Confusion about Differential Geometry Books

  • #1
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I was just browsing through the textbooks forum a few days ago when I came across a post on differential geometry books.
Among the others these two books by the same author seem to be the most widely recommended:

  1. Elementary Differential Geometry (Barret O' Neill)
  2. Semi-Riemannian Geometry with applications to relativity (Barret O'Neill)
I don't know when or if I'll get the time but I think differential geometry is really cool stuff , also it is kind of an inspiration to be able to understand it in the future.
2 seemed to be bursting with cool stuff. But I am in doubt whether 1 is a prerequisite for 2 or can I jump in directly? Have you used any of them?

Thanks in advance
 

Answers and Replies

  • #2
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If you’re interested in General Relativity then books on GR such as Wheeler‘s Gravitation book have chapters on DG to get you up to speed.
 
  • #3
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If you’re interested in General Relativity then books on GR such as Wheeler‘s Gravitation book have chapters on DG to get you up to speed.
Thanks for replying.
I am interested in Differential geometry for the sake of differential geomtery.
 
  • #4
robphy
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You can read samples from Google books.
I pasted the first two pages of each table of contents.

https://www.google.com/books/edition/Elementary_Differential_Geometry/HrriBQAAQBAJ?gbpv=1
1625937598463.png


https://www.google.com/books/editio...try_With_Applicatio/CGk1eRSjFIIC?hl=en&gbpv=1
1625938029088.png
 
  • #5
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I think you misunderstood the question . I am not qualified enough to judge which book is an entry level text and which is not. If you check the original post you'll see that I say the 2nd book is cool. I had already seen the table of contents before I posted. Since both books were written by the same author I am confused whether they have to be read in a particular order. Both have almost minimal prerequisites as quoted in their prefaces but there is the matter of "mathematical maturity". Can one profitably work through one of these on their own without ripping out their hair? Someone who is way more experienced than me can probably clarify for me. Thanks for the reply anyway.:smile:
 
  • #6
robphy
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I think you misunderstood the question . I am not qualified enough to judge which book is an entry level text and which is not. If you check the original post you'll see that I say the 2nd book is cool. I had already seen the table of contents before I posted. Since both books were written by the same author I am confused whether they have to be read in a particular order. Both have almost minimal prerequisites as quoted in their prefaces but there is the matter of "mathematical maturity". Can one profitably work through one of these on their own without ripping out their hair? Someone who is way more experienced than me can probably clarify for me. Thanks for the reply anyway.:smile:

From the chapter headings,
those in the "Elementary Differential Geometry" text look to be entry level
since words like "calculus, surface" are likely more familiar than "manifolds and tensors".
The last chapter is called "Riemannian Geometry".
In the Semi-Riemannian text, "Riemannian" and "Semi-Riemannian" (a variant of Riemannian) are in the Ch3 titles and onward.
So, from looking at the table of contents, I think the titles are appropriate.
The "Elementary" text is a pre-requisite to the "Semi-Riemannian" text.

From reading the introduction and preface,
the Elementary one says
"This book presupposes a reasonable knowledge of elementary calculus and
linear algebra. It is a working knowledge of the fundamentals that is
actually required. The reader will, for example, frequently be called upon
to use the chain rule for differentiation, but its proof need not concern us."

the Semi-Riemmanian one says
"The basic prerequisites for the book are modest: a good working knowledge
of multivariable differential calculus, in f‌irm belief in the existence
and uniqueness theorems of ordinary differential equations, and an
acquaintance with the fundamentals of point set topology and algebra.
Later on. a knowledge of fundamental groups, covering spaces, and Lie groups
is required..."
 
  • #7
martinbn
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Can one profitably work through one of these on their own without ripping out their hair?
One can, but I am guessing you want to know if you can. How could we possibly know?! We don't know you.
 
  • #8
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From the chapter headings,
those in the "Elementary Differential Geometry" text look to be entry level
since words like "calculus, surface" are likely more familiar than "manifolds and tensors".
The last chapter is called "Riemannian Geometry".
In the Semi-Riemannian text, "Riemannian" and "Semi-Riemannian" (a variant of Riemannian) are in the Ch3 titles and onward.
So, from looking at the table of contents, I think the titles are appropriate.
The "Elementary" text is a pre-requisite to the "Semi-Riemannian" text.

From reading the introduction and preface,
the Elementary one says
"This book presupposes a reasonable knowledge of elementary calculus and
linear algebra. It is a working knowledge of the fundamentals that is
actually required. The reader will, for example, frequently be called upon
to use the chain rule for differentiation, but its proof need not concern us."

the Semi-Riemmanian one says
"The basic prerequisites for the book are modest: a good working knowledge
of multivariable differential calculus, in f‌irm belief in the existence
and uniqueness theorems of ordinary differential equations, and an
acquaintance with the fundamentals of point set topology and algebra.
Later on. a knowledge of fundamental groups, covering spaces, and Lie groups
is required..."
Thanks a lot that answers my question
 
  • #9
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9
One can, but I am guessing you want to know if you can. How could we possibly know?! We don't know you.
I was hoping for a generic answer. For example most people wont start studying a field with a book which is considered a "classic" (that's the impression I got for the 2nd book anyway) they'll rather look around for a standard introductory textbook. I was just considered about the reading order. Anyway I'll try read a chapter or two and let you know what works for me.
 
  • #10
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My suspicion is the one with Elementary in the title is the introductory book and the other is more advanced for GR and beyond.
 
  • #11
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I'd read the "Elementary" book first, since it starts with more concrete topics (curves and surfaces in 3-space).
 
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