By David Ervin Blair (auth.)
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Extra info for Contact Manifolds in Riemannian Geometry
B. we M 2n+l prove c M 2n+2 a result hypersurface contact of of Tashiro an a l m o s t that dimensional U(n) an [ 75] t h a t complex which In every manifold over a manifold the C has Now complex orientable G(X,Y) M 2n+2 such a metric = G(JX,JY) structure be structure hypersurface. of the to a f i e l d spaces admits Hermitian let structure is a r e d u c t i o n tangent metric, satisfying is an a l m o s t complex is e q u i v a l e n t of such almost manifold morphisms almost complex, example orientable an almost structure.
E. s t r u c t u r e of a c o m p l e x [h,h] of a t e n s o r field of type (1,2) field with is the structure. h of type given by [h,h] (X,Y) = h 2 [ X , Y ] + [hX,hY] - h [ h X , Y ] - h[X,hY]. All m a n i f o l d s u n d e r our c o n s i d e r a t i o n the N e w l a n d e r - N i r e n b e r g studies . are of class t h e o r e m applies. of c o m p l e x m a n i f o l d s see, C~ , so For d e t a i l e d for e x a m p l e , or 48 Let contact M 2n+l M 2n+l be an almost structure (~0, ~,~) • R .
R , ~ ( X i) = O and aN (Xi,Xj) Thus, since 1 = ~ ( X i D (Xj) - X j N (X i) - ~ (Ix i,xj ]) ) = O. r > n, (~A(d~) n) ( X l , . . , X 2 n + l ) = O, a contra- diction. We have tangent just t o an = ~(Y) = 0 also sufficient of fold. only ~ (~,~,~,g) Mr and be then Mr every tangent to Mr X,Y submanifold d~(X,Y) which dD integral restricted an a s s o c i a t e d vector M 2n+l . X of D, fields then These to b e conditions an are integral sub- as a p r o p o s i t i o n . ~: M r ~ M 2n+l is an are vector = O.