Abstract
For an arbitrary n-dimensional Riemannian manifold N and an integer m ∈ {1,…,n−1} a covariant derivative \(\hat{\nabla}\) on the Grassmann bundle ^ := G m (T N) is introduced which has the property that an m-dimensional submanifold M ⊂ N has parallel second fundamental form if and only if its Gauss map M → ^ is affine. (For N Rn this result was already obtained by J. Vilms in 1972.) By means of this relation a generalization of Cartan's theorem on the total geodesy of a geodesic umbrella can be derived: Suppose, initial data (p,W,b) prescribing a tangent space W ∈ G m (T p N) and a second fundamental form b at p ∈ N are given; for these data we construct an m-dimensional ‘umbrella’ M = M(p,W,b) ⊂ N the rays of which are helical arcs of N; moreover, we present tensorial conditions (not involving \(\hat{\nabla}\)) 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.
Mathematics Subject Classifications (2000): 53B25, 53B20, 53B21.
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Jentsch, T., Reckziegel, H. Submanifolds with Parallel Second Fundamental Form Studied via the Gauss Map. Ann Glob Anal Geom 29, 51–93 (2006). https://doi.org/10.1007/s10455-006-1146-7
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DOI: https://doi.org/10.1007/s10455-006-1146-7