By Tracy Kompelien

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**Extra info for 3-D Shapes Are Like Green Grapes!**

**Sample text**

F)}. F is the normal bundle T(L)1 L of t : L '- (M, go). F) ® T(,F)1. 12. F)H(Af)] 1 and o = ao. F) is precisely Ho. e. F). e. Y E H(M). F)1 C [T(,F)H(Af)]1. The opposite inclusion may be proved in a similar manner. F)H(Af)]1) = (Y - 0(Y)T)1 = ao(HIY), for anyYET(M). 2. F) is degenerate in (T(M), G9). However, the pullback of F to the (total space of the) principal S'-bundle C(M) := [K(M) \ {zero section}]/IR+ turns out to be nondegenerate in (C(M), Fe), where Fe is the Fefferman metric of (M, 0).

F). 47 (or the definition in [179], p. 26. e. £xh = 0 for any X E T(F). 25. 16) is direct. F. F) 9 T(C(M)) = Ker(a) ® Ker(d7r) 2. FOLIATED CR MANIFOLDS 30 then V = XT + fS for some X E T(M) and f E Coo(C(M)), where XT := 0 X. e. F)T + Ker(drr). 16) is proved. F). e. F). F). Then WH = 0. Since n _ 1 n+2{dy+rr`rlo } (for some 1-form rf on M, determined in terms of 9) and (dy)S = 1, it follows that rl(Wv) = 1/(n + 2). e. 19) G9((dir)VH, (dir)WH) + 9((d7r)WH)tl(Vv) = 0. Let us set W = V. e. (dir)VH = 0 hence Vii E Ker(drr) f1 Ker(q) = (0).

T is called the pseudohermitian mean curvature vector of f in (M, 9). 0 Let gp be the bundle metric induced by go on P. 32) +ge(Tv(X, X'), Z) + 9e(Tv(Z, X), X') + 9e(Tv(Z, X'), X), for any X, X' E rx (P) and Z E r°° (PI). 33) +ge(TV(X, Z), Z') + 9B(Tv(X, Z'), Z), for any X E 171 (P) and z, Z'r°°(P1). To illustrate our ideas we give an application of these notions to foliations all of whose leaves are tangent to the characteristic direction T of (M, 9). e. T E r°°(P), let P(O) be the orthogonal complement with respect to go of RT in P.

### 3-D Shapes Are Like Green Grapes! by Tracy Kompelien

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