# [Differential geometry] Book suggestion required

## Main Question or Discussion Point

Hello,

I am a beginner. I am self taught in differential calculus. Can you please suggest me any book, as a beginner, to have a very basic idea and overview on Differential Calculus.

Any free e-book?

Kindly suggest.

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What is it exactly that you want to know about differential geometry. I ask because differential geometry is a pretty broad area.

On one hand, you have the "basic" differential geometry of curves and surfaces. Nice books here are of course Do Carmo: https://www.amazon.com/dp/0132125897/?tag=pfamazon01-20 and O'Neill: https://www.amazon.com/dp/0120887355/?tag=pfamazon01-20 and Millman & Parker: https://www.amazon.com/dp/0132641437/?tag=pfamazon01-20
These books are quite readable, but I do recommend knowing some Linear Algebra beforehand.

Then there is also modern differential geometry. This uses the language of manifolds. The study of this requires quite some more prerequisites. Topology, basic analysis, linear algebra are all needed. A nice book is of course Lee. He has three books on differential geometry, starting with topological manifolds. The second book is smooth manifolds and then Riemannian manifolds.
https://www.amazon.com/dp/1441979395/?tag=pfamazon01-20
https://www.amazon.com/dp/1441999817/?tag=pfamazon01-20
https://www.amazon.com/dp/0387983228/?tag=pfamazon01-20

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Fredrik
Staff Emeritus
Gold Member
Hello,

I am a beginner. I am self taught in differential calculus. Can you please suggest me any book, as a beginner, to have a very basic idea and overview on Differential Calculus.

Any free e-book?

Kindly suggest.
Can you explain what you mean by "differential calculus". Micromass interpreted it as "differential geometry" (which makes sense for several reasons, one of them being that you posted this in the differential geometry forum), but it can also be interpreted as "calculus".

Calculus: Limits, derivatives, integrals, series, etc. Courses with "calculus" in the name tend to focus on the "how to calculate" aspects of these topics.

Analysis: The same topics as in calculus, but now the focus is on the theorems instead of on how to calculate.

Differential geometry: Smooth manifolds, tangent spaces, cotangent spaces, connections, parallel transport, geodesics, tensor fields, differential forms, covariant derivatives, curvature,...

Edit: I'm going to move this thread to the books forum.

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Oops, I totally misread "differential calculus" as "differential geometry". Disregard my post then.

So yeah, the OP should probably clarify what topics he wants to learn. What he already knows. And what level he wants to learn them (rigorous or not so rigorous).

mathwonk
Homework Helper
Books by Serge Lang tend to give the main ideas and overview clearly, including some theoretical rigor, without a lot of examples and problems.

http://www.abebooks.com/servlet/SearchResults?an=serge+lang&sts=t&tn=a+first+course+in+calculus

This next little book was designed for laypersons wanting to know what calculus is about:

https://www.amazon.com/dp/0883856026/?tag=pfamazon01-20

one can get a "very basic idea" of calculus from this fun little book:

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I am reading GTR from Hartle. I think at least some knowledge for differential calculus is necessary. I am looking for some basic ideas.

mathwonk
Homework Helper
you said that. i gave you my suggestions with that in mind.

WannabeNewton
I am reading GTR from Hartle. I think at least some knowledge for differential calculus is necessary. I am looking for some basic ideas.
I'm assuming GTR = General Theory of Relativity. You won't need any differential geometry for Hartle as the book is very light, if even that, on advanced mathematics. For that book you will only need multivariate calculus and linear algebra (it is mathematically at the level of Taylor's mechanics). A book where you definitely will benefit from the knowledge that can potentially gained from the books mentioned above is "General Relativity" - M. Wald (the book is very mathematical and as such puts the mathematical physics of GR far ahead in emphasis than the actual physics of GR). A book that touches on differential geometry at a level higher than Hartle, but is still accessible for someone with only linear algebra and multivariate calculus knowledge, is Schutz "A First Course in General Relativity". This is the book I first attempted when learning GR and it carries nice memories with it

You really need to be clear, by the way, if you mean differential geometry or basic differential calculus!

Hello WannaBeNewton,

Yes, indeed GTR is for Generaltheory of relativity. Yes, I can understand that for Hartle, I don't need differential geometry. I mean differential geometry not differential calculus. I am new to this area. I really want to know the use of diff.geometry in GTR. First, I would like to go with Hartle, but curious about diff.geometry.

WannabeNewton
If you are interested in learning differential geometry then I can suggest a few books to get you started, assuming you know linear algebra and multivariate calculus.

It usually helps to start with books on differential geometry of curves and surfaces. Two books that are great for this are https://www.amazon.com/dp/0132125897/?tag=pfamazon01-20 and https://www.amazon.com/dp/0120887355/?tag=pfamazon01-20

The first book takes the classical route and uses a language that you will later see abstracted in GR. The second book uses differential forms throughout the text so it offers a different and more elegant viewpoint of differential geometry of surfaces that, while very enlightening, will not really show up in introductory GR texts. I would suggest using the first book and only using the second if you have spare time and are interested.

Keep in mind that this is far from the end of the road as this is just differential geometry on embeddings in $\mathbb{R}^{3}$. In order to really understand how differential geometry is used in GR, you will have to learn some point-set topology, then learn differential topology, and finally acquire some knowledge of Riemannian geometry. But as with anything else in physics and math, you have to crawl before you can walk and walk before you can run. Assuming you've already learnt calculus and linear algebra, you've just got some walking to do before you get to the promised land :)

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I'll chime in with another cool book:

https://www.amazon.com/dp/0521468000/?tag=pfamazon01-20

The books mentioned so far tend to fall into two related subject areas, the "differential geometry of curves and surfaces" and "tensor calculus on manifolds". The usual approach in GR books is just to launch into the tensor calculus. Sean Carroll's book does a good, no-nonsense job on this, for example, and is a good followup to Hartle. You don't need the background of "curves and surfaces" to learn and use tensor calculus, but it does help with developing some intuition about curvature. A book that does take this approach is:

https://www.amazon.com/dp/082471749X/?tag=pfamazon01-20

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WannabeNewton
I would personally disagree that Carroll does a "no-nonsense" job with differential geometry; he hand waves a lot of it. If the goal is to learn the mathematics proper (which is what I think the OP wants), then one would need to use a modern text on differential geometry (e.g. the books by Lee). If one wants to learn tensor calculus as used in GR (e.g. be able to do calculations like showing $\mathcal{L}_{\boldsymbol{\psi}}\epsilon_{abcd} = 0$) then yeah books like Carroll will do just fine (or better yet Wald, since Wald does tensor algebra / tensor calculus computations in the exact same way as Carroll but he includes more rigorous background on differential geometry).

I mentioned Do Carmo's differential geometry of curves and surfaces text because it serves as a nice intuitive bridge into Riemannian geometry (of which Do Carmo also has a text on).

I mentioned Do Carmo's differential geometry of curves and surfaces text because it serves as a nice intuitive bridge into Riemannian geometry (of which Do Carmo also has a text on).
Do Carmo's Riemannian geometry text is really nice and laid back. So it's perfect for a first introduction. There are of course many more difficult texts.

WannabeNewton
Do Carmo's Riemannian geometry text is really nice and laid back. So it's perfect for a first introduction. There are of course many more difficult texts.
Lol remember when we first saw Lee call it a "much more leisurely treatment".

Lol remember when we first saw Lee call it a "much more leisurely treatment".
At least he didn't call it "bed-time reading". That was reserved for Spivak, lol.

WannabeNewton
At least he didn't call it "bed-time reading". That was reserved for Spivak, lol.
I don't know about you but if I ever have kids, bed time stories are going to be passages from Spivak's manifold texts xP

I would personally disagree that Carroll does a "no-nonsense" job with differential geometry; he hand waves a lot of it.
To me "hand waving" implies giving the impression you've derived something when you haven't. Carroll doesn't do this. He tells you when he's skipping a rigorous derivation and moves on. That's his approach to the covariant derivitive, for example. Wald proves it's existence. Carroll just says (paraphrasing), we're going to assume it and move on. It's good to have a reference for the existence proof, but it's a distraction on a first pass.

I also like the Do Carmo Riemannian Geometry book mentioned. It's an "easy" read, but I found that I was unprepared for many of the exercises, perhaps because the required background was in his "Curves and Surfaces" book. Also, he doesn't cover semi-Riemannian manifolds.

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WannabeNewton
That's what I meant by hand waving - he doesn't prove much, if any, of the mathematical results. When I first started Carroll this was actually the reason I stopped short and bought Wald instead because I found it very very annoying the way things would be mentioned without proof. I hate taking mathematical results for granted more than anything.

You will find that not many modern introductory texts on differential geometry cover semi-Riemannian manifolds in rigorous detail. One rather advanced book on differential geometry that does go into the subject is Lang's text: https://www.amazon.com/dp/0387943382/?tag=pfamazon01-20

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.You will find that not many modern introductory texts on differential geometry cover semi-Riemannian manifolds in rigorous detail.
I had a vague notion that some of those cool theorems in the latter part of Do Carmo may not apply to the semi-Riemannian case.

I had a vague notion that some of those cool theorems in the latter part of Do Carmo may not apply to the semi-Riemannian case.
Hopf-Rinow Theorem is one of those important results in Riemannian case but does not apply to semi-Riemannian.