foundations differential geometry - michor

[...]... N be a smooth mapping Two vector elds X 2 X(M ) and Y 2 X(N ) are called f -related, if Tf X = Y f holds, i.e if diagram (1) commutes Example If X 2 X(M ) and Y 2 X(N ) and X Y 2 X(M N ) is given (X Y )(x y) = (X (x) Y (y)), then we have: (2) X Y and X are pr1 -related (3) X Y and Y are pr2 -related (4) X and X Y are ins(y)-related if and only if Y (y) = 0, where the mapping ins(y) : M ! M N is given... f -related for i = 1 2, then also 1 X1 + 2 X2 and 1Y1 + 2 Y2 are f -related, and also X1 X2] and Y1 Y2] are f -related Proof The rst assertion is immediate To prove the second we choose h 2 C 1 (N R) Then by assumption we have Tf Xi = Yi f , thus: (Xi(h f ))(x) = Xi(x)(h f ) = (Tx f:Xi (x))(h) = = (Tf Xi )(x)(h) = (Yi f )(x)(h) = Yi (f (x))(h) = (Yi (h))(f (x)) Draft from November 17, 1997 Peter W Michor, ... of local di eomorphisms through IdM with rst non-vanishing derivative m!X = @tmj0 't , and let t be a curve of local di eomorphisms through IdM with rst non-vanishing derivative n!Y = @tn j0 t Draft from November 17, 1997 Peter W Michor, 3.16 3 Vector Fields and Flows, 3.16 27 Then the curve of local di eomorphisms 't t] = t;1 ';1 t 't has rst t non-vanishing derivative (m + n)! X Y ] = @tm+n j0 't... in the compact C 1 -topology, so that by continuity the formula holds for all functions In the following form it implies the claim: k k j0 f ('(t (t x))) = X ;k @ j @ k;j f ('(t (s x)))jt=s=0: @t j t s j =0 Claim 4 Let 't be a curve of local di eomorphisms through IdM with rst non-vanishing derivative k!X = @tk j0't Then the inverse curve of local di eomorphisms ';1 has rst non-vanishing derivative... 1 (E ) which form a frame of E jU Then we P @ have Xi = m fij @uj for fij 2 C 1 (U R) Then f = (fij ) is a (k m)-matrix j =1 valued smooth function on U which has rank k on U So some (k k)-submatrix, say the top one, is invertible at x and thus we may take U so small that this top (k k)-submatrix is invertible; everywhere on U Let g = (gij ) be the inverse of this submatrix, so that f:g = Id We... that if @tjj j0't = 0 for all 1 j < k, then X := k! @tkk j0 't is a well de ned vector eld on M We say that X is the rst non-vanishing derivative at 0 of the curve 't of local di eomorphisms We may paraphrase this as (@tk j0't )f = k!LX f Draft from November 17, 1997 Peter W Michor, 3.16 26 3 Vector Fields and Flows, 3.16 Claim 3 Let 't , t be curves of local di eomorphisms through IdM and let f 2... L(Rn R n ) be given by f (A) := At SA Where is f of constant rank? What is f ;1 (S )? 1.25 Describe TS 2 R 6 Draft from November 17, 1997 Peter W Michor, 1.25 13 2 Submersions and Immersions 2.1 De nition A mapping f : M ! N between manifolds is called a sub- mersion at x 2 M , if the rank of Tx f : Tx M ! Tf (x) N equals dim N Since the rank cannot fall locally (the determinant of a submatrix of the... dimension m Then M can be embedded into R n , if (1) n = 2m + 1 (see Hirsch, 1976, p 55] or Brocker-Janich, 1973, p 73]), (2) n = 2m (see Whitney, 1944]) (3) Conjecture (still unproved): The minimal n is n = 2m ; (m)+1, where (m) is the number of 1's in the dyadic expansion of m Draft from November 17, 1997 Peter W Michor, 1.16 10 1 Di erentiable Manifolds, 1.17 There exists an immersion (see section 2) M... Draft from November 17, 1997 Peter W Michor, 1.19 1 Di erentiable Manifolds, 1.20 0 cos '(R + r cos('=2)) 1 f (r ') := @ sin '(R + r cos('=2)) A r sin('=2) 11 (r ') 2 (;1 1) 02 ) where R is quite big 1.20 Describe an atlas for the real projective plane which consists of three charts (homogeneous coordinates) and compute the chart changings Then describe an atlas for the n-dimensional real projective space... in x 2 Let X 2 XE Then ij X 2 Xloc (Nj ) and is ij -related to X So by lemma 3.14 for j = 1 2 we have ij Flitj X = FltX ij : Now choose xj 2 Nj such that i1 (x1 ) = i2 (x2 ) = x0 2 M and choose vector elds X1 : : : Xn 2 XE such that (X1(x0) : : : Xn(x0)) is a basis of Ex0 Then fj (t1 : : : tn ) := (Flitj1 X1 Draft from November 17, 1997 Peter W Michor, 3.19 FlijnXn )(xj ) t 32 3 Vector Fields and