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1
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Real hypersurfaces in complex two-plane Grassmannians whose normal Jacobi operator is of Codazzi type

100%
Machado C., Dios Pérez J., Jeong I., Suh Y.
Open Mathematics
|
2011
|
tom 9
|
nr 3
578-582
EN
We prove the non-existence of real hypersurfaces in complex two-plane Grassmannians whose normal Jacobi operator is of Codazzi type.
2
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On the structure vector field of a real hypersurface in complex two-plane Grassmannians

100%
Machado C., Pérez J.
Open Mathematics
|
2012
|
tom 10
|
nr 2
451-455
EN
Considering the notion of Jacobi type vector fields for a real hypersurface in a complex two-plane Grassmannian, we prove that if a structure vector field is of Jacobi type it is Killing. As a consequence we classify real hypersurfaces whose structure vector field is of Jacobi type.
3
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Osculating curves in 4-dimensional semi-Euclidean space with index 2

100%
İlarslan K., Kiliç N., Erdem H. A.
Open Mathematics
|
2017
|
tom 15
|
nr 1
562-567
EN
In this paper, we give the necessary and sufficient conditions for non-null curves with non-null normals in 4-dimensional Semi-Euclidian space with indeks 2 to be osculating curves. Also we give some examples of non-null osculating curves in [...] E24 $\mathbb{E}_{2}^{4}$ .
4
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Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster $$\mathfrak{D}^ \bot$$-parallel structure Jacobi operator

100%
Pak E., Suh Y.
Open Mathematics
|
2014
|
tom 12
|
nr 12
1840-1851
EN
Regarding the generalized Tanaka-Webster connection, we considered a new notion of $$\mathfrak{D}^ \bot$$-parallel structure Jacobi operator for a real hypersurface in a complex two-plane Grassmannian G 2(ℂm+2) and proved that a real hypersurface in G 2(ℂm+2) with generalized Tanaka-Webster $$\mathfrak{D}^ \bot$$-parallel structure Jacobi operator is locally congruent to an open part of a tube around a totally geodesic quaternionic projective space ℍP n in G 2(ℂm+2), where m = 2n.
5
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Complete classification of spatial surfaces with parallel mean curvature vector in arbitrary non-flat pseudo-Riemannian space forms

76%
Chen B.-Y.
Open Mathematics
|
2009
|
tom 7
|
nr 3
400-428
EN
Submanifolds with parallel mean curvature vector play important roles in differential geometry, theory of harmonic maps as well as in physics. Spatial surfaces in 4D Lorentzian space forms with parallel mean curvature vector were classified by B. Y. Chen and J. Van der Veken in [9]. Recently, spatial surfaces with parallel mean curvature vector in arbitrary pseudo-Euclidean spaces are also classified in [7]. In this article, we classify spatial surfaces with parallel mean curvature vector in pseudo-Riemannian spheres and pseudo-hyperbolic spaces with arbitrary codimension and arbitrary index. Consequently, we achieve the complete classification of spatial surfaces with parallel mean curvature vector in all pseudo-Riemannian space forms. As an immediate by-product, we obtain the complete classifications of spatial surfaces with parallel mean curvature vector in arbitrary Lorentzian space forms.
6
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A survey on Inverse mean curvature flow in ROSSes

76%
Pipoli G.
Complex Manifolds
|
2017
|
tom 4
|
nr 1
245-262
EN
In this survey we discuss the evolution by inverse mean curvature flow of star-shaped mean convex hypersurfaces in non-compact rank one symmetric spaces. We show similarities and differences between the case considered, with particular attention to how the geometry of the ambient manifolds influences the behaviour of the evolution. Moreover we try, when possible, to give an unified approach to the results present in literature.
7
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Commuting Conditions of the k-th Cho operator with the structure Jacobi operator of real hypersurfaces in complex space forms

76%
Panagiotidou K., Pérez J. d. D.
Open Mathematics
|
2015
|
tom 13
|
nr 1
EN
In this paper three dimensional real hypersurfaces in non-flat complex space forms whose k-th Cho operator with respect to the structure vector field ξ commutes with the structure Jacobi operator are classified. Furthermore, it is proved that the only three dimensional real hypersurfaces in non-flat complex space forms, whose k-th Cho operator with respect to any vector field X orthogonal to structure vector field commutes with the structure Jacobi operator, are the ruled ones. Finally, results concerning real hypersurfaces in complex hyperbolic space satisfying the above conditions are also provided.
8
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Skew Killing spinors

76%
Habib G., Roth J.
Open Mathematics
|
2012
|
tom 10
|
nr 3
844-856
EN
We study the existence of a skew Killing spinor on 2- and 3-dimensional Riemannian spin manifolds. We establish the integrability conditions and prove that these spinor fields correspond to twistor spinors in the two dimensional case while, up to a conformal change of the metric, they correspond to parallel spinors in the three dimensional case.
9
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Dirac and Plateau billiards in domains with corners

52%
Gromov M.
Open Mathematics
|
2014
|
tom 12
|
nr 8
1109-1156
EN
Groping our way toward a theory of singular spaces with positive scalar curvatures we look at the Dirac operator and a generalized Plateau problem in Riemannian manifolds with corners. Using these, we prove that the set of C 2-smooth Riemannian metrics g on a smooth manifold X, such that scalg(x) ≥ κ(x), is closed under C 0-limits of Riemannian metrics for all continuous functions κ on X. Apart from that our progress is limited but we formulate many conjectures. All along, we emphasize geometry, rather than topology of manifolds with their scalar curvatures bounded from below.
10
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Plateau-Stein manifolds

52%
Gromov M.
Open Mathematics
|
2014
|
tom 12
|
nr 7
923-951
EN
We study/construct (proper and non-proper) Morse functions f on complete Riemannian manifolds X such that the hypersurfaces f(x) = t for all −∞ < t < +∞ have positive mean curvatures at all non-critical points x ∈ X of f. We show, for instance, that if X admits no such (not necessarily proper) function, then it contains a (possibly, singular) complete (possibly, compact) minimal hypersurface of finite volume.
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