Product Riemannian Manifolds Research Articles

On Conformal Bi-slant Riemannian Submersions from Locally Product Riemannian Manifolds

On Conformal Bi-slant Riemannian Submersions from Locally Product Riemannian Manifolds

Curves as slant submanifolds of an almost product Riemannian manifold

Curves as slant submanifolds of an almost product Riemannian manifold

The de Rham Cohomology Classes of Hemi-Slant Submanifolds in Locally Product Riemannian Manifolds

This paper aims to discuss the de Rham cohomology of hemi-slant submanifolds in locally product Riemannian manifolds. The integrability and geodesical invariance conditions of the distributions derived from the definition of a hemi-slant submanifold are given. The existence and non-triviality of de Rham cohomology classes of hemi-slant submanifolds are investigated. Finally, an example is presented.

Warped product pointwise bi-slant submanifolds of locally product Riemannian manifolds.

In this paper we introduce the concept of pointwise bi-slant
 submanifolds of locally product Riemannian manifolds and studied warped
 product pointwise bi-slant submanifolds of locally product Riemannian
 manifolds. We obtain some characterization results for warped products
 pointwise bi-slant submanifolds. Also, we provide some non-trivial
 examples of such warped product submanifolds.

Geometrical Aspect of Pointwise Semi-Slant Conformal Submersions

The aim of this paper is to define pointwise semi-slant conformal submersions from locally product Riemannian manifolds onto Riemannian manifolds. We investigated the conditions under which the distributions are integrable and the leaves of the distributions defines totally geodesic foliation. Additionally, we examined the concept of pluriharmonicity of pointwise semi-slant conformal submersions. In support of the results we obtained, we present non-trivial examples.

The Kastler–Kalau–Walze-type theorems about J-Witten deformation

In this paper, we obtain a Lichnerowicz-type formula for [Formula: see text]-Witten deformation and give the proof of the Kastler–Kalau–Walze-type theorems associated with [Formula: see text]-Witten deformation on four-dimensional and six-dimensional almost product Riemannian manifold with (respectively, without) boundary. We give an explanation of the Einstein–Hilbert action for [Formula: see text]-Witten deformation on four-dimensional manifold with boundary.

On a Sequence of Slant Submanifolds in Almost Product Riemannian Setting

We prove that the property of being pointwise slant is transitive on a class of proper pointwise slant submanifolds of almost product Riemannian manifolds, and illustrate this fact with an example. For a given almost product Riemannian manifold $(M_1,g,\varphi_1)$, we consider a sequence of pointwise slant submanifolds $(M_{i+1}\hookrightarrow M_i)_{i\in \mathbb{N}^*}$, and we explicitly determine the relation between the slant functions. Moreover, we state this result in a more general case.

Clairaut semi invariant submersions from locally product Riemannian manifolds

The goal of the present paper is to analyze some geometric features of Clairaut semi invariant Riemannian submersions whose total manifold is a locally product Riemannian manifold and investigate fundamental results on such submersion. We also ensure an explicit example of Clairaut semi invariant Riemannian submersion.

Four-dimensional Riemannian product manifolds with circulant structures

"A 4-dimensional Riemannian manifold equipped with an additional tensor structure, whose fourth power is the identity, is considered. This structure has a circulant matrix with respect to some basis, i.e. the structure is circulant, and it acts as an isometry with respect to the metric. The Riemannian product manifold associated with the considered manifold is studied. Conditions for the metric, which imply that the Riemannian product manifold belongs to each of the basic classes of Staikova-Gribachev's classi cation, are obtained. Examples of such manifolds are given."

The J-twist DJ of the Dirac operator and the Kastler–Kalau–Walze type theorem for six-dimensional manifolds with boundary

In [S. Liu and Y. Wang, A Kastler–Kalau–Walze type theorem for the J-twist DJ of the Dirac operator, preprint (2022), arXiv:2203.10467], the authors proved a Kastler–Kalau–Walze type theorem for the [Formula: see text]-twist [Formula: see text] of the Dirac operator on [Formula: see text]-dimensional and [Formula: see text]-dimensional almost product Riemannian spin manifold with boundary. In this paper, we develop the Kastler–Kalau–Walze type theorem for the [Formula: see text]-twist [Formula: see text] of the Dirac operator on a [Formula: see text]-dimensional almost product Riemannian spin manifold with boundary.

Clairaut Pointwise Slant Submersion from Locally Product Riemannian Manifolds

The goal of the present paper is to analyze some geometric features of Clairaut pointwise slant submersions whose total manifold is a locally product Riemannian manifold. We describe Clairaut pointwise slant submersions from locally product Riemannian manifold onto a Riemannian manifold. We study pointwise slant submersions by providing a consequent which defines the geodesics on the total space of this type submersions. We also give a non-trivial example of the Clairaut pointwise slant submersions whose total manifolds are locally product Riemannian.

A Kastler–Kalau–Walze Type Theorem for the J-Twist D_{J} of the Dirac Operator

In this paper, we give a Lichnerowicz type formula for the J-twist D_{J} of the Dirac operator. And we prove a Kastler–Kalau–Walze type theorem for the J-twist D_{J} of the Dirac operator on three-dimensional and four-dimensional almost product Riemannian spin manifold with boundary.

Joint Design of Transmit Sequence and Receive Filter Based on Riemannian Manifold of Gaussian Mixture Distribution for MIMO Radar

To improve target detection performance in non-Gaussian backgrounds, the joint design of transmit sequence and receive filter for multiple-input-multiple-output (MIMO) radar is studied. By approximating the probability density function of observed non-Gaussian data with the Gaussian mixture model, a Riemannian manifold of Gaussian mixture distribution is developed to depict the complicated background first. Then, maximizing the geometric distance on manifolds, which is converted by maximizing the discrimination between the target and clutter, is proposed as the criterion for the joint design of transmit sequence and receive filter. Thereby, under the constant-modulus constraint, the joint design problem can be transformed into an optimization problem. However, the proposed optimization problem is non-convex and constrained. To solve this problem, a Riemannian optimization framework is provided. By taking the advantage of the underlying geometric and algebraic structure of the constraint space, the original constrained optimization problem in Euclidean space can be transformed into the unconstraint optimization problem over Riemannian product manifolds. Moreover, to obtain the global optimal solution, the Riemannian gradient of the geometric distance cost is derived for the conjugate gradient algorithm. Experiments demonstrate that the proposed method shows advantages in detection performance compared with competitive methods.

Some basic inequalities on golden Riemannian product manifolds with constant curvatures

This article is devoted to prove the basic Chen?s inequalities for slant submanifolds in Riemannian space forms equipped with Golden structure. The equality case and some particular cases of derived inequalities are discussed.

Sequential warped product submanifolds in locally product Riemannian manifolds

In this paper, wepresent sequential warped product submanifolds of locally product Riemannian manifolds and show that there exists a class of non-trivial sequential warped product submanifolds of a locally product Riemannian manifold of the form (M? ??1 M?)??2 MT by giving non-trivial examples. Also, we prove some useful results for such warped products and establish Chen?s inequality which represents a relationship between the squared norm of the second fundamental form for the warping functions ?1 and ?2. Further, some applications of our main result are given.

A study on curvature relations of conformal generic submersions

The present paper enlighten us with the study of the conformal generic submersion whose total space is locally product Riemannian manifold. We investigate;the geometry of the foliations arisen from the definition of conformal generic submersion and provided some decomposition theorem for the total space of the submersion. Meanwhile, harmonicity of conformal generic submersion is also discussed. We mainly established relations between the sectional curvatures of fibres, total space and base manifold and discussed the related consequences. A non-trivial example have also been discussed to make the content wrth.

Energy-Efficient Power Control and Beamforming for Reconfigurable Intelligent Surface-Aided Uplink IoT Networks

Recently, reconfigurable intelligent surface (RIS), a planar metasurface consisting of a large number of low-cost reflecting elements, has received much attention due to its ability to improve both the spectrum and energy efficiencies by reconfiguring the wireless propagation environment. In this paper, we propose an RIS phase shift and BS beamforming optimization technique that minimizes the uplink transmit power of the RIS-aided IoT network. Key idea of the proposed scheme, referred to as Riemannian conjugate gradient-based joint optimization (RCG-JO), is to jointly optimize the RIS phase shifts and the BS beamforming vectors using the Riemannian conjugate gradient technique. By exploiting the product Riemannian manifold structure of the sets of unit-modulus phase shifts and unit-norm beamforming vectors, we convert the nonconvex uplink power minimization problem into the unconstrained problem and then find out the optimal solution over the product Riemannian manifold. From the performance analysis and numerical evaluations, we demonstrate that the proposed RCG-JO technique achieves 94% reduction of the uplink transmit power over the conventional scheme without RIS.

A Riemannian inexact Newton dogleg method for constructing a symmetric nonnegative matrix with prescribed spectrum

This paper is concerned with the inverse problem of constructing a symmetric nonnegative matrix from realizable spectrum. We reformulate the inverse problem as an underdetermined nonlinear matrix equation over a Riemannian product manifold. To solve it, we develop a Riemannian underdetermined inexact Newton dogleg method for solving a general underdetermined nonlinear equation defined between Riemannian manifolds and Euclidean spaces. The global and quadratic convergence of the proposed method is established under some mild assumptions. Then, we solve the inverse problem by applying the proposed method to its equivalent nonlinear matrix equation and a preconditioner for the perturbed normal Riemannian Newton equation is also constructed. Numerical tests show the efficiency of the proposed method for solving the inverse problem.

Minimax Periodic Orbits of Convex Lagrangian Systems on Complete Riemannian Manifolds

In this paper we study the existence of periodic orbits with prescribed energy levels of convex Lagrangian systems on complete Riemannian manifolds. We extend the existence results of Contreras by developing a modified minimax principal to a class of Lagrangian systems on noncompact Riemannian manifolds, namely the so called $\lsh$ Lagrangian systems. In particular, we prove that for almost every $k\in(0,c_u(L))$ the exact magnetic flow associated to a $\lsh$ Lagrangian has a contractible periodic orbit with energy $k$. We also discuss the existence and non-existence of closed geodesics on the product Riemannian manifold $\R\times M$.

Quasi Bi-Slant Submanifolds of Kaehler Manifolds

In this paper, we introduce the new notion of quasi bi-slant submanifolds of almost Hermitian manifolds. Necessary and sufficient conditions for the integrability of distributions which are involved in the definition of such submanifolds of a Kaehler manifold are obtained. We also investigate the necessary and sufficient conditions for these submanifolds of Kaehler manifolds to be totally geodesic and study the geometry of foliations determined by the above distributions. Finally, we obtain the necessary and sufficient conditions for a quasi bi-slant submanifold to be local product Riemannian manifold and also construct some examples of such submanifolds.