New Horizons in Differential Geometry and its Related Fields

The following sections are included:

• PREFACE
• CONTENTS

In this paper we give an affirmative answer to sustain the conjecture about the order of a magnetic curve in a quasi-Sasakian manifold. More precisely, we show that the magnetic curves in a quasi-Sasakian manifold obtained as the product of a Sasakian and a Kähler manifold have maximum order 5. Moreover, we obtain the explicit parametrizations, the periodicity conditions and examples in the study of magnetic curves in 𝕊³ × 𝕊².

We shall construct a homogeneous Sasakian manifold $\tilde{M}$ from a Kähler C-space with its second Betti number one, and study a curvature property of $\tilde{M}$. In addition when M is a Kähler C-space with two isotropy summands, we concretely solve the motion of a charged particle in $\tilde{M}$ and show that if the motion of a charged particle intersects itself, then it is simply closed.

In this paper we give a report on the distribution of lengths of Legendre trajectories on complete simply connected Sasakian space forms following to [3].

We obtain new invariant Einstein metrics on the compact Lie groups Sp(n) which are not naturally reductive. This is achieved by using the quaternionic flag manifolds Sp(k₁ + … + k_p)/Sp(k₁) × … × Sp(k_p) and by imposing certain symmetry assumptions in the set of all left-invariant metrics on Sp(n).

We study (κ, µ)-spaces whose Boeckx invariants satisfy I ≤ −1, from the view-point of submanifold geometry. We give a Lie theoretic proof that these spaces can be realized as homogeneous hypersurfaces in noncompact real two-plane Grassmannians.

The equivalence test is a main part in any classification problem. It helps to prove bounds for the main parameters of the considered combinatorial structures and to study their properties. In this paper, we present algorithms for equivalence of linear codes, based on their relation to multisets of points in a projective geometry.

We present a generalization of Walsh-Hadamard transform that is suitable for applications in Coding Theory, especially for computation of the weight distribution and the covering radius of a linear code over a finite field. The transform used in our research, is a modification of the Vilenkin-Chrestenson transform. Instead of using all the vectors in the considered space, we take a maximal set of nonproportional vectors, which reduces the computational complexity.

The purpose of this paper is to prove that each non-flat totally geodesic surface S² in symmetric spaces SU (4)/SO(4), SU (8)/Sp(4) or SU (4)/S(U (2) × U (2)) can be considered as a non-flat totally geodesic surface in Spin(5)/U (2) which is a totally geodesic submanifold in one of the above three symmetric spaces. This non-flat totally geodesic surface is also a totally real surface with respect to the complex structure on Spin(5)/U (2) ≅ Q³.

The famous theorem due to Hurwitz stated that the normed (division) algebra is isomorphic to one of the following four algebras; the field ℝ of real numbers, the field ℂ of complex numbers, the algebra ℍ of quaternions, and the non-associative algebra 𝕆 of octonions. By using these algebraic structures, we can construct the Hopf fibrations, and Cartan hypersurfaces in a sphere. The purpose of this paper is to give the relationship between these Hopf fibrations and Cartan hypersurfaces.

We study the Bochner curvature tensor field of the total space T *Mof the cotangent bundle of a Riemannian manifold (M, g) which is endowed with a natural diagonal Kähler structure (G, J). We prove that (T *M, G, J) is a Bochner flat Kähler manifold if the proportionality factor between the coefficients of J and G satisfies a condition, in particular, if it is a complex space form. We construct some examples of natural diagonal Kähler structures on T *M and on a tube around the zero section in T *M such that the obtained Kähler manifolds are Bochner flat.

This paper is a survey of isotropicity of space-like or time-like surfaces with zero mean curvature vector in Riemannian, Lorentzian or neutral 4-manifolds. There exist plural conditions to define the isotropicity of such surfaces and the conditions depend on the signatures of the metrics of the surface and the space. However, we can find a condition to define the isotropicity in a common style.

The purpose of this note is to give a survey on recent results concerning geometric properties of Lie hypersurfaces in a complex hyperbolic space, which are homogeneous ones having no focal submanifolds.

We consider that vertex-transitive normal Kähler graphs are candidates of discrete models of homogeneous Kähler manifolds by comparing their adjacency and spherical mean operators. To provide examples of such Kähler graphs we define some product operations of graphs. We study connectivity and bipartiteness of their principal and auxiliary graphs, and investigate commutativity of their adjacency operators. We then get normal Kähler graphs whose principal and auxiliary graphs are connected. Being inspired with examples of such graphs of product type, we give a way of constructing vertex-transitive normal Kähler graphs whose vertex-cardinality are multiples of 4 and whose principal and auxiliary degrees d^(p), d^(a) are odd with d^(p) − d^(a) ≡ 2 (mod 4).