Is Lang's Book on Differential Geometry Suitable for Beginners?

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Discussion Overview

The discussion revolves around the suitability of Serge Lang's book on differential geometry for beginners, particularly in the context of learning differential geometry to gain insights into the foundations of mechanics. Participants explore various textbook recommendations and their appropriateness for someone with limited mathematical background.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant expresses a desire to learn differential geometry from the basics, mentioning previous rudimentary knowledge of topology.
  • Some participants recommend starting with Lang's "Fundamentals of Differential Geometry" and then moving to Spivak's "A Comprehensive Introduction to Differential Geometry."
  • Others argue that Spivak may be too advanced for beginners and suggest alternative texts like "The Geometry of Physics" by T. Frankel or Do Carmo's "Differential Geometry of Curves and Surfaces."
  • A few participants criticize the recommendation of Lang's book, stating it is not suitable for someone with only basic knowledge of topology.
  • There is a suggestion that the initial recommendation might have been a misunderstanding, with some participants speculating that the poster intended to suggest starting with Spivak instead of Lang.
  • Some participants emphasize that Lang's book is too abstract and not practical for beginners, advocating for more accessible texts.
  • There are mentions of other texts such as Singer & Thorpe's "Lecture Notes On Elementary Topology And Geometry" and Pressley's "Elementary Differential Geometry" as suitable alternatives.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the suitability of Lang's book for beginners. There are multiple competing views regarding the recommended texts, with some advocating for Lang and Spivak, while others strongly oppose their use for someone starting out in differential geometry.

Contextual Notes

Some participants note that the discussion reflects differing assumptions about the OP's mathematical background and readiness for advanced texts. There is also a mention of potential trolling behavior by one of the participants, which may have influenced the tone of the discussion.

  • #31
Lavabug said:
Does it have solutions for some of the exercises?
Unfortunately it doesn't the last time I saw it which was in a course on differential topology for physicists. The section on the connections between circuit theory and topology was pretty mind blowing though.
 
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  • #32
Lavabug said:
Does it have solutions for some of the exercises?

I have the second edition and it doesn't contain solutions. Maybe they added some in the third edition but I'm not sure...
 
  • #33
Did this hamper you at all using it as a sole resource for learning the contained subjects? Ie: are the problems more or less straightforward applications of the preceding theory or easy to infer their correctness, or are you left in the dark to toil like with some other books (Goldstein, Landau, etc.)?
 
  • #34
In my opinion the exercises are not that hard. I think there's a good balance between computational exercises and proofs... you souldn't have much trouble figuring out whether your answers are correct or not.
 
  • #35
WannabeNewton said:
This is a joke right? Are you actually trying to help the OP learn or just throw really hard books for no reason at someone looking for an introduction?

Sometimes I think the people replying are just trying to show off and "overkill" instead of actually helping the OP.
 
  • #36
Like in the other thread I recommend to toe dippers with "some" calculus and linear algebra and an interest in physical stuffs Curvature in Mathematics and Physics by Shlomo Sternberg. To the sentimentalist who think three dimensions is more than enough try differential Geometry Of Three Dimensions by C.E. Weatherburn.
 
  • #37
tade said:
Sometimes I think the people replying are just trying to show off and "overkill" instead of actually helping the OP.
In the guy's defense, I didn't feel like I understood differential geometry at all until I borrowed a copy of volume 1 of Spivak's introduction. It opens by defining a manifold as a metric space rather than the more general topological space, but other than that it was fantastic. Plus, I like the way Spivak writes. I don't know about anyone else, but it often reads like he's talking to you rather than attempting to talk around your perceived skill level. I appreciate this.

That being said, Spivak doesn't hold much back either. :-p
 
  • #38
Spivak isn't a problem. Lots of people use Spivak for a first exposition to differential topology. Only Gauss, Riemann, or Weyl would use Lang as an intro to the subject.
 
  • #39
^I don't see any problem looking at Lang (or Kobayashi and Nomizu) early on. It does not cause your face to melt. Lang is one of the few books with infinite dimensional flavor, and as is often the case with Lang, he presents things as they are best understood instead of easiest understood. Still most people would like to also read a more gentle book. Spivak is pretty chatty which others dislike, but I consider it a strength. I dislike the typeset though and if I recall correctly it is unchanged in the third edition.
 
  • #40
The issue I have with Lang is it has no exercises. Otherwise I think it would be a reasonable choice for a dedicated student.
 
  • #41
lurflurf said:
^I don't see any problem looking at Lang (or Kobayashi and Nomizu) early on. It does not cause your face to melt. Lang is one of the few books with infinite dimensional flavor, and as is often the case with Lang, he presents things as they are best understood instead of easiest understood. Still most people would like to also read a more gentle book. Spivak is pretty chatty which others dislike, but I consider it a strength. I dislike the typeset though and if I recall correctly it is unchanged in the third edition.

deluks917 said:
The issue I have with Lang is it has no exercises. Otherwise I think it would be a reasonable choice for a dedicated student.

Man, even my old differential geometry professor said that he looked at Lang and didn't understand much of it because it was so horrible written. And this is a guy who knows differential geometry inside out.
 

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