By G. Hardy

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4 As is shown in [12], a compact surface carries a conformal structure if and only if it is one of the following: (i) P 1 xll? 1 , (ii) a ruled surface over a curve C of genus the covering transformations of Cx P1 ~1 such that respect the ~ natural quadric structure on C x P 1 , (iii) a torus, a bielliptic surface or an elliptic surface with even b 1 and c 21 = o= c 2 , (iv) a surface covered by the bidisc, (v) an Inoue surface s 0 (see [7]), or (vi) a Hopf surface a 2 *;r, where r contains only trans- formations of the form (z 1 ,z 2 ) ~ (az 1 ,az 2 ) or (z 1 ,z 2 ) ~ (az 2 , az 1 ).

E. I I { z E a:n : :L z v 2 < 1 + D I:L ( z v ) 2 I 2 < 2 } • The natural embedding of D into Qn is given by i z~(1 1 z : •• :z n : 1 :r(z) v 2 ). 2 Aut D consists of all transformati ons induced by G leaving D invariant. 4 A quadric structure of Misgiven by an atlas (IVa: Ua-+ Qn)aEJ such that for all a,13EJ the transition map 1Va~V~ 1 : IVa (Ua n u 13 ) ~ IV~3 (Ua n u 13 ) is the restriction of some element of G. 5 If M carries a quadric structure, then so does every unramified covering of M.

J\P (::~ + 52 a ) = w ( P 1 P) +w ( p+ 1 1 p- 1 ) + . e. locally c o, c 1 = o or c 1 < o.

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A Course Of Pure Mathematics by G. Hardy


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