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By Bak A. (ed.)

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Then ~,~ are ambient isometric (by a isometry of CP n) if and only if their directrix curves are equivalent. 26 Proof Suppose the directrix curves are projectively equivalent, ~o - g ~o" say It follows from Theorem 4 and the lemma that g ~0(z) - (adj g)t ~n(-ipf) - (g-1)t ~0(z), so that gig ~0 - ~0" Since ~0 is linearly full g ~ PU(n+I). The converse is trivial. Remark The condition of factoring through RP 2 is crucial here. result is not true in general. [l]). This, together with Theorem 2, is a first step in dealing with the space of minimal immersions of S ~ into Sn.

16) with C a constant. 9) to obtain (n + 2 ) a A a = --(n + 2)c~a" + 3(n -- 1)(a') 2. 1) we get a A a = (A -I~12)a 2. ~(n + 8 ) ~ ) 4--~----~ ] ~ =0. 19) one has 2 ( 4 - n) n2(n +5) 4 T- ~ (a'? - 2-~ = iS ~ - ~ 2 . 20) that a is locally constant on N, which is a contradiction with the definition of U. Hence o~ is constant on M". 1) into 53 consideration, we have (A~)~ = (A -- 1~I2)H, so that either c~ = 0 and M " is minimal or I~12 A and therefore I~12 is constant. But we had at most two different eigenvalues, then because a and lal 2 are constant, such eigenvalues are also constant.

X2k+2 ) = 0. By induction, the first part of the lemma is proved. The proof of the second part is similar, starting from the fact that v2kh = 0 implies that also Y2k+lh = 0. The proof of the following lemma is completely similar to the proof of Lemma 2. Lemma 3 Let M be an equiaffine surface in 0{3 with second fundamental form h. Then v2k+lh = 0 implies that R k + l . vh = 0. The following lemma follows immediately from the skew symmetry of the affine curvature tensor R in its first two components.

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Algebraic K-Theory, Number Theory, Geometry and Analysis: Proceedings of July 26-30, 1982 by Bak A. (ed.)

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