By Masahisa Adachi

This e-book covers basic recommendations within the conception of $C^{\infty }$-imbeddings and $C^{\infty }$-immersions, emphasizing transparent intuitive figuring out and containing many figures and diagrams. Adachi begins with an advent to the paintings of Whitney and of Haefliger on $C^{\infty }$-imbeddings and $C^{\infty }$-manifolds. The Smale-Hirsch theorem is gifted as a generalization of the category of $C^{\infty }$-imbeddings by means of isotopy and is prolonged by means of Gromov's paintings at the topic, together with Gromov's convex integration concept. eventually, as an program of Gromov's paintings, the writer introduces Haefliger's class theorem of foliations on open manifolds. additionally defined here's the Adachi's paintings with Landweber at the integrability of just about advanced constructions on open manifolds. This e-book will be a good textual content for upper-division undergraduate or graduate classes.

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**Example text**

A/"). yetf Using the atlases of I'" and M''. (l'". A/'')\subsetj'(V n. A/'') ,— J r(n. y) e ν" χ A/'' 36 I. C' MANIFOLDS, C' MAPS, AND FlttUR BUNDLES This is called a jet bundle. 1). Set Y = J r(n , p) and G = L r(n,p)\ G acts on Y. Set X = V"xM p . Take a C' atlas ^ = {((/„, φ η) \ a e A ) of V" and an atlas S* = { (W k , ψ λ) \λ e A } of M" . Then with" X t λ = U it χ Η\ , the family {X a e Α, λ e A] is an open cover of X . λ ηΧρ. ,Α)Αβ. β = ^(Λ,^,. ). ;, = ° ψ„ >• Then the { X a λ, g ( ( t ^ j |α, β 6 Λ , Α, μ e Λ } is a system of coordinate transformations in V n χ M p with values in G .

V e V" and ρ € Λ/'1. ν. (('". v) = r }. '", Λ/'') are r-cquiralcni αι v. v in some local coordinate system agree up to order r. The relation ~ is well defined and is an equivalence relation. ·". ('•". M")/ - • We write J[(f) for the cquivalcncc class containing f and we say that J[( f) is the r-jci of f at χ . Set c r '1 / ( ! (! ". A/"). yetf Using the atlases of I'" and M''. (l'". A/'')\subsetj'(V n. A/'') ,— J r(n. y) e ν" χ A/'' 36 I. C' MANIFOLDS, C' MAPS, AND FlttUR BUNDLES This is called a jet bundle.

Then the { X a λ, g ( ( t ^ j |α, β 6 Λ , Α, μ e Λ } is a system of coordinate transformations in V n χ M p with values in G . 1, afiber bundle, which turns out to be the above jet bundle. The total space J r(V" , M p) may be regarded as a C s~' manifold when r < oo. Let / : V" —» M p be a C s map. f) the r-extension of /. The r-extcnsion J r(f) following diagram commute: J'(V" •/ , M") <— J'(n. j is a C'~ r map making the p) V" —ί V" χ M". In the following we take r = 1 . The submanifold S k(n , //)\subset· / ' ( / ' .