Twisted-calibrations and the cone on the Veronese surface.

摘要

This thesis proposes an extension of the methods of calibrated geometries to include non-orientable submanifolds. This is done by "orienting" a non-orientable submanifold N of a Riemannian manifold M with a real Euclidean line bundle.;Real Euclidean line bundles over a smooth manifold are shown to be in one-to-one correspondence with two-sheeted covering spaces. The equivalence classes of Euclidean line bundles are naturally described by a certain cohomology group. Moreover, a given Euclidean line bundle L over a smooth manifold M defines a natural class of submanifolds, called the L-orientable submanifolds. Such a submanifold N is defined by the condition that its orientation bundle be isomorphic to the bundle obtained by restriction of L to N.;For smooth manifolds, the differential forms with values in a Euclidean line bundle are interpreted as ordinary differential forms on the associated two-sheeted cover satisfying an additional "twisting" condition. An analogue of Stokes's theorem for densities is shown to hold for the L-oriented submanifolds described above.;For Riemannian manifolds, we apply the conventional theory of calibrations to the twisted forms on the (Riemannian) double cover. The L-oriented submanifolds which are "twisted-calibrated" satisfy the mass minimizing property (among L-orientable submanifolds) associated to calibrated submanifolds. One consequence of this fact is that a twisted-calibrated submanifold is stable.;Finally, by using the action of SO(3) on the traceless three-by-three symmetric matrices, it is proved that the cone of the Veronese surface is twisted-calibrated and hence stable. In fact, the twisted-calibration is of a special form which shows that the cone minimizes area among a fairly general class of 3-folds.