文档介绍：Corrections to
Introduction to Smooth Manifolds
Version
by John M. Lee
April 18, 2001
• Page 4, second paragraph after Lemma : Omit redundant “the.”
• Page 11, Example : In the third line above the second equation, change “for each j”
to “for each i.”
• Page 12, Example , line 5: Change “manifold” to “smooth manifold.”
• ◦ −1
Page 13, Example : Just before and in the displayed equation, change ϕj (ϕi ) to
◦ −1
ϕi (ϕj ) (twice).
• Page 21, Problem 1-3: Change the definition of σe to σe(x)=−σ(−x). (This is stereographic
projection from the south pole.)
• Page 24, 5th line below the heading: “multiples” is misspelled.
• Page 24, last paragraph before Exercise : There is a subtle problem with the defin-
ition of smooth maps between manifolds given here, because this definition doesn’t obviously
imply that smooth maps are continuous. Here’s how to fix it. Replace the third sentence of
this paragraph by “We say F is a smooth map if for any p ∈ M, there exist charts (U, ϕ)con-
taining p and (V,ψ) containing F (p) such that F (U) ⊂ V and posite map ψ◦ F ◦ϕ−1
is smooth from ϕ(U)toψ(V ). Note that this definition implies, in particular, that every
smooth map is continuous: If W ⊂ N is any open set, for each p ∈ F −1(W )wecanchoosea
coordinate domain V ⊂ W containing F (p), and then the definition guarantees the existence
of a co