On locally conformally flat hypersurfaces with constant scalar curvature

We prove that curves of constant torsion satisfy the \mathcal{C}^{1} -dense h -principle in the space of immersed curves in Euclidean space. In particular, there exists a knot of constant torsion in each isotopy class. Our methods, which involve convex integration and degree theory, quickly establish these results for curves of constant curvature as well.

Abstract For a Lorentzian homogeneous space, we study how algebraic conditions on the isotropy group affect the geometry and curvature of the homogeneous space. More specifically, we prove that a Lorentzian locally homogeneous space is locally isometric to a plane wave if it admits an Ambrose–Singer connection with indecomposable, non‐irreducible holonomy. This generalises several existing results that require a certain algebraic type of the torsion of the Ambrose–Singer connection and moreover is in analogy to the fact that a Lorentzian homogeneous space with irreducible isotropy has constant sectional curvature. In addition, we prove results about Lorentzian connections with parallel torsion and for 2‐symmetric connections.

On Hopf hypersurfaces of the complex quadric with constant principal curvatures

We prove a linear upper bound on the Morse index of closed constant mean curvature (CMC) surfaces in orientable three-manifolds in terms of genus, number of branch points and a Willmore-type energy.

In this paper, we study timelike translation surfaces with constant Gaussian curvature \textbf{(CGC)} in the three-dimensional Minkowski space. Such surfaces are generated as the sum of two timelike space curves and naturally arise in the context of Lorentzian surface geometry. By employing a detailed analytic and geometric approach, we prove that any timelike translation surface with constant Gaussian curvature must be flat. As a consequence, we show that the only timelike translation surfaces satisfying this curvature condition are cylindrical surfaces. Furthermore, we establish that timelike translation surfaces with constant Gaussian curvature cannot be minimal everywhere. As a geometric characterization of the generating curves, we prove that one of the curves must necessarily be either a timelike hyperbola or a straight line. These results provide a complete local classification of timelike translation surfaces with constant Gaussian curvature in Minkowski 3-space and highlight a strong rigidity phenomenon in the timelike Lorentzian setting.

Abstract The superintegrability of five Hamiltonians defined on the 3-dimensional spaces with constant curvature κ , sphere S κ 3 ( κ > 0 ) and hyperbolic space H κ 3 ( κ < 0 ), was recently studied in a previous work. Three of the Hamiltonians were oscillator related, while the other two were the Kepler related. In all the cases the systems had additional nonlinear terms. Now we present a similar study of two new Hamiltonians, neither related to the oscillator nor to the Kepler system, that were previously studied by Evans on the 3-dimensional Euclidean space E 3 . The formalism consider the curvature κ as a parameter. All the mathematical expressions are presented by using κ as a parameter, in such a way that particularizing for κ > 0 , κ = 0, or κ < 0 , the corresponding properties are obtained for the system on the sphere S κ 3 , the Euclidean space E 3 , or the hyperbolic space H κ 3 , respectively.

In this paper, we fully determine all Jacobi-type vector fields in hyperbolic three-space by taking advantage of both its constant negative curvature and its intrinsic compatibility with statistical manifold structures. The study is a natural extension of the results obtained by Wang and Zhang, [16], under the classical Levi-Civita connection.

On Biconservative Hypersurfaces with Constant Scalar Curvature in Pseudo-Riemannian Space Forms

We study existence and nonexistence of diagonal and separating coordinates for Riemannian symmetric spaces of rank 1. We generalize the results of Gauduchon and Moroianu (2020) by showing that a symmetric space of rank 1 has diagonal coordinates if and only if it has constant sectional curvature. This implies that orthogonal separation of variables on a symmetric space of rank 1 is possible only in the constant sectional curvature case. We show that on the complex projective space ℂ P n and on complex hyperbolic space ℂ H n , with n ≥ 2 , separating coordinates necessarily have precisely n ignorable coordinates. In view of results of Boyer et al. (1983, 1985) and later results of Winternitz et al. (1994), this completes the description of separation of variables on ℂ P n for all n and on ℂ H n for n = 2 , 3 .

We study the global geometry of families of tubes of constant mean curvature invariant under screw-motions in homogeneous E ( κ , τ ) -spaces. In particular, we study embeddedness and prove a foliation result. Moreover, we numerically analyze the isoperimetric profile in the compact case. • We prove existence of a continuous family of CMC tubes around any screw motion geodesic in E ( κ , τ ) , that converges to this geodesic. • For κ > 0 , we prove that a subfamily always foliates an open set of ambient space. In some cases the foliation is global. • For κ ≤ 0 , we prove the embeddedness of some tubes. • Unlike all other tubes previously described in the literature, some of these tubes do not admit a dihedral symmetry of order 4. • For Heisenberg space Nil 3 we proof a partial uniqueness result.

On classification of holomorphic two-spheres of constant curvature in the complex Grassmann manifold G(3,6)

This work presents a reinforcement learning framework for controlling a planar threesection continuum robot in environments with static obstacles. Assuming constant curvature for each section, the robot is trained to navigate toward a fixed goal while avoiding collisions with multiple static objects. A custom simulation environment was developed to support three levels of scenario difficulty, easy, medium and hard, each with varying obstacle density and placement. The learning process is driven by the Deep Deterministic Policy Gradient (DDPG) algorithm, which enables smooth and continuous curvature control. Careful attention was paid to the design of the reward function and the network architecture, both of which were critical to achieving stable and reliable policy learning. Performance was evaluated across multiple runs, revealing that the agent successfully generalized its behavior across scenarios of increasing complexity. The proposed framework demonstrates the potential of reinforcement learning as a viable approach to safe and adaptive control in continuum robotic systems, with promising implications for applications such as medical navigation, search and rescue, and inspection in confined environments.

Existence of Constant Mean Curvature Disks in $$\mathbb {R}^3$$ with Capillary Boundary Condition

We study the properties of complete parabolic constant mean curvature spacelike hypersurfaces in generalized Robertson–Walker (GRW) spacetimes I×φPn whose warping function φ fulfills a certain convexity criterion such that −φ is convex, and whose Ricci curvature of the fiber Pn is non-negative. Our approach is based on calculating the Laplacian of an appropriate function. Under appropriate conditions on the constant mean curvature, by using the parabolicity, we obtain a rigidity theorem and some corollaries of spacelike hypersurfaces. As a consequence, we solve new corresponding Calabi–Bernstein-type problems.

We derive a sharp upper bound for the first eigenvalue \lambda_{1,p} of the p -Laplacian on asymptotically hyperbolic (AH) manifolds for 1 < p < \infty . We show that asymptotically constant mean curvature submanifolds within AH manifolds are themselves asymptotically hyperbolic. As a corollary, we show that for any minimal conformally compact submanifold Y^{k+1} within \mathbb{H}^{n+1}(-1) , the first eigenvalue \lambda_{1,p}(Y) satisfies \lambda_{1,p}(Y) = (k/p)^{p} . Finally, we obtain lower bounds for \lambda_{1,2}(Y) for complete, non-compact submanifolds with bounded mean curvature in a large class of AH spaces. In the course of this analysis, we introduce an invariant \hat{\beta}^{Y} for each such submanifold.

Revolution surfaces with zero mean curvature in the Galilean 3-space have been extensively studied in the literature. However, revolution surfaces with non-zero constant mean curvature in this geometric setting have not yet been investigated in a systematic way. In this paper, we address this gap by studying surfaces of revolution in the Galilean 3-space with constant mean curvature. We derive the necessary and sufficient differential conditions for such surfaces and obtain explicit parametrizations of the corresponding families. The results extend the theory beyond the minimal case and reveal geometric features that arise from the degenerate nature of the Galilean metric. Several examples are presented to illustrate the obtained surfaces and to emphasize the qualitative differences between minimal and non-minimal constant mean curvature configurations.

Abstract Accurate and robust positioning of drilling robots within subsea strata is a critical prerequisite for deep-sea resource exploration and geological monitoring. However, conventional underwater and underground positioning methods are often ineffective due to severe signal attenuation in sediment. This paper proposes a novel fused positioning method specifically for subsea stratum drilling robots, which integrates a magnetic beacon system with a MEMS-IMUs array cable. A Piecewise Constant Curvature (PCC) kinematic model is established to reconstruct the spatial morphology of the array cable, providing a continuous position estimate free from temporal integration drift. This estimate is then fused with absolute position and orientation data derived from a magnetic dipole model of the beacon. To handle the high nonlinearity of the magnetic measurements and the complementary error characteristics of the two subsystems, we develop a tightly-coupled sensor fusion framework based on an Iterated Extended Kalman Filter with Leven-berg-Marquardt optimization (IEKF-LM). Simulation and experimental results demonstrate that the proposed system effectively overcomes the limitations of individual sensors. The array cable excels in near-field accuracy, while the magnetic beacon provides superior far-field performance. The IEKF-LM fusion algorithm successfully combines their strengths, achieving high-precision and robust localization in both static and dynamic scenarios simulating in-stratum conditions. The results indicate that the proposed fusion method effectively mitigates the limitations of individual sensor modalities, offering a viable positioning solution for subsea exploration robots.

The objective of this work is to characterize certain geometric aspects of LP-Sasakian (LPS) manifolds admitting a generalized almost Schouten soliton (GASS) and to prove that a such manifold with GASS is of constant scalar curvature. Initially, we examine the solitonic behavior of ϕ-recurrent LPS manifolds with GASS in view of certain curvature conditions. Moreover, we also deliberate the geometric properties of a perfect fluid LPS spacetime with a unit torse-forming vector field (UTVF) in connection with a GASS. Also, the behavior of a GASS is studied in the broader framework of special types of perfect fluid LPS spacetime such as dust fluid, dark fluid, and radiation era. Overall, the main novelty of this work is its study of the geometrical phenomena and characteristics of a GASS on LPS manifolds and their application in a perfect fluid LPS spacetime.

The purpose of this paper is to examine the geometric properties of an almost Ricci soliton on a pseudo Ricci symmetric spacetime that satisfies the Einstein field equation. Among other results, we first determine the soliton constant and the cosmological constant in terms of the scalar curvature and the potential vector field of the spacetime, assuming the scalar curvature is constant. Next, we show that the soliton is shrinking when the spacetime is either dust or dark space, and we prove that the soliton can never be steady on a pseudo Ricci symmetric spacetime with constant scalar curvature whose potential vector field is solenoidal. Furthermore, we establish that the soliton is shrinking when the potential vector field is anti-torqued on a pseudo Ricci symmetric spacetime. We also investigate R-harmonic pseudo Ricci symmetric spacetimes, showing that the associated soliton is shrinking and determining the corresponding cosmological constant. An explicit example is provided to establish the existence of an almost Ricci soliton on a pseudo Ricci symmetric spacetime. Finally, we examine the structure of almost Ricci solitons on conformally flat pseudo Ricci symmetric spacetimes that satisfy the Einstein field equations. In this setting, we compute the soliton constant and scalar curvature when the potential vector field is taken to be an anti-torqued vector field, and we derive the equation of state. We also determine the soliton constant and establish the strong energy condition in the case, where the spacetime admits a Ricci collineation.