Download Global Differential Geometry and Global Analysis 1984: by John K. Beem, Paul E. Ehrlich (auth.), Dirk Ferus, Robert B. PDF

st+P(1) 2 = c -i} In other words, the standard sn(c) ~ S n + P ( c ) , is no more rigid. +a n and > p ~ 1 p imbedding (0 ..... O,a 0 ..... an) ) this is trivial, (3) since for one has a non totally geodesic isom.

Be the basis dual to {e l,e 2} and let Then fl I = ~+2, where f is the CNM-valued function given by { - 4| ~i I (el - ie2,e I - ie2) - 4! 4) In the special case in which n = 3 and W is oriented, NM may be canonically identified with M × ¢ and I~Ii becomes a ~-valued 2-form on M. The expression above for f shows that fI I is the form P introduced by Jensen and Rigoli in E2], so Theorem |.! 2. In the case of arbitrary n, the above expression for f shows that the symmetric 4-form A 1 of Jensen and Rigoli is given by A 1(x 1,xz,x3,x 4) = g(fI 1(x1,x 2),fIl(x3,x4)), where g denotes the extension of the Riemannian metric of W to a complexvalued complex bilinear map.

If we put Z = e I - i e 2 , *V I~Ii is the covariant differential and if of fl I , then I~Ii is holomorphic if and only if *V(CIII ) (Z, Z, Z) = 0. g. E4] page 25) that *v(¢ii1)(z,z,~) = *v(Cii~)(z,~,z) = Let VzZ = ~ Z V~(ClII(Z,Z)) - ¢II I(VZ Z,Z) - ¢II I(Z,VZZ). and VzZ = B Z. 2) the equation 2 with respect to Z, and deduce that ~ + B = 0. 2) cancel, so that *V(cIII)(Z,Z,Z) = ~Z H. 3) 53 The proof of the theorem now follows immediately. We will now obtain a local expression for ~I 1 using moving frames.