Vol. 102, No. 1, 1982

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On the image of the generalized Gauss map of a complete minimal surface in R4

Chi Cheng Chen

Vol. 102 (1982), No. 1, 9–14

The generalized Gauss map of an immersed oriented surface M in R4 is the map which associates to each point of M its oriented tangent plane in G2,4, the Grassmannian of oriented planes in R4. The Grassmannian G2,4 is naturally identified with Q2, the complex hyperquadric

            ∑4  2           3
{[z1,z2,z3,z4]|  zk = 0} in  P (C ).

The normalized Fubini-Study metric on P3(C) with holomorphic curvature 2 induces an invariant metric on Q2G2,4, which corresponds exactly to the metric on the canonical representation of S2(1√ -
2) × S2(1√-
2) in R6 as {X R6x12 + x22 + x32 = (12),x42 + x52 + x62 = (12)}. The product representation above allows us to associate with any map g in Q2 two canonical projections g1, g2. In the case where g is complex analytic map defined on some Riemann surface S0, the projections g1, g2 are complex analytic also. Detailed treatment can be found in the recent work of Hoffman and Osserman.

Mathematical Subject Classification 2000
Primary: 53A10
Secondary: 53C42
Received: 13 July 1981
Published: 1 September 1982
Chi Cheng Chen