## Abstract

In this paper we prove a non-existence of real hypersurfaces in complex hyperbolic two-plane Grassmannians
SU_{2.m}/S(U_{2}·U_{m}), m≥3, whose structure tensors {ɸ_{i}}_{i}=1,2,3 commute with the shape operator.

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Show Summary Details# Real Hypersurfaces in complex hyperbolic
two-plane Grassmannians with commuting
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Juan de Dios Pérez

- Departamento de Geometria y Topologia, Universidad de Granada, 18071-Granada, Spain
**Other articles by this author:**- De Gruyter OnlineGoogle Scholar

- Department of Mathematics, Kyungpook National University, Daegu 702-701, Republic Of Korea
**Other articles by this author:**- De Gruyter OnlineGoogle Scholar

- Department of Mathematics, Kyungpook National University, Daegu 702-701, Republic Of Korea
**Other articles by this author:**- De Gruyter OnlineGoogle Scholar

In this paper we prove a non-existence of real hypersurfaces in complex hyperbolic two-plane Grassmannians
SU_{2.m}/S(U_{2}·U_{m}), m≥3, whose structure tensors {ɸ_{i}}_{i}=1,2,3 commute with the shape operator.

Keywords: Real hypersurfaces; Complex hyperbolic two-plane Grassmannians; Commuting shape operator

[1] J. Berndt and Y.J. Suh, Real hypersurfaces in complex two-plane Grassmannians, Monatshefte für Math. 127 (1999), 1–14. Google Scholar

[2] J. Berndt and Y. J. Suh, Hypersurfaces in noncompact complex Grassmannians of rank two, International J. of Math., World Sci. Publ., 23 (2012), 1250103(35 pages). Google Scholar

[3] P.B. Eberlein, Geometry of nonpositively curved manifolds, University of Chicago Press, Chicago, London 7, 1996. Google Scholar

[4] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Graduate Studies in Math., Amer. Math. Soc. 34, 2001. Google Scholar

[5] S. Helgason, Geometric analysis on symmetric spaces, The 2nd Edition, Math. Survey and Monographs, Amer. Math. Soc. 39, 2008. Google Scholar

[6] Y.J. Suh, Real hypersurfaces in complex two-plane Grassmannians with parallel shape operator, Bull. Austral. Math. Soc. 67 (2003), 493–502. Web of ScienceGoogle Scholar

[7] Y.J. Suh, Hypersurfaces with isometric Reeb flow in complex hyperbolic two-plane Grassmannians, Adv. Appl. Math. 50 (2013), 645—659. Web of ScienceGoogle Scholar

[8] Y.J. Suh, Real hypersurfaces in complex hyperbolic two-plane Grassmannians with Reeb vector field, Adv. Appl. Math. 55 (2014), 131–145. Web of ScienceGoogle Scholar

[9] Y.J. Suh and C. Woo, Real Hypersurfaces in complex hyperbolic two-plane Grassmannians with parallel Ricci tensor, Math. Nachr. 287 (2014), 1524–1529. Web of ScienceGoogle Scholar

**Received**: 2015-05-13

**Accepted**: 2015-07-15

**Published Online**: 2015-08-17

**Citation Information: **Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2015-0046.

©2015 Juan de Dios Pérez et al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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- On the use of semi-closed sets and functions in convex analysis
- Carathéodory solutions of Sturm-Liouville dynamic equation with a measure of noncompactness in Banach spaces
- Absolutely strongly star-Hurewicz spaces
- On the strong metric dimension of the strong products of graphs
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- Pointwise density topology
- Invariance groups of finite functions and orbit equivalence of permutation groups
- Set-valued and fuzzy stochastic integral equations driven by semimartingales under Osgood condition
- Dynamics of differentiation operators on generalized weighted Bergman spaces
- Relative annihilator-preserving congruence relations and relative annihilator-preserving homomorphisms in bounded distributive semilattices
- Vietoris topology on spaces dominated by second countable ones
- Identities arising from higher-order Daehee polynomial bases
- A function space from a compact metrizable space to a dendrite with the hypo-graph topology
- Solutions of minus partial ordering equations over von Neumann regular rings
- New interval oscillation criteria for second-order functional differential equations with nonlinear damping
- Notes on monotonically metacompact generalized ordered spaces
- The classification of modular Lie superalgebras of type M
- On the smoothness of the free boundary in a nonlocal one-dimensional parabolic free boundary value problem
- Two bounds on the noncommuting graph
- A new kind of the solution of degenerate parabolic equation with unbounded convection term
- A signature-based algorithm for computing Gröbner-Shirshov bases in skew solvable polynomial rings
- Properties of k-beta function with several variables
- Commuting Conditions of the k-th Cho operator with the structure Jacobi operator of real hypersurfaces in complex space forms
- Infinite dimension of solutions of the Dirichlet problem
- An extended Prony’s interpolation scheme on an equispaced grid
- Positivity and contractivity in the dynamics of clusters’ splitting with derivative of fractional order
- On Poincaré duality for pairs (G,W)
- Some identities of degenerate special polynomials
- Chaotic and hypercyclic properties of the quasi-linear Lasota equation
- Semilinear problems for the fractional laplacian with a singular nonlinearity
- Existence, decay and blow up of solutions for the extensible beam equation with nonlinear damping and source terms
- On some ideal related to the ideal (v0)
- The Umbral operator and the integration involving generalized Bessel-type functions
- Mild solution of fractional order differential equations with not instantaneous impulses
- On graded P-compactly packed modules
- The classification of partially symmetric 3-braid links
- Linear and nonlinear abstract differential equations of high order
- Real Hypersurfaces in complex hyperbolic two-plane Grassmannians with commuting shape operator
- Exact solutions of supersymmetric Burgers equation with Bosonization procedure
- Only 3-generalized metric spaces have a compatible symmetric topology
- Restricted and quasi-toral restricted Lie-Rinehart algebras
- A class of zero divisor rings in which every graph is precisely the union of a complete graph and a complete bipartite graph
- Torsional rigidity on compact Riemannian manifolds with lower Ricci curvature bounds
- Laws of large numbers for ratios of uniform random variables
- The structure of finite groups and ɵ-pairs of general subgroups
- New criteria for H-tensors and an application
- Hom-structures on semi-simple Lie algebras
- Inequality for power series with nonnegative coefficients and applications
- Generalized co-annihilator of BL-algebras
- Linear relations between modular forms for Г0+(p)
- Weak amenability for the second dual of Banach modules
- On some dicardinal functions in ditopological texture spaces
- Boundedness of Toeplitz type operators associated to Riesz potential operator with general kernel on Orlicz space
- Inequalities of harmonic univalent functions with connections of hypergeometric functions
- Module Connes amenability of hypergroup measure algebras
- Three open problems and a conjecture
- Extremal properties of the set of vector-valued Banach limits
- The estimates of diagonally dominant degree and eigenvalues inclusion regions for the Schur complement of block diagonally dominant matrices
- Some properties of geodesic semi E-b-vex functions
- Computation of double Hopf points for delay differential equations
- Some extensions of a certain integral transform to a quotient space of generalized functions
- Parabolic variational inequalities with generalized reflecting directions
- New properties of conformable derivative
- Coupled fixed point theorems for (α, φ)g-contractive type mappings in partially ordered G-metric spaces
- Double image multi-encryption algorithm based on fractional chaotic time series
- On numerical invariants for knots in the solid torus
- On Vn-semigroups
- W-perfect groups
- Determinants of (–1,1)-matrices of the skew-symmetric type: a cocyclic approach
- All about the ⊥ with its applications in the linear statistical models
- Nonrecursive solution for the discrete algebraic Riccati equation and X + A*X-1A=L
- Generalizations of Nekrasov matrices and applications
- Complete and sufficient statistics and perfect families in orthogonal and error orthogonal normal models
- Computing the numerical range of Krein space operators
- New SOR-like methods for solving the Sylvester equation
- Complexity issues for the symmetric interval eigenvalue problem
- Some fractional integral formulas for the Mittag-Leffler type function with four parameters
- On the numerical solutions of some fractional ordinary differential equations by fractional Adams-Bashforth-Moulton method
- A finite difference method for fractional diffusion equations with Neumann boundary conditions
- Results for Mild solution of fractional coupled hybrid boundary value problems
- Third-order differential subordination and superordination involving a fractional operator
- Upper and lower bounds of integral operator defined by the fractional hypergeometric function
- A detailed analysis for the fundamental solution of fractional vibration equation
- Duplication in a model of rock fracture with fractional derivative without singular kernel
- System of fractional differential equations with Erdélyi-Kober fractional integral conditions
- The general solution for impulsive differential equations with Riemann-Liouville fractional-order q ∈ (1,2)
- Fractional derivative generalization of Noether’s theorem

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