblin of little minds."
Ralph Waldo Emerson

 

[mathwonk]

nonetheless, you should try to understand what doodle bob is getting at here.

 

[MeJennifer]

"Modern Differential Geometry for Physicists" by Isham is a gem IMHO.

 

[mathwonk]

one of the things i remember about the great books on basic physics by feynman was his assertion that he was going to teach the stduehnts as if they were all going to be physicists, even though he knew that was not the case. And he remarked that all their other teachers at cal tech would do the same. There is a reason for that, namely that is the only way to do justice to the subject.

THERE MAY BE EXCEPTIONS, BUT A BOOK ENTITLED math for Physicists strikes me LIKE ONE ENTITLED algebra for dummies, i.e. a slap in the face of the reader, as if he/she is not intelligent enough to apropeciaTE THE COUrsE TAUGHT CORReCTLY.

 

[Doodle Bob]

"A foolish consistency is the hobgoblin of little minds."
Ralph Waldo Emerson

Oh, isn't that nice?

Ragnar, let's get back to the point. The only way to gain deep understanding and intuition with regard to diffl. geometry (and any topic for that matter) is to work on problems, actually pedagogically well-posed problems: i.e. those designed to bring out the various geometrical and analytical ideas involved with the subject.

So, you want a text with not only well-written chapters, but good sets of problems and exercises to work on.

The vast, vast majority of diffl. geometry texts written for/by physicists are for pure content only. From the authors' perspective, geometry is a tool to create models and that's it. This is fine from a purely physics point of view, but not if you're looking for an upgrade on your spatial reasoning skills.

What's wrong with the Russian? you ask. Well, the Russian education system has students working on geometric problems at a comparatively intense level at a very early age. Most texts written for Russian students already assumes a lot of geometrical thinking that most of us pick up during the first few years of studying diffl. geometry.

Finally, those damn Dover books. Look, they're cheap for the most part ecause the copyrights ran out and no one else really cared enough about the book to keep them in print. There are quite a few good Dover texts -- e.g. Bishop, Struik -- but in the end you get what you pay for and I don't think any of these are what you are looking for.

Again, track down the books that have been recommended here. And don't be afraid to wade into the technical details. True understanding comes only from willing to get your hands dirty with specific (and often messy) problems.

 

[Daverz]

the reason to prefer the newer books over the older books is the observatiion that people who learn the old way first tend not to be able or willing to learn the new way afterwards,

Have you actually observed this?

whereas those who learn the modern approach can easily read the old books as well.

I don't think Struik's book is hugely different from some modern undergrad books on differential geometry. Certainly not on the basic concepts.

I think Struik + https://www.amazon.com/dp/0821839888/?tag=pfamazon01-20 would make a nice combination.

 

[mathwonk]

"Have you actually observed this?"yes i have. in fact i presume you have too, the last several years as many people here who think tensors are arrays of indices have persistently declined to learn what they mean in terms of multilinearity.

the point is not that the old books are similar, if they were you could read the two types in any order.

the point is the modern books are more conceptual, and require a new way of thinking, which people disinclined to learn new things refuse to acquire in their old age.

 

[mathwonk]

i recall this insight on the reluctance of people trained in old methods to upgrade, was first pointed out to me by the famous complex algebraic geometer Boris Moishezon, when i was a relatively young man in 1976. He had been obliged by his teacher to indulge the biases of the older seminar members against modern methods when speaking.

In particular he had been asked by his advisor to present Hironaka's proof of the resolution of singularities, without using the concept of quotient ring. I still remember his plaintive cry, "how is this possible?!"I listened to his advice.

 

[mathwonk]

here is some complementary advice from richard courant: After stressing the necessity of acquiring the modern higher appreciation for general "methods" over the ancient one of focus on individual examples, he says:

"The point of view of school mathematics tempts one to linger over details and to lose ones grasp of general relationships and systematic methods. On the other hand, in the "higher" point of view there lurks the opposite danger of getting out of touch with concrete details, so that one is left helpless when faced with the simplest cases of individual difficulty...

The reader must find his own way of meeting this dilemma. In this he can only succeed by repeatedly thinking out particular cases for himself and acquiring a firm grasp of the applications of general principles in particular cases; here lies the chief task of anyone who wishes to pursue the study of Science."

pages 2-3, Differential and Integral Calculus, vol. 1.perhaps this is what you mean to address by recommending one older and one newer book.

 

[mathwonk]

i want to remind people that michael spivak's books on diff geom, especially vol 1 for manifolds and their de rham cohomology, and vol 2 for the classical theory of curvature, explained in modern language, is still unique and unmatched in my opinion.

there are to my knowledge no other books that literally give you riemann and gauss's own words and then explain them patiently and clearly in modern language.

 

[Daverz]

the point is the modern books are more conceptual

Sorry, I don't buy that. The basic concepts were worked out early in the last century. There's nothing magical about the formalism added in the last 60 years that makes it "more conceptual". In fact, quite the opposite if you make a fetish of modern formalism.

You comments about the older view of tensors is a telling strawman. The old definition of tensors is still very useful and did not become magically inoperative.

 

[mathwonk]

please rethink this. I believe you are missing the point.

 

[quantum123]

The best book is still The book that you are going to write yourself. It starts with just blank pages.
I always learn a subject as if I am going to teach it later. So compile the notes from all the books you have read, give your own taste and approach and even originality, there you are - a book written by you for your students.

But I agree with this: books written for physicists is really not a good deal. They are not very clear. But some physicist think that it is good enough.

 

[mathwonk]

good point. think about the goal of differential geometry, to understand curves and curved surfaces. the basic problem is how to emasure the curvature.

This is done by comparing the curves and surfaces you have with the simplest curved objects, namely circles and spheres.

Essentially a curve (which is not a circle) has the same curvature at a point as a circle tangent to it at the point and which also crosses it there. the curvature of a circle is measured by its radius, or ratehr the reciprocal of the radius, since that gets larger as the circle gets tighter.

another way to compoare the curve to a circle, if the curve is in the plane, is to send each point of the curve to the end point of the unit vector at the origin, which is parallel to the normal vector to the given curve at each point.

We also have a choice of length of the vector, instead of a unit vector. If we choose the length of the normal vectors to map to, so that the derivative of the map is one wrt arc lnegth, at our point, then we are mapping to the circle with the same curvature as our curve.

This method of comparing curvature by compoaring arclength generaliuzes to surfaces, as Gauss realized.a surface in three space is compared with a sphere centered at the origin. we simply map each point of the surface to the endpoint of a vector of fixed length, emanating (I love that word) from the origin, and parallel to the normal vector to the surface at the point.

then we compare areas under this map, by taking the determinant of the derivative (called the jacobian determinant). The sphere has the same curvature as the surface at a given point, when the area change is one, i.e. the area determinant is one. This happens when the radius vector from the origin has appropriate length.

the curvature is apparently measured by the square of the reciprocal of the curvature of a geodesic circle on this sphere, but I am no tpositive since I ma making all this up. I.e. the usual definition of Gauss curvature is the valoue at the point of the jacobioan determinant, maybe divided by pi or somehting.after all this is my book. but see how simple it all is? the crap in diff geom books with huge arrays of indices and coordinates is just an attempt to assign numbers to these ideas. What does it profit a man to learn to manipulate all those indices and lose his soul? no wait that's from the hebrew bible - I meant: what good is it to throw around words like connection, and chern class, or contravariant, or lie derivative, and not understand the meaning of curvature?

Spivak starts you out just like this: even as we take a limit of two points to define a tangent line, so dow e take a limit of three points to define a circle of curvature. then he goes staright to gauss, and finalkly tyranslates it for you into fancy notation but with a conceptual underpinning.

he introduces all the may versions opf modern curvature, and checks each one in the "test case". Namely he checks that each one gives zero for flat euclidean space. then he shows the acid test, that zero curvature implies locally flat in the sense of locally equivalent (isometric) to euclidean space. at least as i recall. it has been 30 years since i read spivak, and i only spent 2 days reading it (vol 2).

and he gives you many extras, like how to compute the dimension of a lie group, how to work with de rham cohomology, how to drive a significant global result from a trivial local one (sheaf theory)...

 

[mathwonk]

abstract BS on manifolds

Now to do all this "intrinsically", without the coordinates of 3 space, someone had the brilliant but very cumbersome idea to equip the surface with a different coordinate system at every point, e.g. a tangent plane at each point, and a projection from each tangent plane down onto the surface, at least locally.

Then to measure lengths, angles, and distances, one needs a dot product, but one needs a different one in every tangehnt space, i.e. one needs a dot product at every point. To make it less explanatory to the uninitiated, and claim the blessing of deities, we call this choice a "Riemannian metric".

Then we have everything you have in euclidean space, but we have a different one at each point! we have families of diferential forms, one form, 2 forms, etc... we correctly recognize these as "sections" of various "bundles" (of vector and covector spaces).

so we mesmerize the innocent by telling them a differential one form is a section of the cotangent bundle, and since they already know what determinants are, we use a new word to frighten them, calling a differential k form a section of the kth exterior algebra on the cotangent bundle.

by now they are quivering in fear, and we introduce curvature in terms of "connections", or even "Koszul connections", maybe to make them think of the monsters in ghostbusters, but which is just a way of taking derivatives.

spheres are completely forgotten, we have insured that no one will any longer grasp how simple curvature is. we can rest, our work is done.

 

[quantum123]

Interesing ..
2 points:-
1) Does it extend to N dimension? Can you extend this to pseudo-riemannian?
2) Do you mean that it is better to have an ambient space for the surface to embedd and that sphere is in the ambient space? This is not very popular among the relativity people. The idea of ambient space is considered unnecssary. Intrinsic is considered beautiful.

 

[mathwonk]

2) that is the first big result of gauss, called the "theorema egregium" or "big theorem". namely that the curvature of a surface embedded in three space, and measured using the gauss map to the sphere, is exactly the same as the intrinsic version of the curvature, defined using only the riemannian metric induced by the embedding. Or in plain english "the metric determines the curvature". so this intrinsic point of view is due to gauss and riemann. the current general relativity people are only following their 150 year old lead. unfortunately they often decline to use modern truly intrinsic mathematical language for these concepts, but stick with exactly the same outmoded tools as used by riemann and einstein.

the purpose of spivaks first two volumes is to teach in vol 1 the modern language of bundels etc, then show in vol2 how they enablle one to better understand and appreciate the intrinsic conceprs riemann was trying to express without modern terminology available. inded the importance of his ideas was one thing that led people to try to express it more naturally.

1) and yes riemann showed how to generalize gauss' notion of curvature of two dimensional surfaces, to a notion of curvature in n dimensions, essentially by acting on all 2 dimensional surfaces in the larger manifold. that is riemanns big contribution,a nd the one which apparently caused gauss to leave his lecture exhulting on riemann's marvellous and fertile originaity.

i don't know what exactly pseudo riemannian means (probably relax the positive definiteness), but i will suggest the an swer is yes one can do something similar.

and i can say all this without knowing essentially anythign abut differential geometry, merely knowing how to look at something, and what books to look at.
read spivak if you want to understand the subject.

 

[mathwonk]

i do not mean ambiently embedded surfaces are better, just easier to understand. Note the key point is that the tangent vectors in euclidean space can all be translated to the origin, i.e,. "the tangent bundle of R^n is 'trivial'".

so all one needs for a gauss map is to be embedded in a manifold with trivial tangent bundle. and every manifold does embed in euclidean space. moreover complex and real tori also have trivial tangent bundles, so all smooth subvarieties of tori also have gauss maps, and these were used crucially in the study of curves in their jacobians, by andreotti and mayer.

Even in manifolds without trivial tangent bundles, gauss maps, i.e. maps induced on tangent bundles, i.e. derivatives, have been used in study of moduli spaces and maps between them, especially by Carlson and Griffiths, and (earlier but less clearly) in my thesis.

 

[cliowa]

Getting back to the title of this thread: I heard a lot of good stuff about Bishop and Crittendens https://www.amazon.com/dp/0821829238/?tag=pfamazon01-20.

Has anybody else heard of it?

 

[mathwonk]

it is a very famous old book, with excellent content, but i have never heard it compared with the ones recommended here in terms of readability. but if you can read it, it will serve you well.