CHARACTERIZING GAUSS MAPS

[摘要] The definition of the generalized Gauss map is as follows: given an immersed manifold f : M(;;n) (--->) (//R)(;;n+k), its Gauss map is the map (gamma)(,f) : M (--->) G(,n,n+k) defined by (gamma)(p) = f(,*)(T(,p)M). As defined the Gauss map is not only invariant under all Euclidean motions but under all affine motions as well. In this thesis we study the problem of determining whether a smooth map (gamma) : M(;;n) (--->) G(,n,n+k) is the Gauss map for some immersed manifold in (//R)(;;n+k). Using the method of moving frames, certain necessary and sufficient conditions are found on (gamma) for it to locally be a Gauss map. These conditions specifically involve a tensor induced from the differential of (gamma) called the pseudo affine first fundamental form of (gamma), which is analagous to the affine first fundamental form of an immersed manifold. These conditions are then applied in the study of particular cases. In the case of curves and hypersurfaces in (//R)(;;N), and surfaces in (//R)(;;4), the generic Gauss-like map is shown to locally be a Gauss map. For the case of surfaces in (//R)(;;5), the generic Gauss-like maps are found to satisfy integrability conditions if and only if they are locally Gauss maps. These conditions depend on first and second order information of (gamma). Finally, a certain class of generic Gauss-like maps of codimension two manifolds are studied. For this class of maps, sufficient conditions are found for such analytic maps to be local Gauss maps. These conditions depend on first order information of (gamma).

[发布机构] Rice University