Search Results (32)

We investigate the longstanding question of whether compact Levi-flat hypersurfaces exist in the complex projective plane ${\mathbb{CP}}^2$. While the nonexistence of closed real-analytic Levi-flat hypersurfaces in ${\mathbb{CP}}^n$ for $n>2$ is well known, the case $n=2$ remains open. By combining techniques from the classification of homogeneous CR-manifolds with projective foliation geometry, we prove that no homogeneous Levi-flat hypersurfaces exist in ${\mathbb{CP}}^2$, thus partially resolving the problem under natural symmetry assumptions. Full article

The product manifold ${\mathbb{S}}^3\times {\mathbb{S}}^3$, which belongs to the homogenous six-dimensional nearly Kähler manifolds, admits two structures, the almost complex structure J and the almost product structure P. The investigation of embeddings of different classes of CR submanifolds of ${\mathbb{S}}^3\times {\mathbb{S}}^3$ was started some time ago by investigating three-dimensional CR submanifolds. It resulted that the almost product structure P is very important for the study of CR submanifolds of ${\mathbb{S}}^3\times {\mathbb{S}}^3$, since submanifolds characterized by different actions of the almost product structure on base vector fields often appear as a result of the study of some specific types of CR submanifolds. Therefore, the investigation of four-dimensional CR submanifolds of ${\mathbb{S}}^3\times {\mathbb{S}}^3$ is initiated in this article. The main result is the classification of four-dimensional CR submanifolds of ${\mathbb{S}}^3\times {\mathbb{S}}^3$, whose almost complex distribution ${\mathcal{D}}_1$ is almost product orthogonal on itself. First, it was proved that such submanifolds have a non-integrable almost complex distribution, and then it was proved that these submanifolds are locally product manifolds of curves and three-dimensional CR submanifolds of ${\mathbb{S}}^3\times {\mathbb{S}}^3$ of the same type, and they were therefore constructed in this way. Full article

We reformulate holographic space–time models in terms of Hilbert bundles over the space of the time-like geodesics in a Lorentzian manifold. This reformulation resolves the issue of the action of non-compact isometry groups on finite-dimensional Hilbert spaces. Following Jacobson, I view the background geometry as a hydrodynamic flow, whose connection to an underlying quantum system follows from the Bekenstein–Hawking relation between area and entropy, generalized to arbitrary causal diamonds. The time-like geodesics are equivalent to the nested sequences of causal diamonds, and the area of the holoscreen (The holoscreen is the maximal $d-2$ volume (“area”) leaf of a null foliation of the diamond boundary. I use the term area to refer to its volume.) encodes the entropy of a certain density matrix on a finite-dimensional Hilbert space. I review arguments that the modular Hamiltonian of a diamond is a cutoff version of the Virasoro generator $L_0$ of a $1+1$-dimensional CFT of a large central charge, living on an interval in the longitudinal coordinate on the diamond boundary. The cutoff is chosen so that the von Neumann entropy is $\ln {\mathrm{D}}_{\diamond },$ up to subleading corrections, in the limit of a large-dimension diamond Hilbert space. I also connect those arguments to the derivation of the ’t Hooft commutation relations for horizon fluctuations. I present a tentative connection between the ’t Hooft relations and $U\left(1\right)$ currents in the CFTs on the past and future diamond boundaries. The ’t Hooft relations are related to the Schwinger term in the commutator of the vector and axial currents. The paper in can be read as evidence that the near-horizon dynamics for causal diamonds much larger than the Planck scale is equivalent to a topological field theory of the ’t Hooft CR plus small fluctuations in the transverse geometry. Connes’ demonstration that the Riemannian geometry is encoded in the Dirac operator leads one to a completely finite theory of transverse geometry fluctuations, in which the variables are fermionic generators of a superalgebra, which are the expansion coefficients of the sections of the spinor bundle in Dirac eigenfunctions. A finite cutoff on the Dirac spectrum gives rise to the area law for entropy and makes the geometry both “fuzzy” and quantum. Following the analysis of Carlip and Solodukhin, I model the expansion coefficients as two-dimensional fermionic fields. I argue that the local excitations in the interior of a diamond are constrained states where the spinor variables vanish in the regions of small area on the holoscreen. This leads to an argument that the quantum gravity in asymptotically flat space must be exactly supersymmetric. Full article

This paper aims to develop a general theory of quaternion Kahlerian statistical manifolds and to study quaternion CR-statistical submanifolds in such ambient manifolds. It extends the existing theories of quaternion submanifolds and totally real submanifolds. Additionally, the work examines quaternion Kahlerian statistical submersions, including illustrative examples. The exploration also includes an analysis of the total space and fibers under certain conditions with example(s) in support. Moreover, Chen–Ricci inequality on the vertical distribution is derived for quaternion Kahlerian statistical submersions from quaternion Kahlerian statistical manifolds. Full article

The concept of pointwise slant submanifolds of a Kähler manifold was presented by Chen and Garay. This research extends this notion to a more general setting, specifically in a locally conformal Kähler manifold. We study the pointwise pseudo-slant warped products of the form ${\mathsf{\Sigma }}^{\theta }{\times \hspace{0.17em}}_f{\mathsf{\Sigma }}^{\perp }$ in a locally conformal Kähler manifold. Using the concept of pointwise pseudo-slant, we establish the necessary and sufficient condition for it to be characterized as a warped product submanifold. In addition, we derive several results on pointwise pseudo-slant warped products that expand previously proven main ones. Further, some examples of such submanifolds and their warped products are also given. Full article

The utilization of 20CrNiMo/Incoloy 825 composite materials as high-pressure pipe manifold steel can not only improve the strength and hardness of the steel, but also improve its corrosion resistance. However, research on the heat treatment of 20CrNiMo/Incoloy 825 composite materials is still scarce. Thus, the aim of this study was to investigate the influence of solid solution treatment on the microstructure and properties of 20CrNiMo/Incoloy 825 composite materials. Firstly, the composite materials were subjected to solid solution treatment at temperatures ranging from 850 to 1100 °C with varied holding times of 1 h, 4 h, and 6 h. Microstructural analysis revealed that the solid solution treatment temperature had a more pronounced effect than the treatment time on the interface decarburization layer, carburization layer, and grain size. It was observed that the carburized layer thickness decreased while the decarburized layer thickness increased with an increase in the solid solution treatment temperature, oil cooling was found to enhance the hardness of the base layer of the composite materials, and the size of the original austenite grains of 20CrNiMo steel and Incoloy 825 increased with an increase in the solid solution treatment temperature. Secondly, the tensile properties, microhardness, and fracture morphology were evaluated after the composite materials underwent solid solution treatment at temperatures between 950 °C and 1100 °C for 1 h. The results indicated that increasing the solution temperature initially led to an increase in tensile strength and elongation after fracture, followed by a decrease; furthermore, the hardness of Incoloy 825 exhibited a declining trend, while the hardness of 20CrNiMo first decreased then increased. Thirdly, the shear properties and interfacial element diffusion of the composite materials were analyzed following solid solution treatment in a temperature range of 950 °C to 1100 °C for 1 h. The findings demonstrated that higher solid solution treatment temperatures induced full diffusion of Cr, Ni, and Fe atoms at the interface and softened the matrix, leading to an increase in the thickness of the diffusion layer and toughening of the composite interface. Therefore, the shear strength increased with an increase in the solid solution treatment temperature. Finally, the optimal solid solution treatment process for 20CrNiMo/Incoloy 825 composite materials was determined to be 1050 °C/1 h oil cooling, following which the composite materials had good comprehensive mechanical properties. Full article

In this paper, we conduct a thorough study of CR-warped product submanifolds in a Kaehler manifold, utilizing a semi-symmetric metric connection within the framework of warped product geometry. Our analysis yields fundamental and noteworthy results that illuminate the characteristics of these submanifolds. Additionally, we investigate the implications of our findings on the homology of these submanifolds, offering insights into their topological properties. Notably, we present a compelling proof demonstrating that, under a specific condition, stable currents cannot exist for these warped product submanifolds. Our research outcomes contribute significant knowledge concerning the stability and behavior of CR-warped product submanifolds equipped with a semi-symmetric metric connection. Furthermore, this work establishes a robust groundwork for future explorations and advancements in this particular field of study. Full article

The reachable set of controlled dynamical systems is the set of all reachable states from an initial condition over a certain time horizon, subject to operational constraints and exogenous disturbances. In astrodynamics, rapid approximation of reachable sets is invaluable for trajectory planning, collision avoidance, and ensuring safe and optimal performance in complex dynamics. Leveraging the connection between minimum-time trajectories and the boundary of reachable sets, we propose a sampling-based method for rapid and efficient approximation of reachable sets for finite- and low-thrust spacecraft. The proposed method combines a minimum-time multi-stage indirect formulation with the celebrated primer vector theory. Reachable sets are generated under two-body and circular restricted three-body (CR3B) dynamics. For the two-body dynamics, reachable sets are generated for (1) the heliocentric phase of a benchmark Earth-to-Mars problem, (2) two scenarios with uncertainties in the initial position and velocity of the spacecraft at the time of departure from Earth, and (3) a scenario with a bounded single impulse at the time of departure from Earth. For the CR3B dynamics, several cislunar applications are considered, including L1 Halo orbit, L2 Halo orbit, and Lunar Gateway 9:2 NRHO. The results indicate that low-thrust spacecraft reachable sets coincide with invariant manifolds existing in multi-body dynamical environments. The proposed method serves as a valuable tool for qualitatively analyzing the evolution of reachable sets under complex dynamics, which would otherwise be either incoherent with existing grid-based reachability approaches or computationally intractable with a complete Hamilton–Jacobi–Bellman method. Full article

In this paper, we investigate the pointwise hemi-slant submanifolds of a locally conformal Kähler manifold and their warped products. Moreover, we derive the necessary and sufficient conditions for integrability and totally geodesic foliation. We establish characterization theorems for pointwise hemi-slant submanifolds. Several fundamental results that extend the CR submanifold warped product in Kähler manifolds are proven in this study. We also provide some non-trivial examples and applications. Full article

The main goal of this research paper is to investigate contact CR-warped product submanifolds within Sasakian space forms, utilizing a semi-symmetric metric connection. We conduct a comprehensive analysis of these submanifolds and establish several significant results. Additionally, we formulate an inequality that establishes a relationship between the squared norm of the second fundamental form and the warping function. Lastly, we present a number of geometric applications derived from our findings. Full article

The primary objective of this paper is to explore contact CR-warped product submanifolds of Sasakian space forms equipped with a semi-symmetric metric connection. We thoroughly examine these submanifolds and establish various key findings. Furthermore, we derive an inequality relating the Ricci curvature to the mean curvature vector and warping function. Full article

This paper mainly devotes to the study of some solitons such as Ricci and Yamabe solitons and also their combination called Ricci-Yamabe solitons. In the geometry of solitons, a fundamental question is to identify the conditions under which these solitons can be trivial. Firstly, in this paper we study some extensive results on generic contact CR-submanifolds of Sasakian manifolds endowed with concurrent vector fields. Then some applications of solitons such as Ricci and Ricci-Yamabe solitons on such submanifolds with concurrent vector fields in the same ambient manifold have been discussed. Full article

Recently, we studied CR-slant warped products $B_1{\times }_f{M}_{\perp }$, where $B_1=M_T\times M_{\theta }$ is the Riemannian product of holomorphic and proper slant submanifolds and $M_{\perp }$ is a totally real submanifold in a nearly Kaehler manifold. In the continuation, in this paper, we study $B_2{\times }_f{M}_{\theta }$, where $B_2=M_T\times M_{\perp }$ is a CR-product of a nearly Kaehler manifold and establish Chen’s inequality for the squared norm of the second fundamental form. Some special cases of Chen’s inequality are given. Full article

In this article, we derived an equality for $\mathcal{CR}$-warped product in a complex space form which forms the relationship between the gradient and Laplacian of the warping function and second fundamental form. We derived the necessary conditions of a $\mathcal{CR}$-warped product submanifolds in K$\ddot{a}$hler manifold to be an Einstein manifold in the impact of gradient Ricci soliton. Some classification of $\mathcal{CR}$-warped product submanifolds in the K$\ddot{a}$hler manifold by using the Euler–Lagrange equation, Dirichlet energy and Hamiltonian is given. We also derive some characterizations of Einstein warped product manifolds under the impact of Ricci Curvature and Divergence of Hessian tensor. Full article