By Hassler Whitney
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Within the sequence of volumes which jointly will represent the guide of Differential Geometry a slightly entire survey of the sector of differential geometry is given. different chapters will either take care of the elemental fabric of differential geometry and with study effects (old and recent). All chapters are written by way of specialists within the quarter and include a wide bibliography.
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Extra info for Complex analytic varieties
Without loss of generality we may assume that E 0 = (*<> £»), A0 is a (k , A:)-matrix and |. l0| # 0. Then there exists an ε > 0 such that \A\ / 0 if the absolute value of each entry of A - A0 is less than ε . Now let U C M(p , η ; R) consist of all [ρ , /z)-matrices of the form Ε = (a o) • where A is a (k , /c)-matrix such that the absolute value of each entry of A - A0 is less than ε . Then we have Ε e M(p , n \ k) if and only if D = CA~ lB. because the rank of (h 0 \(A B\ =( \ A - I p_ k)\C D) Α Β \ νΛ'Λ + C xb + D) .
N) . , of R" is both the right and left unit of J'(n , n) with respect to this product. Denote by L'(n) a subset of J r(n,n) consisting of all invcrtiblc elements. 4. ( i ) l' (n) = GL(T/ , R ) . (ii) Z/(//) = (7Tr t)-\L\n)). (iii) L'(n) is a Lie group for each r. I < /· < x . (iv) L r(n) has the homotopv ivpe of GL(ZJ. R) (cf. Thom and Lcvinc [Bll]). B. Singular sets. /'(//, p) —* j'(n , ρ). ,, , ('K),/ /-(<•), , ,(0 . 41. A jet f >r ) e J'(n. p) is regular if its representative J in C'(n, p) has the maximal rank at 0.
The main problem here is the existence of embeddings of Γ in Μ and their classification up to isotopy for a given pair of manifolds Γ and Μ . Throughout this chapter we shall work in the ( " v category. §1. Embeddings and isotopies We first define isotopies of embeddings and then state the theorems of Whitney and Haefiiger. Let V be an /(-dimensional c x manifold and lei Μ be an /(/-dimensional manifold. 2. Let / and g be embeddings. We say that /' and g are isotopic and write f ~ g if there exists a C map /·' : I ' χ / — · M .
Complex analytic varieties by Hassler Whitney