Cayley submanifolds of Calabi-Yau 4-folds

Details Cayley submanifolds of Calabi-Yau 4-folds Cayley submanifolds of Calabi-Yau 4-folds

2004-08-16

arxiv.org/abs/math/0408206v2

Mathematics

Differential Geometry

v1: Plain LaTeX, 60 pages. v2, 48 pages: This is a quite modified version of the first one. The residue formula in the first v

Scientific paper

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 $\sin^2\theta$ and $\cos^2\theta$ respectively, where $\theta$ 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 $\bigwedge^2_+NM$ is naturally isomorphic to $\bigwedge^2_+TM$. (3) If $N$ is Ricci-flat (not necessarily CY) and $M$ is a Cayley submanifold, then $ p_1(\bigwedge^2_+NM)= p_1(\bigwedge^2_+TM)$ still holds, but $p_1(\bigwedge^2_-NM)- p_1(\bigwedge^2_-TM)$ 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 $\Phi:TM \to NM$, $\Phi(X)=(JX)^{\bot}$, 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 $J$_Lagrangian points by a Lelong-Poincar\'{e} type formula. We study the almost complex structure $\Jw$ on $M$ induced by $F$.

Affiliated with

Also associated with

No associations

Rating

Cayley submanifolds of Calabi-Yau 4-folds does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Cayley submanifolds of Calabi-Yau 4-folds, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Cayley submanifolds of Calabi-Yau 4-folds will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-580138