Research

6 pages, 270 KiB

Open AccessArticle

by Teresa Arias-Marco and José Manuel Fernández-Barroso

Symmetry 2019, 11(10), 1316; https://doi.org/10.3390/sym11101316 - 20 Oct 2019

Viewed by 2054

Working on closed Riemannian manifolds the first author and Schueth gave a list of curvature properties which cannot be determined by the eigenvalue spectrum of the Laplace–Beltrami operator. Following Kac, it is said that such properties are inaudible. Here, we add to that [...] Read more.

Working on closed Riemannian manifolds the first author and Schueth gave a list of curvature properties which cannot be determined by the eigenvalue spectrum of the Laplace–Beltrami operator. Following Kac, it is said that such properties are inaudible. Here, we add to that list the dimension of the manifold minus three new properties namely k-D’Atri for

k = 3, \dots, dim M - 1

. Full article

(This article belongs to the Special Issue Geometry of Submanifolds and Homogeneous Spaces)

19 pages, 322 KiB

Open AccessArticle

New Characterizations of the Clifford Torus and the Great Sphere

by Sun Mi Jung, Young Ho Kim and Jinhua Qian

Symmetry 2019, 11(9), 1076; https://doi.org/10.3390/sym11091076 - 27 Aug 2019

Cited by 2 | Viewed by 2351

Abstract

In studying spherical submanifolds as submanifolds of a round sphere, it is more relevant to consider the spherical Gauss map rather than the Gauss map of those defined by the oriented Grassmannian manifold induced from their ambient Euclidean space. In that sense, we [...] Read more.

In studying spherical submanifolds as submanifolds of a round sphere, it is more relevant to consider the spherical Gauss map rather than the Gauss map of those defined by the oriented Grassmannian manifold induced from their ambient Euclidean space. In that sense, we study ruled surfaces in a three-dimensional sphere with finite-type and pointwise 1-type spherical Gauss map. Concerning integrability and geometry, we set up new characterizations of the Clifford torus and the great sphere of 3-sphere and construct new examples of spherical ruled surfaces in a three-dimensional sphere. Full article

(This article belongs to the Special Issue Geometry of Submanifolds and Homogeneous Spaces)

8 pages, 233 KiB

Open AccessArticle

Geodesic Chord Property and Hypersurfaces of Space Forms

by Dong-Soo Kim, Young Ho Kim and Dae Won Yoon

Symmetry 2019, 11(8), 1052; https://doi.org/10.3390/sym11081052 - 16 Aug 2019

Viewed by 2044

Abstract

In the Euclidean space

E^{n}

, hyperplanes, hyperspheres and hypercylinders are the only isoparametric hypersurfaces. These hypersurfaces are also the only ones with chord property, that is, the chord connecting two points on them meets the hypersurfaces at the same angle at [...] Read more.

In the Euclidean space

E^{n}

, hyperplanes, hyperspheres and hypercylinders are the only isoparametric hypersurfaces. These hypersurfaces are also the only ones with chord property, that is, the chord connecting two points on them meets the hypersurfaces at the same angle at the two points. In this paper, we investigate hypersurfaces in nonflat space forms with the so-called geodesic chord property and classify such hypersurfaces completely. Full article

(This article belongs to the Special Issue Geometry of Submanifolds and Homogeneous Spaces)

12 pages, 249 KiB

Open AccessArticle

Necessary and Sufficient Optimality Conditions for Vector Equilibrium Problems on Hadamard Manifolds

by Gabriel Ruiz-Garzón, Rafaela Osuna-Gómez and Jaime Ruiz-Zapatero

Symmetry 2019, 11(8), 1037; https://doi.org/10.3390/sym11081037 - 12 Aug 2019

Cited by 13 | Viewed by 2534

Abstract

The aim of this paper is to show the existence and attainability of Karush–Kuhn–Tucker optimality conditions for weakly efficient Pareto points for vector equilibrium problems with the addition of constraints in the novel context of Hadamard manifolds, as opposed to the classical examples [...] Read more.

The aim of this paper is to show the existence and attainability of Karush–Kuhn–Tucker optimality conditions for weakly efficient Pareto points for vector equilibrium problems with the addition of constraints in the novel context of Hadamard manifolds, as opposed to the classical examples of Banach, normed or Hausdorff spaces. More specifically, classical necessary and sufficient conditions for weakly efficient Pareto points to the constrained vector optimization problem are presented. The results described in this article generalize results obtained by Gong (2008) and Wei and Gong (2010) and Feng and Qiu (2014) from Hausdorff topological vector spaces, real normed spaces, and real Banach spaces to Hadamard manifolds, respectively. This is done using a notion of Riemannian symmetric spaces of a noncompact type as special Hadarmard manifolds. Full article

(This article belongs to the Special Issue Geometry of Submanifolds and Homogeneous Spaces)

8 pages, 255 KiB

Open AccessArticle

On Formality of Some Homogeneous Spaces

by Aleksy Tralle

Symmetry 2019, 11(8), 1011; https://doi.org/10.3390/sym11081011 - 05 Aug 2019

Viewed by 1721

Abstract

Let

G / H

be a homogeneous space of a compact simple classical Lie group G. Assume that the maximal torus

T_{H}

of H is conjugate to a torus

T_{β}

whose Lie algebra

t_{β}

is the kernel of the [...] Read more.

Let

G / H

be a homogeneous space of a compact simple classical Lie group G. Assume that the maximal torus

T_{H}

of H is conjugate to a torus

T_{β}

whose Lie algebra

t_{β}

is the kernel of the maximal root

β

of the root system of the complexified Lie algebra

g^{c}

. We prove that such homogeneous space is formal. As an application, we give a short direct proof of the formality property of compact homogeneous 3-Sasakian spaces of classical type. This is a complement to the work of Fernández, Muñoz, and Sanchez which contains a full analysis of the formality property of

S O (3)

-bundles over the Wolf spaces and the proof of the formality property of homogeneous 3-Sasakian manifolds as a corollary. Full article

(This article belongs to the Special Issue Geometry of Submanifolds and Homogeneous Spaces)

5 pages, 210 KiB

Open AccessArticle

Sasaki-Einstein 7-Manifolds, Orlik Polynomials and Homology

by Ralph R. Gomez

Symmetry 2019, 11(7), 947; https://doi.org/10.3390/sym11070947 - 23 Jul 2019

Cited by 2 | Viewed by 2035

Abstract

In this article, we give ten examples of 2-connected seven dimensional Sasaki-Einstein manifolds for which the third homology group is completely determined. Using the Boyer-Galicki construction of links over particular Kähler-Einstein orbifolds, we apply a valid case of Orlik’s conjecture to the links [...] Read more.

In this article, we give ten examples of 2-connected seven dimensional Sasaki-Einstein manifolds for which the third homology group is completely determined. Using the Boyer-Galicki construction of links over particular Kähler-Einstein orbifolds, we apply a valid case of Orlik’s conjecture to the links so that one is able to explicitly determine the entire third integral homology group. We give ten such new examples, all of which have the third Betti number satisfy

10 \leq b_{3} (L_{f}) \leq 20

. Full article

(This article belongs to the Special Issue Geometry of Submanifolds and Homogeneous Spaces)

19 pages, 829 KiB

Open AccessArticle

Multivariate Optimal Control with Payoffs Defined by Submanifold Integrals

by Andreea Bejenaru and Constantin Udriste

Symmetry 2019, 11(7), 893; https://doi.org/10.3390/sym11070893 - 08 Jul 2019

Cited by 6 | Viewed by 1745

Abstract

This paper adapts the multivariate optimal control theory to a Riemannian setting. In this sense, a coherent correspondence between the key elements of a standard optimal control problem and several basic geometric ingredients is created, with the purpose of generating a geometric version [...] Read more.

This paper adapts the multivariate optimal control theory to a Riemannian setting. In this sense, a coherent correspondence between the key elements of a standard optimal control problem and several basic geometric ingredients is created, with the purpose of generating a geometric version of Pontryagin’s maximum principle. More precisely, the local coordinates on a Riemannian manifold play the role of evolution variables (“multitime”), the Riemannian structure, and the corresponding Levi–Civita linear connection become state variables, while the control variables are represented by some objects with the properties of the Riemann curvature tensor field. Moreover, the constraints are provided by the second order partial differential equations describing the dynamics of the Riemannian structure. The shift from formal analysis to optimal Riemannian control takes deeply into account the symmetries (or anti-symmetries) these geometric elements or equations rely on. In addition, various submanifold integral cost functionals are considered as controlled payoffs. Full article

(This article belongs to the Special Issue Geometry of Submanifolds and Homogeneous Spaces)

5 pages, 236 KiB

Open AccessArticle

The Existence of Two Homogeneous Geodesics in Finsler Geometry

by Zdeněk Dušek

Symmetry 2019, 11(7), 850; https://doi.org/10.3390/sym11070850 - 01 Jul 2019

Cited by 1 | Viewed by 1707

Abstract

The existence of a homogeneous geodesic in homogeneous Finsler manifolds was positively answered in previous papers. However, the result is not optimal. In the present paper, this result is refined and the existence of at least two homogeneous geodesics in any homogeneous Finsler [...] Read more.

The existence of a homogeneous geodesic in homogeneous Finsler manifolds was positively answered in previous papers. However, the result is not optimal. In the present paper, this result is refined and the existence of at least two homogeneous geodesics in any homogeneous Finsler manifold is proved. In a previous paper, examples of Randers metrics which admit just two homogeneous geodesics were constructed, which shows that the present result is the best possible. Full article

(This article belongs to the Special Issue Geometry of Submanifolds and Homogeneous Spaces)

13 pages, 276 KiB

Open AccessArticle

Slant Curves and Contact Magnetic Curves in Sasakian Lorentzian 3-Manifolds

by Ji-Eun Lee

Symmetry 2019, 11(6), 784; https://doi.org/10.3390/sym11060784 - 12 Jun 2019

Cited by 9 | Viewed by 2345

Abstract

In this article, we define Lorentzian cross product in a three-dimensional almost contact Lorentzian manifold. Using a Lorentzian cross product, we prove that the ratio of

κ

and

τ - 1

is constant along a Frenet slant curve in a Sasakian Lorentzian three-manifold. [...] Read more.

In this article, we define Lorentzian cross product in a three-dimensional almost contact Lorentzian manifold. Using a Lorentzian cross product, we prove that the ratio of

κ

and

τ - 1

is constant along a Frenet slant curve in a Sasakian Lorentzian three-manifold. Moreover, we prove that

γ

is a slant curve if and only if M is Sasakian for a contact magnetic curve

γ

in contact Lorentzian three-manifold M. As an example, we find contact magnetic curves in Lorentzian Heisenberg three-space. Full article

(This article belongs to the Special Issue Geometry of Submanifolds and Homogeneous Spaces)

9 pages, 229 KiB

Open AccessArticle

On a New type of Tensor on Real Hypersurfaces in Non-Flat Complex Space Forms

by George Kaimakamis and Konstantina Panagiotidou

Symmetry 2019, 11(4), 559; https://doi.org/10.3390/sym11040559 - 18 Apr 2019

Cited by 4 | Viewed by 1685

Abstract

In this paper the notion of

^{*}

-Weyl curvature tensor on real hypersurfaces in non-flat complex space forms is introduced. It is related to the

^{*}

-Ricci tensor of a real hypersurface. The aim of this paper is to provide two classification theorems [...] Read more.

In this paper the notion of

^{*}

-Weyl curvature tensor on real hypersurfaces in non-flat complex space forms is introduced. It is related to the

^{*}

-Ricci tensor of a real hypersurface. The aim of this paper is to provide two classification theorems concerning real hypersurfaces in non-flat complex space forms in terms of

^{*}

-Weyl curvature tensor. More precisely, Hopf hypersurfaces of dimension greater or equal to three in non-flat complex space forms with vanishing

^{*}

-Weyl curvature tensor are classified. Next, all three dimensional real hypersurfaces in non-flat complex space forms, whose

^{*}

-Weyl curvature tensor vanishes identically are classified. The used methods are based on tools from differential geometry and solving systems of differential equations. Full article

(This article belongs to the Special Issue Geometry of Submanifolds and Homogeneous Spaces)

11 pages, 287 KiB

Open AccessArticle

Chen Inequalities for Warped Product Pointwise Bi-Slant Submanifolds of Complex Space Forms and Its Applications

by Akram Ali and Ali H. Alkhaldi

Symmetry 2019, 11(2), 200; https://doi.org/10.3390/sym11020200 - 11 Feb 2019

Cited by 4 | Viewed by 1959

Abstract

In this paper, by using new-concept pointwise bi-slant immersions, we derive a fundamental inequality theorem for the squared norm of the mean curvature via isometric warped-product pointwise bi-slant immersions into complex space forms, involving the constant holomorphic sectional curvature c, the Laplacian [...] Read more.

In this paper, by using new-concept pointwise bi-slant immersions, we derive a fundamental inequality theorem for the squared norm of the mean curvature via isometric warped-product pointwise bi-slant immersions into complex space forms, involving the constant holomorphic sectional curvature c, the Laplacian of the well-defined warping function, the squared norm of the warping function, and pointwise slant functions. Some applications are also given. Full article

(This article belongs to the Special Issue Geometry of Submanifolds and Homogeneous Spaces)

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