COASSOCIATIVE CONES RULED BY 2-PLANES

摘要

It is shown that coassociative cones in R7 that are r-oriented and ruled by 2-pianes are equivalent to CR-holomorphic curves in the oriented Grassmanian of 2-planes in R? The geometry of these CR-holomorphic curves is studied and related to holomorphic curves in S6 This leads to an equivalence between associative cones on one side and the coassociative cones whose second fundamental form has an O(2) symmetry on the other. It also provides a number of methods for explicitly constructing coassociative 4-folds. One method leads to a family of coassociative 4-folds whose members are neither cones nor are ruled by 2-planes, This family directly generalizes the original family of examples provided by Harvey and Lawson when they introduced coassociative geometry.