Submanifolds in Hyper - K . ~ Hler Geometry Robert Bryant

A calibration ¢ is a differential form on a Riemannian manifold with two additional properties. First, the form should be closed under exterior differentiation. Second, it should be less than or equal to the volume form on each oriented submanifold (of the same dimension as the degree of the form ¢). Each calibration ¢ determines a geometry of submanifolds, namely those oriented submanifolds for which ¢ restricts to be exactly the volume form. Such submanifolds are called ¢-submanifolds. The Fundamental Lemma of the theory of calibrations .says that each ¢-submanifold is homologically area minimizing. A Kahler form provides the most important classical example of a calibration. In this case the ¢-submanifolds are just the complex submanifolds of dimension one. One of the most interesting nonclassical examples of a calibration, introduced in Harvey and Lawson [HL], is a 4-form on euclidean R8 called the Cayley 4-form. This 4-form has an elegant description in terms of the algebra of octonians 0, and is fixed by the subgroup Spin(7) of the group of all orthogonal transformations on O. As such, it would appear unlikely that would have higher dimensional generalizations. The purpose of this paper is to provide the higher dimensional analogue. The Cayley form on R8 == 0 can also be considered in a very natural way as a 4-form on H2, the quatemionic plane. After choosing to distinguish the scalar quatemion K, the 4-form can be expressed as