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CURVATURE AND COMPLEX SINGULARITIES 461
As in the real case we may consider the holomorphic Gauss mapping
y M----) G(n,N)
with y*E T(M). The Hermitian connection and curvature are induced from
those in E G(n, N). There is a commutative diagram
m.
G(n, N)
G(2n, 2N)
where YR is the usual real Gauss mapping, and we deduce from (3.6) and the
discussion at the end of c that the Gauss Bonnet integrand is
y*Pf((P) y*c,(l)) c,(1); i.e.,
(3.10) C e KdM C,(M).
NOW a natural question is whether the Gaussian image y(M) has dimension n,
and using (3.7) we shall prove:
For S C (" + a complex-analytic surface
(3.11) c2(1s) >- 0
with equality holding only if (i) S is a plane, (ii) S is a developable ruled surface;
(4) or (iii) S is a cone.
Proof. We shall adhere to the notations in the proof of (3.7). If c2(1t) 0
and alternative ii) holds, then we may choose a local moving frame {el, e; ea,
., e, + .} so that
From
de1 (.Ollel, and thus 0.112 (121 0,
de2 (.022e2 + (.o23e3 + (.o24e4.
(023 h213(.01 -+- h223(.02 and
we deduce that (.023 h223(.02 Similarly (.024 h224(.02, and it follows that the
Gaussian image T(S) is a curve. The fibres of T:S 7(S) are defined by
o2 0; these are curves along which
dz (.o el, del 0911 el.
We infer that these fibres are straight lines.
If alternative (ii) in (3.7) holds, then we may choose a moving flame so that