By Arutyunov A.V., Jacimovic V.
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Extra info for 2-Normal Processes in Controlled Dynamical Systems
Show that there exists an angle function Â on an open subset U Â S1 if and only if U ¤ S1 . U; Â / is a smooth coordinate chart for S1 with its standard smooth structure. (Used on pp. ) 1-9. Complex projective n-space, denoted by CP n , is the set of all 1-dimensional complex-linear subspaces of C nC1 , with the quotient topology inherited from the natural projection W C nC1 X f0g ! CP n . Show that CP n is a compact 2n-dimensional topological manifold, and show how to give it a smooth structure analogous to the one we constructed for RP n .
Iv) Countably many of the sets U˛ cover M . (v) Whenever p; q are distinct points in M; either there exists some U˛ containing both p and q or there exist disjoint sets U˛ ; Uˇ with p 2 U˛ and q 2 Uˇ . U˛ ; '˛ / is a smooth chart. 22 1 Smooth Manifolds Fig. 9 The smooth manifold chart lemma Proof. V /, with V an open subset of Rn , as a basis. W /, there is a third basis set containing p and contained in the intersection. W / is itself a basis set (Fig. 9). U˛ \ Uˇ /, and (ii) implies that this set is also open in Rn .
4 that the n-sphere Sn Â RnC1 is a topological n-manifold. We put a smooth structure on Sn as follows. 4. For any distinct indices i and j , the transition map 'i˙ ı 'j˙ 1 is easily computed. In the case i < j , we get Á p 1 'i˙ ı 'j˙ u1 ; : : : ; un D u1 ; : : : ; ubi ; : : : ; ˙ 1 juj2 ; : : : ; un (with the square root in the j th position), and a similar formula holds when i > j . When i D j , an even simpler computation gives 'iC ı 'i 1 D 'i ı 'iC 1 D « ˚ IdBn . Thus, the collection of charts Ui˙ ; 'i˙ is a smooth atlas, and so defines a smooth structure on Sn .
2-Normal Processes in Controlled Dynamical Systems by Arutyunov A.V., Jacimovic V.