Osaka Journal of Mathematics

Parallel submanifolds of the real 2-Grassmannian

Tillmann Jentsch

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Abstract

A submanifold of a Riemannian symmetric space is called parallel if its second fundamental form is parallel. We classify parallel submanifolds of the Grassmannian $\mathrm{G}^{+}_{2}(\mathbb{R}^{n+2})$ which parameterizes the oriented 2-planes of the Euclidean space $\mathbb{R}^{n+2}$. Our main result states that every complete parallel submanifold of $\mathrm{G}^{+}_{2}(\mathbb{R}^{n+2})$, which is not a curve, is contained in some totally geodesic submanifold as a symmetric submanifold. The analogous result holds if the ambient space is the Riemannian product of two Euclidean spheres of equal curvature or the non-compact dual of one of the previously considered spaces. We also give a characterization of parallel submanifolds with curvature isotropic tangent spaces of maximal possible dimension in any symmetric space of compact or non-compact type.

Article information

Source
Osaka J. Math., Volume 51, Number 2 (2014), 285-337.

Dates
First available in Project Euclid: 8 April 2014

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1396966251

Mathematical Reviews number (MathSciNet)
MR3192544

Zentralblatt MATH identifier
1314.53092

Subjects
Primary: 53C35: Symmetric spaces [See also 32M15, 57T15] 53C40: Global submanifolds [See also 53B25] 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]

Citation

Jentsch, Tillmann. Parallel submanifolds of the real 2-Grassmannian. Osaka J. Math. 51 (2014), no. 2, 285--337. https://projecteuclid.org/euclid.ojm/1396966251


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