Parallel submanifolds of the real 2-Grassmannian

Details Parallel submanifolds of the real 2-Grassmannian Parallel submanifolds of the real 2-Grassmannian

2011-07-28

arxiv.org/abs/1107.5761v3

Mathematics

Differential Geometry

41 pages. Submitted to Osaka Journal of Mathematics. This version contains a new result on the existence of parallel submanifo

Scientific paper

A submanifold of a Riemannian symmetric space is called parallel if its second fundamental form is a parallel section of the appropriate tensor bundle. We classify parallel submanifolds of the Grassmannian $\rmG^+_2(\R^{n+2})$ which parameterizes the oriented 2-planes of the Euclidean space $\R^{n+2}$\,. Our main result states that every complete parallel submanifold of $\rmG^+_2(\R^{n+2})$\,, which is not a curve, is contained in some totally geodesic submanifold as a symmetric submanifold. This result holds also if the ambient space is the non-compact dual of $\rmG^+_2(\R^{n+2})$\,.

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