Intro to differential geometry with worked examples

[dyn]

Hi. I am looking for the most basic intro to differential geometry with plenty of worked examples. I want it to cover the following - differential forms , pull-backs , manifolds , tensors , metrics , Lie derivatives and groups and killing vectors. Problems with solutions would also be good as I am self-studying. I already have the book " Geometrical methods of mathematical physics" by Schutz. Thanks.

 

[George Jones]

Maybe Fecko:

Another book worth looking at is Differential Geometry and Lie Groups for Physicists by Marian Fecko,

https://www.amazon.com/dp/0521845076/?tag=pfamazon01-20.

This book is not as rigorous as the books by Lee and Tu, but it more rigorous and comprehensive than the book by Schutz. Fecko treats linear connections and associated curvature, and connections and curvature for bundles. Consequently, Fecko can be used for a more in-depth treatment of the math underlying both GR and gauge field theories than traditionally is presented in physics courses.

Fecko has an unusual format. From its Preface,

A specific feature of this book is its strong emphasis on developing the general theory through a large number of simple exercises (more than a thousand of them), in which the reader analyzes "in a hands-on fashion" various details of a "theory" as well as plenty of concrete examples (the proof of the pudding is in the eating).

The book is reviewed at the Canadian Association of Physicists website,

http://www.cap.ca/BRMS/Reviews/Rev857_554.pdf.

From the review

There are no problems at the end of each chapter, but that's because by the time you reached the end of the chapter, you feel like you've done your homework already, proving or solving every little numbered exercise, of which there can be between one and half a dozen per page. Fortunately, each chapter ends with a summary and a list of relevant equations, with references back to the text. ...

A somewhat idiosyncratic flavour of this text is reflected in the numbering: there are no numbered equations, it's the exercises that are numbered, and referred to later.

Personal observations based on my limited experience with my copy of the book:

1) often very clear, but sometimes a bit unclear;
2) some examples of mathematical imprecision/looseness, but these examples are not more densely distributed than in, say, Nakahara;
3) the simple examples are often effective.

 

[dextercioby]

If you think Fecko is too difficult, perhaps Nakahara's (2nd edition) could serve as an alternative. The examples are from physics.

 

[NumericalFEA]

https://www.amazon.com/dp/0486634337/?tag=pfamazon01-20

 

[Daverz]

That Dover book is not what he's looking for.

...I should expand on that. Going by his list of topics, what he needs to look for are books with "differentical forms" in the title. Most will also cover differential forms on manifolds. Like this book by do Carmo:

https://www.amazon.com/dp/3540576185/?tag=pfamazon01-20

 

[SteamKing]

Hi. I am looking for the most basic intro to differential geometry with plenty of worked examples. I want it to cover the following - differential forms , pull-backs , manifolds , tensors , metrics , Lie derivatives and groups and killing vectors. Problems with solutions would also be good as I am self-studying. I already have the book " Geometrical methods of mathematical physics" by Schutz. Thanks.

You can always try the Schaum's Outline of Differential Geometry:

https://www.amazon.com/dp/0070379858/?tag=pfamazon01-20

This series contains worked examples and plenty of practice problems. IDK if it covers all the topics on your list, but it will get you started with the basics.

 

[SteamKing]

You can get an intro to tensors in this companion volume in the Schaum's series:

https://www.amazon.com/dp/B00AO1JM5O/?tag=pfamazon01-20

 

[dyn]

Is tensor calculus part of differential geometry or are they separate subjects ? Is General Relativity taught in 2 different ways ; one using tensors and one using differential forms or are they both combined in GR ?

 

[dextercioby]

Tensor calculus is a part of differential geometry. GR is taught using tensors at the standard level. Using differential forms (Cartan calculus) is a fancy but useful way to write it, expecially when one thinks of GR's extensions: PGT, EC, SUGRA, etc.

 

[SteamKing]

Tensors can show up in other areas of math and physics besides diff. geometry. One place I seem to stumble over them occasionally is in finding the principal axes of a general 3-D body. There are other applications in mechanics, involving stress analysis.

A lot of geometry has been formulated these days using matrix methods, to facilitate doing numerical calculations with computers, and it seems you run into tensors as a consequence of this also.