2004/08/16 by Salavessa, Isabel M. C., Vale, Ana Pereira do
#53C25 #53C38 #53C55 #57R45 #Differential Geometry (math.DG) #FOS: Mathematics #Primary: 53C42 #Secondary: 57R20
paper · doi:10.48550/arxiv.math/0408206
Our main results are: (1) The complex a Lagrangian points of a non-complex Lagrangian 2n-dimensional submanifold F:M\ra N, immersed with parallel mean curvature and with equal Kaehler angles into a Kaehler-Einstein manifold (N,J,g) of complex dimension 2n, are zeros of finite order of sin2θ and cos2θ respectively, where θ is the common J-Kaelher angle. (2) If M is a Cayley submanifold of a Calabi-Yau (CY) manifold N of complex dimension 4, then \bigwedge2+NM is naturally isomorphic to \bigwedge2+TM. (3) If N is Ricci-flat (not necessarily CY) and M is a Cayley submanifold, then p1(\bigwedge2+NM)= p1(\bigwedge2+TM) still holds, but p1(\bigwedge2-NM)- p1(\bigwedge2-TM) may describe a residue on the J-complex points, in the sense of Harvey and Lawson. We describe this residue by a PDE on a natural morphism Φ:TM → NM, Φ(X)=(JX)\bot, with singularities at the complex points. We give an explicit formula of this residue in a particular case. When (N,I,J,K,g) is an hyper-Kaehler manifold and M is an I-complex closed 4-submanifold, the first Weyl curvature invariant of M may be described as a residue on the J-Kaehler angle at the JLagrangian points by a Lelong-Poincaré type formula. We study the almost complex structure \Jw on M induced by F.