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The purpose of this paper is to investigate the existence of Ricci solitons and the nature of curvature inheritance as well as collineations on the Robinson–Trautman (briefly, RT) spacetime. It is shown that under certain conditions RT spacetime admits almost-Ricci soliton, almost- η -Ricci soliton, almost-gradient η -Ricci soliton. As a generalization of curvature inheritance [K. L. Duggal, Curvature inheritance symmetry in Riemannian spaces with applications to fluid space times, J. Math. Phys. 33(9) (1992) 2989–2997] and curvature collineation [G. H. Katzin, J. Livine and W. R. Davis, Curvature collineations: A fundamental symmetry property of the space-times of general relativity defined by the vanishing Lie derivative of the Riemann curvature tensor, J. Math. Phys. 10(4) (1969) 617–629], in this paper, we introduce the notion of generalized curvature inheritance and examine if RT spacetime admits such a notion. It is shown that RT spacetime also realizes the generalized curvature (resp., Ricci, Weyl conformal, concircular, conharmonic, Weyl projective) inheritance. Finally, several conditions are obtained, under which RT spacetime possesses curvature (resp., Ricci, conharmonic, Weyl projective) inheritance as well as curvature (resp., Ricci, Weyl conformal, concircular, conharmonic, Weyl projective) collineation, and we have also introduced the concept of generalized Lie inheritance and showed that RT spacetime realizes such a notion. [ABSTRACT FROM AUTHOR]

In this paper we have found a number of commutator formulas between the weighted divergence, the weighted Laplacian, the weighted Lichnerowicz Laplacian and the Lie derivtive on closed orientable gradient Ricci shrinkers, then using them we have generalized a Theorem of Cao and Zhu about the necessary condition for linear stability of a gradient Ricci shrinker. Recentlly they have also extended our results. [ABSTRACT FROM AUTHOR]

In this article, we study the Ricci soliton on slant submanifolds of trans-Sasakian manifolds with a quarter symmetric non-metric connection. Moreover, we derive a lower-bound-type inequality for the slant submanifolds of trans-Sasakian manifolds with a quarter symmetric non-metric connection in terms of gradient Ricci solitons. We also characterize anti-invariant, invariant, quasi-umbilical submanifolds of trans-Sasakian manifolds with a quarter symmetric non-metric connection for which the same inequality case holds. Finally, we deduce the above inequalities in terms of a scalar concircular field on submanifolds of trans-Sasakian manifolds with a quarter symmetric non-metric connection. [ABSTRACT FROM AUTHOR]

The investigation of Ricci solitons is the focus of this work. We have proved triviality results for compact gradient Ricci soliton under certain restriction. Later, a rigidity result is derived for a compact gradient shrinking Ricci soliton. Also, we have estimated the conjugate radius for non-compact gradient shrinking Ricci soliton with superharmonic potential. Moreover, an upper bound for the conjugate radius of Ricci soliton with concircular potential vector field is determined. Finally, it is proved that a non-compact gradient Ricci soliton with a pole and non-negative Ricci curvature is non-shrinking. [ABSTRACT FROM AUTHOR]

The main goal of this manuscript is to investigate the properties of N(k)-contact metric manifolds admitting a Z*-tensor. We prove the necessary conditions for which N(k)- contact metric manifolds endowed with a Z*-tensor are Einstein manifolds. In this sequel, we accomplish that an N(k)-contact metric manifold endowed with a Z*-tensor satisfying Z*(G1, ζ).R* = 0 is either locally isometric to the Riemannian product En+1(0)×Sn(4) or an Einstein manifold. We also prove the condition for which an N(k)-contact metric manifold endowed with a Z*-tensor is a Sasakian manifold. To validate some of our results, we construct a non-trivial example of an N(k)-contact metric manifold. [ABSTRACT FROM AUTHOR]

In this paper, we have established the existence of conformal Ricci soliton as self similar solution of conformal Ricci flow and characterized the potential function f of almost conformal gradient Ricci soliton on a Sasakian manifold in terms of the non-dynamical scalar field p. We have found the necessary condition for the potential function f to be constant. Furthermore, a relation between λ and the potential function f has been established. Finally, we prove a sufficient condition for an almost conformal Ricci soliton to be an almost conformal gradient Ricci soliton and characterize it in terms of shrinking, steady or expanding soliton. [ABSTRACT FROM AUTHOR]

We found conditions on an $ n $-dimensional Ricci soliton $ \left(M, g, \mathbf{u}, \lambda \right) $ to be trivial. First, we showed that under an appropriate upper bound on the squared length of the covariant derivative of the potential field $ \mathbf{u} $, the Ricci soliton $ \left(M, g, \mathbf{u}, \lambda \right) $ reduces to a trivial soliton. We also showed that appropriate upper and lower bounds on the Ricci curvature $ Ric\left(\mathbf{u}, \mathbf{u}\right) $ of a compact Ricci soliton $ \left(M, g, \mathbf{u}, \lambda \right) $ with potential field $ \mathbf{u} $ geodesic vector field makes it a trivial soliton. We showed that if the Ricci operator $ S $ of the Ricci soliton $ \left(M, g, \mathbf{u}, \lambda \right) $ is invariant under the potential field $ \mathbf{u} $, then $ \left(M, g, \mathbf{u}, \lambda \right) $ is trivial and the converse is also true. Finally, it was shown that if the potential field $ \mathbf{u} $ of a connected Ricci soliton $ \left(M, g, \mathbf{u}, \lambda \right) $ is a concurrent vector field, then the Ricci soliton is shrinking.

In this paper, we conduct a thorough investigation into holomorphic Riemannian maps from nearly Kaehler manifolds. We determine the necessary and sufficient conditions for these maps to be harmonic. In addition, we explore the scenario where the total manifold admits a Ricci soliton, analyze the properties of the holomorphic Riemannian maps in this context. We also establish several conditions under which the fibers of these maps admit Ricci solitons and trivial Ricci solitons. [ABSTRACT FROM AUTHOR]

We study Ricci solitons and -Ricci solitons on generalized Sasakian-space-forms (GSSF) with parallel -Ricci tensor. We find that if GSSF with Kenmotsu metric admits a Ricci soliton or a -Ricci soliton, then and . Moreover, the Ricci soliton is expanding, and the -Ricci soliton is steady. Further, we provide some examples. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to study Ricci solitons on Sasaki–Kenmotsu manifolds. It is shown that if the characteristic vector fields and are recurrent torse-forming vector fields on the Sasaki–Kenmotsu metric as a Ricci soliton, then both and are concurrent and Killing vector fields. We classify and characterize a Sasaki–Kenmotsu manifold admitting holomorphically planar conformal vector field. Also, we prove that an field on a Sasaki–Kenmotsu manifold is solenoidal. Moreover, in a Sasaki–Kenmotsu manifold admitting field with , the field is the eigen vector of the Ricci operator with eigen value [ABSTRACT FROM AUTHOR]

In this paper, we initiate the study of impact of the existence of a unit vector ν , called a concurrent-recurrent vector field, on the geometry of a Riemannian manifold. Some examples of these vector fields are provided on Riemannian manifolds, and basic geometric properties of these vector fields are derived. Next, we characterize Ricci solitons on 3-dimensional Riemannian manifolds and gradient Ricci almost solitons on a Riemannian manifold (of dimension n) admitting a concurrent-recurrent vector field. In particular, it is proved that the Riemannian 3-manifold equipped with a concurrent-recurrent vector field is of constant negative curvature - α 2 when its metric is a Ricci soliton. Further, it has been shown that a Riemannian manifold admitting a concurrent-recurrent vector field, whose metric is a gradient Ricci almost soliton, is Einstein. [ABSTRACT FROM AUTHOR]

In this work, we study a new type of semi-symmetric non-metric connection on hyperbolic Kenmotsu manifold. Some Riemannian curvature's characteristics on hyperbolic Kenmotsu manifold are investigated. The properties of semi-symmetric, locally φ-symmetric and Ricci semi-symmetric hyperbolic Kenmotsu manifold endowed with a new type of semi-symmetric non-metric connection are evaluated. A semi-symmetric and Ricci semi-symmetric hyperbolic Kenmotsu manifold with a semi-symmetric nonmetric connection is also demonstrated, the Ricci soliton of data (g1, ξ♭, λ) is shrinking. Finally, we demonstrate our results with a 3-dimensional example. [ABSTRACT FROM AUTHOR]

The main object of the present paper is to discuss about the quarter symmetric metric connection on α-Sasakian Manifold with respect to Ricci Soliton. In this paper, firstly, we discuss the quarter symmetric metric connection on α-Sasakian manifold. Secondly, we elaborate the results of quarter symmetric metric connection on α-Sasakian manifold which admits Ricci Soliton and also flourish a non-trivial example of α-sasakian manifold and validate some of our results. Thirdly, we classify certain curvature properties of Ricci α-Sasakian manifold in regard to Quarter symmetric metric connection. Finally, we show Ricci Soliton on submanifold of α-Sasakian manifold in term of Levi-Civita and quarter symmetric metric connection. [ABSTRACT FROM AUTHOR]

In this study, we adapt the Schouten-van Kampen (SVK) connection on trans-Sasakian 3-manifolds. Then we consider semi-symmetric conditions on trans-Sasakian 3-manifolds with respect to the SVK connection admitting conformal Ricci soliton. [ABSTRACT FROM AUTHOR]

We found conditions on an n-dimensional Ricci soliton (M, g, u, λ) to be trivial. First, we showed that under an appropriate upper bound on the squared length of the covariant derivative of the potential field u, the Ricci soliton (M, g, u, λ) reduces to a trivial soliton. We also showed that appropriate upper and lower bounds on the Ricci curvature Ric (u, u) of a compact Ricci soliton (M, g, u, λ) with potential field u geodesic vector field makes it a trivial soliton. We showed that if the Ricci operator S of the Ricci soliton (M, g, u, λ) is invariant under the potential field u, then (M, g, u, λ) is trivial and the converse is also true. Finally, it was shown that if the potential field u of a connected Ricci soliton (M, g, u, λ) is a concurrent vector field, then the Ricci soliton is shrinking. [ABSTRACT FROM AUTHOR]

We provide a condition for a 2-Killing vector field on a compact Riemannian manifold to be Killing and apply the result to doubly warped product manifolds. We establish a connection between the property of a vector field on a doubly warped product manifold and its components on the factor manifolds to be Killing or 2-Killing. We also prove that a Killing vector field on the doubly warped product gives rise to a Ricci soliton factor manifold if and only if it is an Einstein manifold. If a component of a Killing vector field on the doubly warped product is of a gradient type, then, under certain conditions, the corresponding factor manifold is isometric to the Euclidean space. Moreover, we provide necessary and sufficient conditions for a doubly warped product to reduce to a direct product. As applications, we characterize the 2-Killing vector fields on the doubly warped spacetimes, particularly on the standard static spacetime and on the generalized Robertson–Walker spacetime. [ABSTRACT FROM AUTHOR]

In this paper, we show that a complete shrinking general Ricci flow soliton system with condition is compact if and only if is bounded on , where is the 2-form with components . We also prove that a complete shrinking general Ricci flow system soliton has finite fundamental group. [ABSTRACT FROM AUTHOR]

The aim of the paper is to study *-Ricci tensor in generalized Sasakian space form. We study the generalized Sasakian space form admitting *-conformal η-Ricci soliton and analyse the behaviour of the soliton. Also, we prove *-Ricci semi-symmetric and Pseudo *-Ricci semisymmetric generalized Sasakian space forms are *-Ricci at. [ABSTRACT FROM AUTHOR]

The main aim of the proposed paper is to investigate the lifts of Kenmotsu manifolds that admit NSNMC in the tangent bundle. We investigate several properties of the lifts of the curvature tensor, the conformal curvature tensor, and the conharmonic curvature tensor of Kenmotsu manifolds that admit NSNMC in the tangent bundle. We also study and discover that the lift of the Kenmotsu manifold that admit NSNMC is regular in the tangent bundle. Additionally, we find that the data provided by the lift of Ricci soliton on the lift of Ricci semi-symmetric Kenmotsu manifold that admits NSNMC in the tangent bundle are expanding. [ABSTRACT FROM AUTHOR]

The prime objective of the approach is to give geometric classifications of kk-almost Ricci solitons associated with paracontact manifolds. Let M2n+1(φ,ξ,η,g){M}^{2n+1}\left(\varphi ,\xi ,\eta ,g) be a paracontact metric manifold, and if a KK-paracontact metric gg represents a kk-almost Ricci soliton (g,V,k,λ)\left(g,V,k,\lambda ) and the potential vector field VV is Jacobi field along the Reeb vector field ξ\xi , then either k=λ−2nk=\lambda -2n, or gg is a kk-Ricci soliton. Next, we consider KK-paracontact manifold as a kk-almost Ricci soliton with the potential vector field VV is infinitesimal paracontact transformation or collinear with ξ\xi . We have proved that if a paracontact metric as a kk-almost Ricci soliton associated with the non-zero potential vector field VV is collinear with ξ\xi and the Ricci operator QQ commutes with paracontact structure φ\varphi , then it is Einstein of constant scalar curvature equals to −2n(2n+1)-2n\left(2n+1). Finally, we have deduced that a para-Sasakian manifold admitting a gradient kk-almost Ricci soliton is Einstein of constant scalar curvature equals to −2n(2n+1)-2n\left(2n+1).

In this article, we study the Ricci soliton on slant submanifolds of trans-Sasakian manifolds with a quarter symmetric non-metric connection. Moreover, we derive a lower-bound-type inequality for the slant submanifolds of trans-Sasakian manifolds with a quarter symmetric non-metric connection in terms of gradient Ricci solitons. We also characterize anti-invariant, invariant, quasi-umbilical submanifolds of trans-Sasakian manifolds with a quarter symmetric non-metric connection for which the same inequality case holds. Finally, we deduce the above inequalities in terms of a scalar concircular field on submanifolds of trans-Sasakian manifolds with a quarter symmetric non-metric connection.

Let $ (M, g) $ be an $ n $-dimensional (pseudo-)Riemannian manifold and $ TM $ be its tangent bundle $ TM $ equipped with the complete lift metric $ ^{C}g $. First, we define a Ricci quarter-symmetric metric connection $ \overline{\nabla } $ on the tangent bundle $ TM $ equipped with the complete lift metric $ ^{C}g $. Second, we compute all forms of the curvature tensors of $ \overline{\nabla } $ and study their properties. We also define the mean connection of $ \overline{\nabla } $. Ricci and gradient Ricci solitons are important topics studied extensively lately. Necessary and sufficient conditions for the tangent bundle $ TM $ to become a Ricci soliton and a gradient Ricci soliton concerning $ \overline{\nabla } $ are presented. Finally, we search conditions for the tangent bundle $ TM $ to be locally conformally flat with respect to $ \overline{\nabla } $.

The object of this paper is to study mixed quasi-Einstein manifolds of dimension n (for short, M (QE) n ) and their geometric properties. We discuss some geometric properties of M (QE) n such as Ricci semisymmetric, concircular Ricci pseudosymmetric and W i Ricci pseudosymmetric. Furthermore, we discuss the mixed quasi-Einstein manifolds with Ricci soliton and Riemann soliton, respectively. Finally, we construct an example of mixed quasi-Einstein manifold. [ABSTRACT FROM AUTHOR]

Let be an -dimensional (pseudo-)Riemannian manifold and be its tangent bundle equipped with the complete lift metric . First, we define a Ricci quarter-symmetric metric connection on the tangent bundle equipped with the complete lift metric . Second, we compute all forms of the curvature tensors of and study their properties. We also define the mean connection of . Ricci and gradient Ricci solitons are important topics studied extensively lately. Necessary and sufficient conditions for the tangent bundle to become a Ricci soliton and a gradient Ricci soliton concerning are presented. Finally, we search conditions for the tangent bundle to be locally conformally flat with respect to . [ABSTRACT FROM AUTHOR]

This paper examines almost Kenmotsu manifolds (briefly, AKMs) endowed with the almost Ricci–Yamabe solitons (ARYSs) and gradient ARYSs. The condition for an AKM with ARYS to be η -Einstein is established. We also show that an ARYS on Kenmotsu manifold becomes a Ricci–Yamabe soliton under certain restrictions. In this series, it is proven that a (2 n + 1) -dimensional (κ , μ) ′ -AKM equipped with a gradient ARYS is either locally isometric to ℍ n + 1 (− 4) × ℝ n or the Reeb vector field and the soliton vector field are codirectional. The properties of three-dimensional non-Kenmotsu AKMs endowed with a gradient ARYS are studied. [ABSTRACT FROM AUTHOR]

On a (pseudo-)Riemannian manifold, we consider an operator associated to a vector field and to an affine connection, which extends, in a certain way, the Hessian of a function, study its properties and point out its relation with statistical structures and gradient Ricci solitons. In particular, we provide the necessary and sufficient condition for it to be a Codazzi tensor field, and moreover, to give rise to a statistical structure. [ABSTRACT FROM AUTHOR]

The object of the present paper is to characterize the class of Kenmotsu manifolds which admits conformal η -Ricci soliton. Here, we have investigated the nature of the conformal η -Ricci soliton within the framework of Kenmotsu manifolds. It is shown that an η -Einstein Kenmotsu manifold admitting conformal η -Ricci soliton is an Einstein one. Moving further, we have considered gradient conformal η -Ricci soliton on Kenmotsu manifold and established a relation between the potential vector field and the Reeb vector field. Next, it is proved that under certain condition, a conformal η -Ricci soliton on Kenmotsu manifolds under generalized D-conformal deformation remains invariant. Finally, we have constructed an example for the existence of conformal η -Ricci soliton on Kenmotsu manifold. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The objective of this paper is to study the nature of Ricci soliton admitting various type of contact metric manifolds such as Kenmostu manifold, LP Sasakian manifold and LCS manifold. In this paper, it is proved that a Kemnotsu Ricci soliton is expanding, whereas the LP-Sasakian Ricci soliton is shrinking. Further, the conditions have been obtained on (LCS)n Ricci soliton to be expanding, shrinking and steady and the results are verified by suitable examples. It is also proved that the possible values of soliton constant is the set of all even integers Z2n and the set of negative integres Z respectively for Kenmostu Ricci soliton and LP Sasakian Ricci Soliton and if the (LCS) Ricci soliton is expanding then soliton constant lies on the interval (0;1) and for shrinking it lies on (1; 0). The Projectively flat cases for the above manifolds are also discussed to be expanding, shrinking and steady. Finally, we study these Ricci solitons admitting Ricci semi-symmetric condition R:S = 0 and prove that the soliton constant is an eigen value of metric tensor g with respect to associated vector field. Keywords: Ricci soliton, Contant metric manifolds, Kenmostu manifold. [ABSTRACT FROM AUTHOR]

Ricci solitons (RS) have an extensive background in modern physics and are extensively used in cosmology and general relativity. The focus of this work is to investigate Ricci almost solitons (RAS) on Lorentzian manifolds with a special metric connection called a semi-symmetric metric u -connection (SSM -connection). First, we show that any quasi-Einstein Lorentzian manifold having a SSM -connection, whose metric is RS, is Einstein manifold. A similar conclusion also holds for a Lorentzian manifold with SSM -connection admitting RS whose soliton vector Z is parallel to the vector u. Finally, we examine the gradient Ricci almost soliton (GRAS) on Lorentzian manifold admitting SSM -connection. [ABSTRACT FROM AUTHOR]

In this article, we have studied the behavior of the potential function along some geodesic in a Ricci soliton under some curvature restriction. In particular, we have showed that under some curvature restriction, the potential function reduced to a parabola along some geodesic. Furthermore, we have investigated the change of intersecting angles between the potential vector field and a geodesic in a Ricci soliton. Further, we have deduced the condition when the potential function becomes convex in a shrinking Ricci soliton. Finally, we have concluded the paper by showing the non-existence of convex potential in an expanding Ricci soliton having non-negative Ricci curvature. [ABSTRACT FROM AUTHOR]

The necessary requirements for half-lightlike and coisotropic lightlike submanifolds to be a Ricci soliton are obtained. Some examples of Ricci soliton half-lightlike and Ricci soliton coisotropic lightlike submanifolds are given. The Ricci soliton equation is investigated on totally geodesic, totally umbilical, and irrotational lightlike submanifolds. [ABSTRACT FROM AUTHOR]

In this paper, we study conformal submersions from Ricci solitons to Riemannian manifolds with non-trivial examples. First, we study some properties of the O'Neill tensor A in the case of conformal submersion. We also find a necessary and sufficient condition for conformal submersion to be totally geodesic and calculate the Ricci tensor for the total manifold of such a map with different assumptions. Further, we consider a conformal submersion F : M → N from a Ricci soliton to a Riemannian manifold and obtain necessary conditions for the fibers of F and the base manifold N to be Ricci soliton, almost Ricci soliton and Einstein. Moreover, we find necessary conditions for a vector field and its horizontal lift to be conformal on N and (Ker F ∗) ⊥ , respectively. Also, we calculate the scalar curvature of Ricci soliton M. Finally, we obtain a necessary and sufficient condition for F to be harmonic. [ABSTRACT FROM AUTHOR]

We study Ricci solitons on Golden Riemannian manifolds(GRM) with constant sectional curvatures. We define and study Golden Ricci soliton on Golden Riemannian manifolds(GRM). Further, we study Ricci soliton and Golden Ricci soliton on GRM with potential vector fields as concurrent, Killing, conformal Killing, and homothetic. Also, we provide examples of Ricci soliton and Golden Ricci soliton on GRM. [ABSTRACT FROM AUTHOR]

The prime object in this article is to study an almost η-Ricci–Yamabe soliton and gradient almost η-Ricci–Yamabe soliton within the framework of almost Kenmotsu manifolds. It is shown that normal almost Kenmotsu manifold admitting an almost η-Ricci–Yamabe soliton or gradient η-Ricci–Yamabe soliton is locally isometric to hyperbolic space ℍ2n+1(−1). Next, we prove that if a (κ, μ) almost Kenmotsu manifold admits an almost η-Ricci–Yamabe soliton, then the manifold is η-Einstein. Besides, we find the condition for non-normal almost Kenmotsu manifolds acknowledging gradient almost η-Ricci–Yamabe soliton. Moreover, an almost η-Ricci–Yamabe soliton on (κ, μ)′-almost Kenmotsu manifold has been studied. Lastly, we construct an example of a gradient almost η-Ricci–Yamabe soliton on a 3-dimensional Kenmotsu manifold. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Ricci solitons on tangent bundles with respect to the complete lift of a projective semi-symmetric connection using vertical and complete lifts of torqued vector fields. [ABSTRACT FROM AUTHOR]

In this paper we discussed the geometry of CR-submanifolds of a SQSasakian manifold. Next, we considered Chaki pseudo parallel as well as Deszcz pseudo parallel CR-submanifolds of SQ-Sasakian manifolds. Further we studied almost Ricci soliton and almost Yamabe soliton with torse forming vector field on a CR-submanifold of a SQ-Sasakian manifold using semi-symmetric metric connection. [ABSTRACT FROM AUTHOR]

In the present paper, we have discussed Ricci soliton and gradient Ricci soliton in spacetime. Through these discussions, some new valuable things were found which will further enrich the Ricci soliton and gradient Ricci soliton in spacetime. [ABSTRACT FROM AUTHOR]

In this paper, first we define Clairaut Riemannian map between Riemannian manifolds by using a geodesic curve on the base space and find necessary and sufficient conditions for a Riemannian map to be Clairaut with a nontrivial example. We also obtain necessary and sufficient condition for a Clairaut Riemannian map to be harmonic. Thereafter, we study Clairaut Riemannian map from Riemannian manifold to Ricci soliton with a nontrivial example. We obtain scalar curvatures of rangeF* and (rangeF*) ⊥ by using Ricci soliton. Further, we obtain necessary conditions for the leaves of rangeF* to be almost Ricci soliton and Einstein. We also obtain necessary condition for the vector field β to be conformal on rangeF* and necessary and sufficient condition for the vector field β to be Killing on (rangeF*)⊥, where β is a geodesic curve on the base space of Clairaut Riemannian map. Also, we obtain necessary condition for the mean curvature vector field of rangeF* to be constant. Finally, we introduce Clairaut antiinvariant Riemannian map from Riemannian manifold to Kähler manifold, and obtain necessary and sufficient condition for an antiinvariant Riemannian map to be Clairaut with a nontrivial example. Further, we find necessary condition for rangeF* to be minimal and totally geodesic. We also obtain necessary and sufficient condition for Clairaut antiinvariant Riemannian maps to be harmonic. [ABSTRACT FROM AUTHOR]

The Ricci equation and the soliton solutions are obtained in this paper for two coupled pendula in order to optimise its locomotion and structure. The robot is swinging repeatedly like a rope with successive movement steps along a mobile support. The cnoidal theory and a genetic algorithm are used to solve the problem via the Ricci solitons and the pseudospherical reduction of the rheological Zener equations. The Bäcklund transform is applied to Ricci equation to generate pseudospherical surfaces. [ABSTRACT FROM AUTHOR]

We provide a condition for a 2-Killing vector field on a compact Riemannian manifold to be Killing and apply the result to doubly warped product manifolds. We establish a connection between the property of a vector field on a doubly warped product manifold and its components on the factor manifolds to be Killing or 2-Killing. We also prove that a Killing vector field on the doubly warped product gives rise to a Ricci soliton factor manifold if and only if it is an Einstein manifold. If a component of a Killing vector field on the doubly warped product is of a gradient type, then, under certain conditions, the corresponding factor manifold is isometric to the Euclidean space. Moreover, we provide necessary and sufficient conditions for a doubly warped product to reduce to a direct product. As applications, we characterize the 2-Killing vector fields on the doubly warped spacetimes, particularly on the standard static spacetime and on the generalized Robertson–Walker spacetime.

We propose fundamental inequalities for contact pseudo-slant submanifolds of (ϵ) -para Sasakian space form employing generalized normalized δ -Casorati curvature. We characterize submanifolds for which equality cases hold and illustrate the main result with some applications. Further, we have considered a certain type of submanifold for a Ricci soliton and after computing its scalar curvature, developed an inequality to find correlations between intrinsic or extrinsic invariants. [ABSTRACT FROM AUTHOR]

In this paper, we establish two gap theorems for ends of smooth metric measure space (M^n, g,e^{-f}dv) with the Bakry-Émery Ricci tensor \operatorname {Ric}_{f}\!\ge -(n-1) in a geodesic ball B_{o}(R) with radius R and center o\in M^n. When \operatorname {Ric}_{f}\ge 0 and f has some degeneration outside B_{o}(R), we show that there exists an \epsilon =\epsilon (n,\sup _{B_{o}(1)}|f|) such that such a space has at most two ends if R\le \epsilon. When \operatorname {Ric}_{f}\ge \frac 12 and f(x)\le \frac 14d^2(x,B_{o}(R))+c for some constant c>0 outside B_{o}(R), we can also get the same gap conclusion. [ABSTRACT FROM AUTHOR]

Suppose (M n , g , f) is a complete shrinking gradient Ricci soliton. Assume that | R i c | < n - 2 2 n , where n ≥ 3 , then it has only one end. Similar results hold for the expanding gradient Ricci soliton. [ABSTRACT FROM AUTHOR]

It is well known that C H n has the structure of a solvable Lie group with left invariant metric of constant holomorphic sectional curvature. In this paper we give the full classification of all possible left invariant Riemannian metrics on this Lie group. We prove that each of those metrics is of constant negative scalar curvature, only one of them being Einstein (up to isometry and scaling). [ABSTRACT FROM AUTHOR]