By Michael Spivak
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Extra info for Comprehensive Introduction To Differential Geometry, 2nd Edition, Volume 4
We have previously mentioned the differential of the projection p' at x was such that: dpl 5 dx' . The function x I+ is identified to the image xi of point x. > Problem. Give the expression of the linear form w, in the dual basis (dx') of the basis (ei) which is such that: dxf(ej= ) 6;,. Answer. The image of h under w, is Lecture 0 = o,(x,,)h [ putting the real wi(x, ) = w, (el ) 1. i From dx i ( h )= hi it follows: o, (A) = m, (xo ) dxl(h). A difSentia1 vne-fonn on U is a mapping w which links each point X E U with a linear form w, defined on F,that is: D w : U ( c R n ) + (R")' : x HW , In other words and more explicitly: D * A digerenth1one-form on U is a mapping w : U ( c R n ) 4L(Rn;R ) = (Rn)' : x I+ w, such that o,:Rn+R:hw~,(x)dx'(h).
The reader will see that T(g 0 f ) = Tg 0 Tf. 7 immersioni and submersion D at x is a mapping f : U ( c E) + F of class C? such that f ' ( x ) is injective (one-to-one). A submersion at x is a mapping f : U(c E ) + F of class C? such that f l ( x ) is surjective. 3 DIFFERENTIATION OF An i-mion R" INTO BANACH Let F be a Banach space, U be an open of R n , xo be a point of U, f be a hfferentiable mapping U(cR n ) +F : x I+ f (x) The space R", with its vector structure, is a Banach space with the norm D The differential o f f at x, IS the linear mapping dfxo : R n- , F : h e d J X D ( h ) such that Make explicit this mapping and first remember that f(x) and df,(h) are vectors of Banach F.
Prove that two mappings f, g tangent to a third h, at xo, are tangent at this point. Seeing that lim llf (XI - Mx)l[ = Il+ -xoll and lim llg(x) - 4 x ) ) l = o , o lix - xo 1I then the equality If (XI- ntx)ll < - Ik- xo l Ilf (x) - W X ) ~+ C(X)- g(x]I llx - I Ilx - xoll XO implies the third equivalence property. 2. Differentiable mapping at a point D A mapping f : U ( c E) -+ F : x Hf (x) is differentiable at point xo of U if there is a continuous linear mapping I : u + F : x ~ +e ( ~ ) such that the mapping U ( C E ) + F : X H f(x,) + P ( x - x , ) is tangent to f a t xo.
Comprehensive Introduction To Differential Geometry, 2nd Edition, Volume 4 by Michael Spivak