LOCAL HERMITIAN GEOMETRY OF COMPLEX SURFACES IN ＄P^{3}＄ : TOTALLY PARABOLIC SURFACES
摘要
A study of Hermitian geometry of holomorphically immersed complex surfaces in P^{3} is given. A normalization of the complex second fundamental form is given and is used to extract the global invariants, complex principal curvatures. We give formulae expressing the Chern forms in terms of the Kaehler form and the complex principal curvatures. For compact surfaces in P^{3} we obtain a pair of Gauss-Bonnet type formulae. As an application of our Gauss-Bonnet formulae we prove that an algebraic surface in P^{3} with an immersive osculating map is a quadric. The notion of a parabolic surface in P^{3} is introduced. Compact parabolic surfaces do not exist since they would have to satisfy the critical Chern number equality, (c_{1})^{2}=3c_{2} . An infinite family of local parabolic surfaces are constructed using the method of prolongation. In the process we also give an infinite family of non-left-invariant integrable distributions on U(4) .