Frame bundle approach to generalized minimal submanifolds

Abstract

We extend the notion of $r$-minimality of a submanifold in arbitrary codimension to $u$-minimality for a multi-index $u\in {\mathbb{N}}^q$, where $q$ is the codimension. This approach is based on the analysis on the frame bundle of orthonormal frames of the normal bundle to a submanifold and vector bundles associated with this bundle. The notion of $u$-minimality comes from the variation of ${\sigma }_u$-symmetric function obtained from the family of shape operators corresponding to all possible bases of the normal bundle. We obtain the variation field, which gives alternative definition of $u$--minimality. Finally, we give some examples of $u$-minimal submanifolds for some choices of $u$.

20 pages; major revision; all mistakes corrected

References in corpus (1)

• Extrinsic curvatures of distributions of arbitrary codimension