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Example text

11. A p a r t i t i o n of u n i t y (of c l a s s cp) on a cp-manifold M c o n s i s t s of an open covering { u . } of M and a system of functions $ . 1 3 : M + R satisfying the conditions: (i) For aLZ x E M one has: I$. 1 (x) t 0 . (ii) The support of $ , i s contained i n U.. } of (iii) The covering 1 M i s finite. (XI = 1. 1 {vk } i s another covering of t o { u , } i f each v M, then i s contained i n some it i s said t o be subordinated u.. 1 k 1 Each compact differentiable manifold admits a differentiable partition of unity.

3. m i s a C -manifold, whose a t l a s c o n t a i n s j u s t one lo. Euclidean space Wn chart. 2O. Sphere n-1 S C n IR ,n Sn- 1 > 2, = {x E nn! 1x1 = 11 i s again a m c - manifold, whose a t l a s c o n t a i n s a t l e a s t two c h a r t s ( U ,,$ , ) I , j = 1,2 t h e -1 1 1 maps $ . , j = 1 , 2 , being t h e s t e r e o g r a p h i c p r o j e c t i o n s onto two copies of n-1 B . m Further c i r c l e S1 i s a C -manifold,whose a t l a s contains one c h a r t given by t h e angle coordinate 8 = 0-l : n1 -f S 1 .

We are going t o show t h a t with some c o n s t a n t C > 0 one has: I lqll SUP 2 C ([O,a]) O

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