Twisted-calibrations and the cone on the Veronese surface
Murdoch, Timothy Armstrong
Bryant, Robert L.
Doctor of Philosophy
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.