Submanifolds in Hyper - K . ~ Hler Geometry Robert Bryant
نویسنده
چکیده
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
منابع مشابه
Bryant and Reese
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 fo...
متن کاملar X iv : m at h - ph / 0 60 50 74 v 2 2 J un 2 00 6 THE BRYANT - SALAMON G 2 - MANIFOLDS AND HYPERSURFACE GEOMETRY
We show that two of the Bryant-Salamon G2-manifolds have a simple topology, S \S or S \CP . In this connection, we show there exists a complete Ricci-flat (non-flat) metric on Sn \ Sm for some n − 1 > m. We also give many examples of special Lagrangian submanifolds of T ∗Sn with the Stenzel metric. Hypersurface geometry is essential for these arguments.
متن کاملCalibrated Embeddings in the Special Lagrangian and Coassociative Cases
Every closed, oriented, real analytic Riemannian 3-manifold can be isometrically embedded as a special Lagrangian submanifold of a Calabi-Yau 3-fold, even as the real locus of an antiholomorphic, isometric involution. Every closed, oriented, real analytic Riemannian 4-manifold whose bundle of self-dual 2-forms is trivial can be isometrically embedded as a coassociative submanifold in a G2-manif...
متن کاملar X iv : m at h - ph / 0 60 50 74 v 1 2 9 M ay 2 00 6 BRYANT - SALAMON ’ S G 2 - MANIFOLDS AND THE HYPERSURFACE GEOMETRY
We show that two of Bryant-Salamon’s G2-manifolds have a simple topology, S \ S or S \ CP . In this connection, we show there exists a complete Ricci-flat (non-flat) metric on Sn \ Sm for some n − 1 > m. We also give many examples of special Lagrangian submanifolds of T ∗Sn with the Stenzel metric. The hypersurface geometry is essential in the argument.
متن کاملA Pseudo-group Isomorphism between Control Systems and Certain Generalized Finsler Structures
The equivalence problem for control systems under non-linear feedback is recast as a problem involving the determination of the invariants of submanifolds in the tangent bundle of state space under fiber preserving transformations. This leads to a fiber geometry described by the invariants of submanifolds under the general linear group, which is the classical subject of centro-affine geometry. ...
متن کامل