Thread: Reccomendations - Book of Geometry for a Physicist

I'm looking for a book to help me learn a bit more about the actual maths that underpins differentiable manifolds.

I already know the basics of Riemann manifolds having read Schutz Book on General Relativity, but I want to extend and deepen my understanding of the maths.

I found another book by him that I think looks ok.

http://www.amazon.co.uk/Geometrical-...4044084&sr=1-2

Can anyone say if this is a good place to start? It's got lie derivatives and group theory in it aswell.

Or is their another book which might be better?

I know basic calculus and matrix algebra, but thats about it really. I'm not a mathematician, so any suggestions will probably have to be quite tame.

Cheers, sox.

Originally Posted by sox

I'm looking for a book to help me learn a bit more about the actual maths that underpins differentiable manifolds.

I already know the basics of Riemann manifolds having read Schutz Book on General Relativity, but I want to extend and deepen my understanding of the maths.

I found another book by him that I think looks ok.

http://www.amazon.co.uk/Geometrical-...4044084&sr=1-2

Can anyone say if this is a good place to start? It's got lie derivatives and group theory in it aswell.

Or is their another book which might be better?

I know basic calculus and matrix algebra, but thats about it really. I'm not a mathematician, so any suggestions will probably have to be quite tame.

Cheers, sox.

Try a book with a title like "Differentiable Manifolds". Three are books by Hu and Frank Warner like that. Both are good. I don't know if they qualify as "tame". There is also a set (5 volume) by Michael Spivak, entitles "Differential Geometry" that is quite good. You might take a look at volume 1.

On the other hand any book on differential geometry will assume familiarity with the elements of point set topology. I don't know the Schutz Book, but normally one needs basic topology and theory of manifolds before going to Riemannian geometry. So, you might want to make sure that you understand advanced calculus and a bit of topology before going to the differential geometry books.

Hi sox: I am familiar with the Schutz book you link to, and it is quite good. He does cover some elementary concepts needed to progress from, say, chapter 1, but only in very sketchy form.

The general development is logical, but you should be aware he uses some rather unusual notation.

Also reasonably accessible is Bishop & Goldberg, but again, some notation is a little non-standard.

Differential geometry is not an easy subject, at least I don't find it so, and I agree with Rocket that a solid grounding in multivariable calculus is essential, as is some knowledge of point-set topology. While it is true that both books have short sections on these, they are really just "refreshers".

In short, this is not a subject I would advise someone with little math background to dive into on a technical level

Cheers for the reccomendations both of you. I have done multivariable calculus (86% in moi exam 8) )

Do either of you know anything about twistor theory? The reason i'm wanting to find a more mathematically rigorous treatment of manifolds and vector spaces etc is because I'm wanting to get my teeth sunk into an introductory book on twistor theory.

Would the books that have been suggested help towards that goal? Help prepare for tackling twistor theory? Or am I looking in the wrong area to start with?

Sorry to be a pain, I'm just not certain about what all the different areas of maths are and trying to wrok out the difference between topology and differential geometry and stuff is a bit over my head at the minute.

Cheers, sox

Originally Posted by sox

Cheers for the reccomendations both of you. I have done multivariable calculus (86% in moi exam 8) )

Do either of you know anything about twistor theory? The reason i'm wanting to find a more mathematically rigorous treatment of manifolds and vector spaces etc is because I'm wanting to get my teeth sunk into an introductory book on twistor theory.

Would the books that have been suggested help towards that goal? Help prepare for tackling twistor theory? Or am I looking in the wrong area to start with?

Sorry to be a pain, I'm just not certain about what all the different areas of maths are and trying to wrok out the difference between topology and differential geometry and stuff is a bit over my head at the minute.

Cheers, sox

I don't know much about it beyond that it is an approach due to Roger Penrose for a theory of quantum gravity, that is a bit outside the maiinstream. But much of Penrose's stuf is outside the mainstream, so one should not worry about that aspect overmuch. Penrose is a very bright guy.

My impression is that you are between 3 and 5 years of fairly intense study of mathematics away from this sort of thing.

Topology and differential geometry are rather different branches of mathematics, but like all mathematics there are deep connections.

My recommendation is that before you start worrying about things like twistors you first get a firm grounding in pre-requisite mathematics -- point set topology, rigorous real and complex analysis, analysis on differential manifolds and some abstract algebra.

Don't try to get fancy too quickly. You will only get in way over your head.

Originally Posted by DrRocket

Penrose is a very bright guy.

And ya know what? I am sitting on his father's office chair right now!

Long story short: my PhD supervisor was herself a student of Lionel P. (Roger's father) and she passed the chair on to me during an office clean out.

Maybe I should offer to give it back to Roger - I have often agonized about this. It seems the decent thing to do, but I love the connection

I am, of course, breaking my own "rules". Ah well.......

Originally Posted by Guitarist

Originally Posted by DrRocket

Penrose is a very bright guy.

And ya know what? I am sitting on his father's office chair right now!

Long story short: my PhD supervisor was herself a student of Lionel P. (Roger's father) and she passed the chair on to me during an office clean out.

Maybe I should offer to give it back to Roger - I have often agonized about this. It seems the decent thing to do, but I love the connection

I am, of course, breaking my own "rules". Ah well.......

Lionel Penrose was primarily a geneticist, though he was pretty versatile. His sons are uniformly impressive.

Originally Posted by DrRocket

Lionel Penrose was primarily a geneticist, though he was pretty versatile.

This is of course true, but seems to miss an important point which is, vaguely, relevant to this forum.

Prior to the 1960's, nobody knew how to do human genetics in an ethical and effective way (of course the Nazis had tried, but failed. Not because of WW2, but because their programme was scientifically flawed).

So for the first half of the 20th century, human genetics was literally the application of mathematical methods to observational data. Penrose Sr. was a gifted mathematician. His immediate predecessors in the Galton Chair of Human Genetics were JBS Haldane and RA Fisher, both of whom were trained as "proper" mathematicians.

Those of us who were privileged to train in the same department more recently were still brought up in this spirit, even though, by then, human genetics had become very much an experimental science. That is, we were taught a due and proper regard for the place of mathematics in science and especially in genetics.

Hence my involvement, albeit in an amateurish sort of way, with this sub-forum.

I bought that Schutz book.

I'm only a few pages in (literally - i'm on page 5) and have been reading about open sets in and the Haussdorf property. Is this sort of the beginning of point-set topology?

Originally Posted by sox

I bought that Schutz book.

I'm only a few pages in (literally - i'm on page 5) and have been reading about open sets in and the Haussdorf property. Is this sort of the beginning of point-set topology?

That is the beginning of topology of . but not general point-set topology. The Hausdorff property for is not very exciting, since is a metric space.

Didn't understand any of that really but thanks for answering my question anyway!

Originally Posted by sox

Didn't understand any of that really but thanks for answering my question anyway!

Point-set topology is also called general topology and it is the study of topological spaces defined on abstract sets.

A topology on a set S is a class of subsets of S that contains S and the empty set, and is closed under arbitrary unions and finite intersections. The elements of that class are called "open sets".

A metric space is a set with a distance function that satisfies the properties that

d(x,x)= 0 for any x

d(x,y) = 0 iff x=y and

d(x,z) <\= d(xy)+d(y,z) for any x,y,z

The set of all elements z such that d(x,z)<r is the ball of radius r about x. The topology generated by all such sets is the metric topology on the set.

An important property of some metric spaces is that they are "complete". The fact that the real line is complete is what makes most of calculus work.

Normed vector spaces are metric spaces. Complete normed vector spaces are called Banach spaces. If the norm comes from an inner product the space is called a Hilbert space -- very important for quantum mechanics. is a Hilbert space.

Maybe it's too late to chime in since you already bought a book. My expertise is in geometry motivated by physics. So here are my recommendations:

John Milnor: "Topology From a Differentiable View Point"

Raoul Bott and Loring Tu: "Differential Forms in Algebraic Topology"

John Lee: "Introduction to Smooth Manifolds"

Start with Milnor. Then try to read Bott and Tu. These are both highly regarded classics. Bott and Tu will be a bit challenging though. John Lee's book may serve as a useful supplement.

If I were to pick just one book out of the three to recommend, it would be Bott and Tu.

For Lie Groups, try getting Frank Adams, "Lectures on Lie Groups".

For Riemannian Geometry, try the graduate level Do Carmo book.

Originally Posted by salsaonline

Maybe it's too late to chime in since you already bought a book. My expertise is in geometry motivated by physics. So here are my recommendations:

John Milnor: "Topology From a Differentiable View Point"

Raoul Bott and Loring Tu: "Differential Forms in Algebraic Topology"

John Lee: "Introduction to Smooth Manifolds"

Start with Milnor. Then try to read Bott and Tu. These are both highly regarded classics. Bott and Tu will be a bit challenging though. John Lee's book may serve as a useful supplement.

If I were to pick just one book out of the three to recommend, it would be Bott and Tu.

For Lie Groups, try getting Frank Adams, "Lectures on Lie Groups".

For Riemannian Geometry, try the graduate level Do Carmo book.

Those are good books, but they might be just a wee bit difficult for someone who has not yet had a good course in advanced calculus -- say for instance at least including proofs of the inverse function theorem and the implicit function theorem. Sard's theorem ((in Milnor's little book) would not even make sense to someone who has not seen the Lebesgue integral.

Milnor's book is very good, and I checked on it -- apparently Princeton is now publishing it. I am a bit surprised at the price. I have older copy published by the University of Virginia. It was a bit cheaper when I bought it.

I got it for $14. Are you sure we're talking about the same book?

Originally Posted by salsaonline

I got it for $14. Are you sure we're talking about the same book?

I am sure that my copy was quite a bit less than that, way back when, but when I checked I found it at about $25 (e.g. Amazaon), which strikes me a pretty high for such a small old paperback. It is worth the price in terms of content, but still the price strikes me a rather high for what is really just a very short set of lecture notes.

The prices of mathematics books generally strike me as rather outrageous these days. Ten times or more for a paperback what I used to pay for a good hardback -- comparing identical titles.

Milnor is not only a stupendous mathematician, he is also an excellent writer and speaker. I attended a colloquim lecture that he gave years ago. I recall that he started out with very elementary ideas and did not loose me until the very end. I also recall that he appeared to have lost a tremendous amount of weight -- his belt had about 2 feet of free end to it (it almost looked like it would have gone around him twice).

Originally Posted by DrRocket

A metric space is a set with a distance function that satisfies the properties that

d(x,x)= 0 for any x

d(x,y) = 0 iff x=y and

d(x,z) <\= d(xy)+d(y,z) for any x,y,z

Well isn't that interesting; when I stated this in the past you stated that I didn't know what I was talking about and was wrong.

Originally Posted by Ellatha

Originally Posted by DrRocket

A metric space is a set with a distance function that satisfies the properties that

d(x,x)= 0 for any x

d(x,y) = 0 iff x=y and

d(x,z) <\= d(xy)+d(y,z) for any x,y,z

Well isn't that interesting; when I stated this in the past you stated that I didn't know what I was talking about and was wrong.

I doubt that. You fail to provide a specific reference.

Originally Posted by DrRocket

Originally Posted by Ellatha

Originally Posted by DrRocket

A metric space is a set with a distance function that satisfies the properties that

d(x,x)= 0 for any x

d(x,y) = 0 iff x=y and

d(x,z) <\= d(xy)+d(y,z) for any x,y,z

Well isn't that interesting; when I stated this in the past you stated that I didn't know what I was talking about and was wrong.

I doubt that. You fail to provide a specific reference.

http://www.thescienceforum.com/viewt...hlight=#223065

Originally Posted by Ellatha

Originally Posted by DrRocket

Originally Posted by Ellatha

Originally Posted by DrRocket

A metric space is a set with a distance function that satisfies the properties that

d(x,x)= 0 for any x

d(x,y) = 0 iff x=y and

d(x,z) <\= d(xy)+d(y,z) for any x,y,z

Well isn't that interesting; when I stated this in the past you stated that I didn't know what I was talking about and was wrong.

I doubt that. You fail to provide a specific reference.

http://www.thescienceforum.com/viewt...hlight=#223065

All that you have demonstrated, once again, is that you indeed don't know what you are talking about.

The link that you provided is to a discussion regarding the notion of "conitnuous" which you misconstrue as a synomym for "connected" and which is not particularly relatied to the specialization of topological spaces to metric spaces.

Originally Posted by DrRocket

Originally Posted by Ellatha

Originally Posted by DrRocket

Originally Posted by Ellatha

Originally Posted by DrRocket

A metric space is a set with a distance function that satisfies the properties that

d(x,x)= 0 for any x

d(x,y) = 0 iff x=y and

d(x,z) <\= d(xy)+d(y,z) for any x,y,z

Well isn't that interesting; when I stated this in the past you stated that I didn't know what I was talking about and was wrong.

I doubt that. You fail to provide a specific reference.

http://www.thescienceforum.com/viewt...hlight=#223065

All that you have demonstrated, once again, is that you indeed don't know what you are talking about.

The link that you provided is to a discussion regarding the notion of "conitnuous" which you misconstrue as a synomym for "connected" and which is not particularly relatied to the specialization of topological spaces to metric spaces.

Wrong.

Originally Posted by Ellatha

Wrong.

You will have great difficulty getting beyond the snotty little kid stage until you can recognize when you are wrong.

I give up on you.

I'm considering leaving university to pursue some personal study in mathematics. One area of which is topology.

Looking back on this thread I was given advice on which areas I should brush up on before I looked at differential geometry one of which was topology. But are their any prerequisites for learning topology?

If I do go down this route I plan on reviewing all my undergraduate maths first. Would this be sufficient foundations for tackling a book on topology?

Cheers again, Sox.

Originally Posted by sox

I'm considering leaving university to pursue some personal study in mathematics. One area of which is topology.

Looking back on this thread I was given advice on which areas I should brush up on before I looked at differential geometry one of which was topology. But are their any prerequisites for learning topology?

If I do go down this route I plan on reviewing all my undergraduate maths first. Would this be sufficient foundations for tackling a book on topology?

Cheers again, Sox.

You need point set topology before algebraic topology. There are no prerequisites for point set topology.

Point set topology is also known as general topology. The books by Kelley or Dugundji are good texts.

I would recommend against leaving your university.

Originally Posted by sox

I'm considering leaving university to pursue some personal study in mathematics.

Don't do it! A degree in any subject is far, far better that no degree at all. So-called "personal studies"

Looking back on this thread I was given advice on which areas I should brush up on before I looked at differential geometry one of which was topology. But are their any prerequisites for learning topology?

As DrRocket implies, topology comes in at least two flavours: point set topology and algebraic topology. The first requires, at least in my opinion, some basic grounding in elementary set theory, the second is something of a nightmare for the mathematical naif.

Differential geometry is something else again; it is a hard subject, even at the introductory level, and I seriously doubt it can be self-taught.

If I do go down this route I plan on reviewing all my undergraduate maths first. Would this be sufficient foundations for tackling a book on topology?.

No idea. My guess would be NO, but I have no idea what your undergrad math comprised.

Ah don't worry. I do already have an undergraduate degree.

Originally Posted by sox

Ah don't worry. I do already have an undergraduate degree.

If you drop out, the chances of going back for a graduate degree are small. You wiil forget how to live on no money.

Despite what Guitarist said, don't worry about prerequisites for point set topology. All the background that you need can be written on two pages. I had essentially no prerequisites when I learned it -- and we had no lectures, the students doing all the proofs from scratch.

For a number of reasons I'm not really enjoying university and so have been thinking very seriously about "going it alone" and studying my own program of material.

This program is influenced by both my interests, and the advice I have received on this thread and also my thread on group theory.

I am interested in pursuing a PhD in quantum gravity.

I would appreciate constructive criticism on the following program:

Prior Revision
Quantum Physics
Linear Algebra
Complex Analysis

New Material - Mathematics
Point Set Topology, leading to Algebraic and Geometric Topology
Differential Geometry
Representation Theory with the aim of understanding Lie Groups
If time I would also like to look at Solitons and Instantons.

Revision
General Relativity
Quantum field theory including QED
The Standard Model - just an overview to keep me aware of the general ideas involved

New Material - Physics
Quantum Gravity, specifically the material contained in "Quantum Gravity" by Rovelli.
Depending on how things go I might also look at String net condensates, but I have no idea how advanced this would be.

I'm budgeting at least 8 months for all of that, probably more depending on how difficult the topology and geometry is.

So, what do you think? Please dont be too harsh if I've been overly optimistic...

Originally Posted by sox

I'm budgeting at least 8 months for all of that, probably more depending on how difficult the topology and geometry is.

So, what do you think?

You are looking at several years, not months. It would take that long in graduate school.

What is it about school that causes you to not enjoy it ?

A few things.

In no particular order.

Alot of the lecturers are rubbish.
Alot of the material we are given is rushed or not fully explained.
We haven't actually been taught any physics, only how to manipulate equations.
I do not learn by listening, I learn by reading, yet I have to spend 15+ hours a week in lectures copying down course material, just to go home and do the same number of hours of reading just to digest the material for the first time.

The largest single problem lies with myself though. I spent the exam period studying for the exams, but having 7 classes, found that by the time I had read for a few classes I had forgotten what I'd read at the beginning before I even managed getting to the next classes. That was very frustrating.

Finally (and perhaps most importantly), I find myself in a situation where my interest is changing. I still enjoy physics, but I am interested in the mathematics aswell now where before I was not.

Alot of those reasons are somewhat personal and it might appear like im one of those people who is in denial and wants to blame everyone else for how things are turning out.

But I know myself I've worked hard so I dont feel any guilt at the back of my mind.

Originally Posted by sox

A few things.

In no particular order.

Alot of the lecturers are rubbish.
Alot of the material we are given is rushed or not fully explained.
We haven't actually been taught any physics, only how to manipulate equations.
I do not learn by listening, I learn by reading, yet I have to spend 15+ hours a week in lectures copying down course material, just to go home and do the same number of hours of reading just to digest the material for the first time.

The largest single problem lies with myself though. I spent the exam period studying for the exams, but having 7 classes, found that by the time I had read for a few classes I had forgotten what I'd read at the beginning before I even managed getting to the next classes. That was very frustrating.

Finally (and perhaps most importantly), I find myself in a situation where my interest is changing. I still enjoy physics, but I am interested in the mathematics aswell now where before I was not.

Alot of those reasons are somewhat personal and it might appear like im one of those people who is in denial and wants to blame everyone else for how things are turning out.

But I know myself I've worked hard so I dont feel any guilt at the back of my mind.

I did not like lecture classes in graduate school either. Fortunately few of the classes were of that mode. Most were predominantly student participation -- typically 85 -100 %.

A full-time load was 3 classes, not 7. (7 was typical as an undergraduate). Formal in-class tests were rare.

Maybe you should consider another school.

Therein lies the problem. I can't afford to do another degree after this one, hence the self study idea.

Got another 7 classes to do this term too.

Originally Posted by sox

Therein lies the problem. I can't afford to do another degree after this one, hence the self study idea.

Got another 7 classes to do this term too.

There are fellowships and assistanceships for graduate students. One has to live frugally, but they cover the cost of going to school and living expenses.

I never considered actually paying to go to graduate school.

Obviously I've been waaaay too optimistic with the timescale for learning all that material, but what did you think of the structure and order?

Originally Posted by sox

Obviously I've been waaaay too optimistic with the timescale for learning all that material, but what did you think of the structure and order?

I think you will revise it once you start, but it is an OK rough initial plan, rxcept for one thing -- representation theory.

To understand representation theory you will need to understand, in addition to what you have kisted, quite a bit of functional analysis, which in turn will require measure and integration. Representation theory of Lie groups is a very large and very deep subject.

Somewhere along the way you will need the basics of abstract algebra, probably
before you tackle introductory algebraic topology.