Search Results (229)

In total, geodesic surfaces and their generalizations, namely totally umbilical and parallel surfaces, are well-known topics in Submanifold Theory and have been intensively studied in three-dimensional ambient spaces, both Riemannian and Lorentzian. In this paper, we prove the non-existence of parallel and totally umbilical (in particular, totally geodesic) surfaces for three-dimensional Lorentzian Lie groups, which admit a four-dimensional isometry group, but are neither of Bianchi–Cartan–Vranceanu-type nor homogeneous plane waves. Consequently, the results of the present paper complete the investigation of these fundamental types of surfaces in all homogeneous Lorentzian manifolds, whose isometry group is four-dimensional. As a byproduct, we describe a large class of flat surfaces of constant mean curvature in these ambient spaces and exhibit a family of examples. Full article

In this study, theorems and proofs related to spherical and focal curves are presented in the BCV-Sasakian space. An approximate solution to the differential equation characterizing spherical curves in the BCV-Sasakian manifold ${\mathfrak{M}}^3$ is obtained using the Taylor matrix collocation method. The general equations of canal and tubular surfaces are provided within this geometric framework. Additionally, the curvature properties of the tubular surface constructed around a non-vertex focal curve are computed and analyzed. All of these results are presented for the first time in the literature within the context of the BCV-Sasakian geometry. Thus, this study makes a substantial contribution to the differential geometry of contact metric manifolds by extending classical concepts into a more generalized and complex geometric structure. Full article

This paper explores novel inequalities for statistical submanifolds within the framework of the Norden golden-like statistical manifold. By leveraging the intrinsic properties of statistical manifolds and the structural richness of Norden golden geometry, we establish fundamental relationships between the intrinsic and extrinsic invariants of submanifolds. The methodology involves deriving generalized Chen-type and $\delta (2,2)$ curvature inequalities using curvature tensor analysis and dual affine connections. A concrete example is provided to verify the theoretical framework. The novelty of this work lies in extending classical curvature inequalities to a newly introduced statistical structure, thereby opening new perspectives in the study of geometric inequalities in information geometry and related mathematical physics contexts. Full article

The current work establishes the geometrical bearing for hypersurfaces in a Golden Riemannian manifold with constant golden sectional curvature with respect to k-almost Newton-conformal Ricci solitons. Moreover, we extensively explore the immersed r-almost Newton-conformal Ricci soliton and determine the sufficient conditions for total geodesicity with adequate restrictions on some smooth functions using mathematical operators. Furthermore, we go over some natural conclusions in which the gradient k-almost Newton-conformal Ricci soliton on the hypersurface of the Golden Riemannian manifold becomes compact. Finally, we establish a Schur’s type inequality in terms of k-almost Newton-conformal Ricci solitons immersed in Golden Riemannian manifolds with constant golden sectional curvature. Full article

This paper is about a problem at the intersection of differential geometry, spectral analysis and the theory of manifolds. The study of finite-type subvarieties was initiated by Chen in the 1970s, with the aim of obtaining improved estimates for the mean total curvature of compact subvarieties in Euclidean space. The concept of a finite-type subvariety naturally extends that of a minimal subvariety or surface, the latter being closely related to variational calculus. In this work, we classify factorable surfaces in the Lorentz–Heisenberg space ${\mathbb{H}}_3$, equipped with a flat metric satisfying ${\Delta }^I{r}_i={\lambda }_i{r}_i$, which satisfies algebraic equations involving coordinate functions and the Laplacian operator with respect to the surface’s first fundamental form. Full article

The main objective of this paper is to investigate the properties related to the sectional curvatures of a Kähler golden manifold, an almost Hermitian golden manifold whose almost complex golden structure is parallel with respect to the Levi–Civita connection. Under certain conditions, we prove that a Kähler golden manifold with constant sectional curvature is flat. We introduce the concepts of $\Phi$-holomorphic sectional curvature and $\Phi$-holomorphic bi-sectional curvature on a Kähler golden manifold, and compare them respectively with the holomorphic sectional curvature and holomorphic bi-sectional curvature on a Kähler manifold. Full article

This research paper aims to study the Q-curvature tensor on Kenmotsu manifolds endowed with the Schouten–van Kampen connection. Using the Q-curvature tensor, whose trace is the well-known Z-tensor, we characterized Kenmotsu manifolds by introducing the notion of $\zeta$-$\tilde{Q}$ flat and $\varphi$-$\tilde{Q}$ flat manifolds and novel tensor conditions, such as $\tilde{Q}(\xi ,X)\tilde{Q}=0,$$\tilde{Q}(\xi ,X)\tilde{R}=0,$$\tilde{Q}(\xi ,X)\tilde{C}=0,$$\tilde{Q}(\xi ,X)\tilde{S}=0$, $\tilde{Q}(\xi ,X)\tilde{H}=0$, and $\tilde{Q}(\xi ,X)\tilde{P}=0$, with the Schouten–van Kampen connection. To validate some of our results, we constructed a non-trivial example of Kenmotsu manifolds endowed with the Schouten–van Kampen connection. Full article

This paper undertakes a detailed study of $\eta$-Ricci–Bourguignon solitons on $ϵ$-Kenmotsu manifolds, with particular focus on three special types of Ricci tensors: Codazzi-type, cyclic parallel and cyclic $\eta$-recurrent tensors that support such solitonic structures. We derive key curvature conditions satisfying Ricci semi-symmetric $(R·E=0)$, conharmonically Ricci semi-symmetric $(C(\xi ,{\beta }_X)·E=0)$, $\xi$-projectively flat $(P({\beta }_X,{\beta }_Y)\xi =0)$, projectively Ricci semi-symmetric $(L·P=0)$ and $W_5$-Ricci semi-symmetric $(W(\xi ,{\beta }_Y)·E=0)$, respectively, with the admittance of $\eta$-Ricci–Bourguignon solitons. This work further explores the role of torse-forming vector fields and provides a thorough characterization of $\varphi$-Ricci symmetric indefinite Kenmotsu manifolds admitting $\eta$-Ricci–Bourguignon solitons. Through in-depth analysis, we establish significant geometric constraints that govern the behavior of these manifolds. Finally, we construct explicit examples of indefinite Kenmotsu manifolds that satisfy the $\eta$-Ricci–Bourguignon solitons equation, thereby confirming their existence and highlighting their unique geometric properties. Moreover, these solitonic structures extend soliton theory to indefinite and physically meaningful settings, enhance the classification and structure of complex geometric manifolds by revealing how contact structures behave under advanced geometric flows and link the pure mathematical geometry to applied fields like general relativity. Furthermore, $\eta$-Ricci–Bourguignon solitons provide a unified framework that deepens our understanding of geometric evolution and structure-preserving transformations. Full article

The manifolds studied are almost contact complex Riemannian manifolds, known also as almost contact B-metric manifolds. They are equipped with a pair of pseudo-Riemannian metrics that are mutually associated to each other using an almost contact structure. Furthermore, the structural endomorphism acts as an anti-isometry for these metrics, called B-metrics, if its action is restricted to the contact distribution of the manifold. In this paper, some curvature properties of a special class of these manifolds, called Sasaki-like, are studied. Such a manifold is defined by the condition that its complex cone is a holomorphic complex Riemannian manifold (also called a Kähler–Norden manifold). Each of the two B-metrics on the considered manifold is specialized here as an $\eta$-Ricci–Bourguignon almost soliton, where $\eta$ is the contact form, i.e., has an additional curvature property such that the metric is a self-similar solution of a special intrinsic geometric flow. Almost solitons are generalizations of solitons because their defining condition uses functions rather than constants as coefficients. The introduced (almost) solitons are a generalization of some well-known (almost) solitons (such as those of Ricci, Schouten, and Einstein). The soliton potential is chosen to be collinear with the Reeb vector field and is therefore called vertical. The special case of the soliton potential being solenoidal (i.e., divergence-free) with respect to each of the B-metrics is also considered. The resulting manifolds equipped with the pair of associated $\eta$-Ricci–Bourguignon almost solitons are characterized geometrically. An example of arbitrary dimension is constructed and the properties obtained in the theoretical part are confirmed. Full article

We study the rigidity of conformal solitons, give a sufficient and necessary condition that guarantees that every closed conformal soliton is gradient conformal soliton, and prove that complete conformal solitons with a nonpositive Ricci curvature must be trivial under an integral condition. In particular, by using a p-harmonic map from a complete gradient conformal soliton in a smooth Riemannian manifold, we classify complete noncompact nontrivial gradient conformal solitons under some suitable conditions, and similar results are given for gradient Yamabe solitons and gradient k-Yamabe solitons. Full article

Recent interest among geometers in f-structures of K. Yano is due to the study of topology and dynamics of contact foliations, which generalize the flow of the Reeb vector field on contact manifolds to higher dimensions. Weak metric structures introduced by the author and R. Wolak as a generalization of Hermitian and Kähler structures, as well as f-structures, allow for a fresh perspective on the classical theory. In this paper, we study a new f-structure of this kind, called the weak $\beta$-Kenmotsu f-structure, as a generalization of K. Kenmotsu’s concept. We prove that a weak $\beta$-Kenmotsu f-manifold is a locally twisted product of the Euclidean space and a weak Kähler manifold. Our main results show that such manifolds with $\beta =const$ and equipped with an $\eta$-Ricci soliton structure whose potential vector field satisfies certain conditions are $\eta$-Einstein manifolds of constant scalar curvature. Full article

This paper investigates compact Riemannian hypersurfaces immersed in $(n+1)$-dimensional Riemannian or Lorentzian manifolds that admit concircular vector fields, also known as closed conformal vector fields (CCVFs). We focus on the support function of the hypersurface, which is defined as the component of the conformal vector field along the unit-normal vector field, and derive an expression for its Laplacian. Using this, we establish integral formulae for hypersurfaces admitting CCVFs. These results are then extended to compact Riemannian hypersurfaces isometrically immersed in Riemannian or Lorentzian manifolds with constant sectional curvatures, highlighting the crucial role of CCVFs in the study of hypersurfaces. We apply these results to provide characterizations of compact Riemannian hypersurfaces in Euclidean space ${\mathbb{R}}^{n+1}$, Euclidean sphere $S^{n+1}$, and de Sitter space $S_1^{n+1}$. Full article

This paper explores the geometry of 3-dimensional quasi Sasakian manifolds under CL-transformations. We construct both infinitesimal and $CL$-transformation and demonstrate that the former does not necessarily yield projective killing vector fields. A novel invariant tensor, termed the $CL$-curvature tensor, is introduced and shown to remain invariant under $CL$-transformations. Utilizing this tensor, we characterize $CL$-flat, $CL$-symmetric, $CL$-$\phi$ symmetric and $CL$-$\phi$ recurrent structures on such manifolds by mean of differential equations. Furthermore, we investigate conditions under which a Ricci soliton exists on a CL-transformed quasi Sasakian manifold, revealing that under flat curvature, the structure becomes Einstein. These findings contribute to the understanding of curvature dynamics and soliton theory within the context of contact metric geometry. Full article

In this paper, we examine torse-forming vector fields to characterize extrinsic spheres (that is, totally umbilical hypersurfaces with nonzero constant mean curvatures) in Riemannian and Lorentzian manifolds. First, we analyze the properties of these vector fields on Riemannian manifolds. Next, we focus on Ricci solitons on Riemannian hypersurfaces induced by torse-forming vector fields of Riemannian or Lorentzian manifolds. Specifically, we show that such a hypersurface in the manifold with constant sectional curvature is either totally geodesic or an extrinsic sphere. Full article