Jentsch, Tillmann (2006) Submanifolds with parallel second fundamental form studied via the Gauß map. PhD thesis, Universität zu Köln.
For an arbitrary n-dimensional riemannian manifold N and an integer m between 1 and n-1 a covariant derivative on the Graßmann bundle G_m(TN) is introduced which has the property that an m-dimensional submanifold M of N has parallel second fundamental form if and only if its Gauß map (defined on M with values in G_m(TN)) is affine. (For the case that N is the euclidian space this result was already obtained by J.Vilms in 1972.) By means of this relation a generalization of E. Cartan's theorem on the total geodesy of a geodesic umbrella can be derived: Suppose, initial data (p,W,b) prescribing an m-dimensional tangent space W and a second fundamental form b at p in N are given; for these data we construct an m-dimensional ``umbrella'' M=M(p,W,b) in N, the rays of which are helical arcs of N; moreover we present tensorial conditions (not involving the covariant derivative on G_m(TN)) which guarantee that the umbrella M has parallel second fundamental form. These conditions are as well necessary, and locally every submanifold with parallel second fundamental form can be obtained in this way.
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