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I've been recommended Fundamentals of Differential Geometry by Serge Lang several times, but cannot find anywhere a description of prerequisites necessary for the book. Can someone summarize what the prerequisites are?

In addition, what level is the book at? Would it be a first-year graduate level text, comparable to texts on Riemannian Geometry by Boothby or do Carmo, for instance?

DC2974
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    Possibly helpful is Alan U. Kennington's amazon.com review. There are no published reviews at JSTOR search. If you have access (I don't), there might be something relevant in the Math. Reviews entry. Freely available is the zbMATH review. See also this MSE answer and the comments to this MSE question. – Dave L. Renfro Jul 24 '24 at 13:15
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    Lang treats (exclusively) Banach manifolds, so you need to start with background in calculus on Banach spaces. – Ted Shifrin Jul 24 '24 at 15:25
  • @TedShifrin What are some references for calculus on Banach spaces? And are there any prerequisites for that? – DC2974 Jul 25 '24 at 03:42
  • If you know finite-dimensional multivariable analysis well (which is 100% a prerequisite for any differential geometry text), then it’s not too bad to adapt to thinking about continuous linear maps on Banach spaces. Standard texts covering this material (including the inverse and implicit function theorems) are Lang’s own Real Analysis (in some version) and Dieudonné’s Treatise on Analysis, vol. 1. – Ted Shifrin Jul 25 '24 at 03:47
  • also, see Loomis and Sternberg’s Advanced Calculus, and Henri Cartan’s Differential Calculus in Banach Spaces (he also has a book about (Banach-valued) differential forms). – peek-a-boo Jul 25 '24 at 06:25
  • @TedShifrin By multivariable analysis do you mean real analysis of several variables? – DC2974 Jul 25 '24 at 07:43
  • What are some references for calculus on Banach spaces? --- The most accessible such book I know of is Introduction to Functional Analysis: Banach Spaces and Differential Calculus by Leopoldo Nachbin (1981), which I give some details about in this MSE answer. – Dave L. Renfro Jul 25 '24 at 08:24

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