Totally geodesic submanifolds of the complex and the quaternionic 2-Grassmannians

Klein, Sebastian

doi:10.1090/s0002-9947-09-04699-6

Cited by 37 publications

(33 citation statements)

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“…Therefore P is a totally geodesic submanifold of Q m * . By applying the classification theorem for totally geodesic submanifolds in the complex quadric Q m [5], [6] to its non-compact dual Q m * , we know that P is locally congruent to the complex (m − 1)-dimensional complex hyperbolic quadric described in Section 3, and M is a tube of radius r around P . Case 2: k = √ 2.…”

Section: Proof Of the Mainmentioning

confidence: 99%

Contact real hypersurfaces in the complex hyperbolic quadric

Klein

Suh

2019

Annali di Matematica

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We give a new proof of the classification of contact real hypersurfaces with constant mean curvature in the complex hyperbolic quadric Q m * = SO o m,2 /SO m SO 2 , where m ≥ 3. We show that a contact real hypersurface M in Q m * for m ≥ 3 is locally congruent to a tube of radius r∈R + around the complex hyperbolic quadric Q m−1 * , or to a tube of radius r ∈ R + around the A-principal m-dimensional real hyperbolic spacefor all vector fields X on M 2m−1 , meaning that (φ, ξ, η) is a almost contact structure, and moreover this structure is adapted to the Riemannian metric g by g(φX, φY ) = g(X, Y ) − η(X) η(Y ) and η(X) = g(X, ξ) for all vector fields X, Y on M 2m−1 .Clearly, if M 2m−1 has an almost contact metric structure (φ, ξ, η, g) so that η∧dη m−1 = 0 holds, then M 2m−1 is a contact manifold. Conversely, if M 2m−1 is a contact manifold, then for any contact form η on M 2m−1 there exists an almost contact metric structure (φ, ξ, η, g) with this η by a result due to Sasaki [12, Theorem 4].Let us now consider a real hypersurface M of a Kähler manifoldM of complex dimension m. Then M has real dimension 2m − 1, and the complex structure J and the Riemannian metric g ofM induce an almost contact metric structure (φ, ξ, η, g) on M: Let N be a unit 2010 Mathematics Subject Classification: Primary 53C40. Secondary 53C55.

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Section: Proof Of the Mainmentioning

confidence: 99%

Contact real hypersurfaces in the complex hyperbolic quadric

Klein

Suh

2019

Annali di Matematica

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“…Here, k ≥ 1. The totally geodesic submanifolds of irreducible Riemannian symmetric spaces M of noncompact type with rk(M ) = 2 were classified by Klein in [5], [6], [7] and [8]. From Wolf's and Klein's classifications we obtain i(M ) for all irreducible Riemannian symmetric spaces M of noncompact type with rk(M ) ≤ 2.…”

Section: Further Applicationsmentioning

confidence: 99%

Maximal totally geodesic submanifolds and index of symmetric spaces

Berndt¹,

Olmos

2016

J. Differential Geom.

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Let M be an irreducible Riemannian symmetric space. The index i(M ) of M is the minimal codimension of a totally geodesic submanifold of M .In [1] we proved that i(M ) is bounded from below by the rank rk(M ) of M , that is, rk(M ) ≤ i(M ). In this paper we classify all irreducible Riemannian symmetric spaces M for which the equality holds, that is, rk(M ) = i(M ). In this context we also obtain an explicit classification of all non-semisimple maximal totally geodesic submanifolds in irreducible Riemannian symmetric spaces of noncompact type and show that they are closely related to irreducible symmetric R-spaces. We also determine the index of some symmetric spaces and classify the irreducible Riemannian symmetric spaces of noncompact type with i(M ) ∈ {4, 5, 6}.

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“…For the principal orbits of the action of S(U (n+1)×U (1)), consider the totally geodesic G 2 (C n+1 ) arising as an exceptional orbit of this action. This is known to be complex with respect to the Kähler structure and quaternionic with respect to the quaternionic-Kähler structure [18]. Hence, a short calculation using the explicit expression for the Riemannian curvature tensor yields that K ξ (T G 2 (C n+1 )) ⊂ G 2 (C n+1 ), whence G 2 (C n+1 ) and the tubes around it are curvature-adapted.…”

Section: Complex Two-plane Grassmanniansmentioning

confidence: 99%

Curvature-adapted submanifolds of symmetric spaces

Murphy¹

2012

Indiana Univ. Math. J.

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Abstract. Curvature-adapted submanifolds have been extensively studied in complex and quaternionic space forms. This paper extends their study to a wider class of ambient spaces. We generalize Cartan's theorem classifying isoparametric hypersurfaces of spheres to any compact symmetric space. Our second objective is to investigate such hypersurfaces in some specific symmetric spaces. We classify those with constant principal curvatures in the Octonionic planes. Various classification results for hypersurfaces in complex two-plane Grassmannians are also obtained.

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Totally geodesic submanifolds of the complex and the quaternionic 2-Grassmannians

Cited by 37 publications

References 3 publications

Contact real hypersurfaces in the complex hyperbolic quadric

Contact real hypersurfaces in the complex hyperbolic quadric

Maximal totally geodesic submanifolds and index of symmetric spaces

Curvature-adapted submanifolds of symmetric spaces

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