Infinitesimal Contact Transformations Research Articles

Geometry of almost contact metrics as almost ∗-Ricci solitons

In this paper, we give some characterizations by considering ∗-Ricci soliton as a Kenmotsu metric. We prove that if a Kenmotsu manifold represents an almost ∗-Ricci soliton and the potential vector field [Formula: see text] is a Jacobi along the Reeb vector field, then it is a steady ∗-Ricci soliton. Next, we show that a Kenmotsu matric endowed an almost ∗-Ricci soliton which is Einstein metric if it is [Formula: see text]-Einstein or the potential vector field [Formula: see text] is collinear to the Reeb vector field or [Formula: see text] is an infinitesimal contact transformation.

A study on conformal Ricci solitons and conformal Ricci almost solitons within the framework of almost contact geometry

The goal of this paper is to find some important Einstein manifolds using conformal Ricci solitons and conformal Ricci almost solitons. We prove that a Kenmotsu metric as a conformal Ricci soliton is Einstein if it is an $\eta$-Einstein or the potential vector field $V$ is infinitesimal contact transformation or collinear with the Reeb vector field $\xi$. Next, we prove that a Kenmotsu metric as gradient conformal Ricci almost soliton is Einstein if the Reeb vector field leaves the scalar curvature invariant. Finally, we have embellished an example to illustrate the existence of conformal Ricci soliton and gradient almost conformal Ricci soliton on Kenmotsu manifold.

Conformal vector field and gradient Einstein solitons on η-Einstein cosymplectic manifolds

In this paper, we characterize the [Formula: see text]-Einstein cosymplectic manifolds with the gradient Einstein solitons and the conformal vector fields. It is proven that if an [Formula: see text]-Einstein cosymplectic manifold [Formula: see text] of dimension [Formula: see text] with [Formula: see text] admits a gradient Einstein soliton, then either [Formula: see text] is Ricci flat or the gradient of Einstein potential function is pointwise collinear with the Reeb vector field of [Formula: see text]. Also, we prove that if [Formula: see text] admits a conformal vector field [Formula: see text], then either [Formula: see text] is Ricci flat or [Formula: see text] is homothetic. We also establish that a conformal vector field [Formula: see text] on [Formula: see text] is a strict infinitesimal contact transformation, provided the scalar curvature of [Formula: see text] is constant. Finally, the non-trivial examples of cosymplectic manifolds admitting the gradient Einstein solitons are given.

Geometry of paracontact metric as an almost Yamabe solitons

In this offering exposition, we intend to study paracontact metric manifold [Formula: see text] admitting almost Yamabe solitons. First, for a general paracontact metric manifold, it is proved that [Formula: see text] is Killing if the vector field [Formula: see text] is an infinitesimal contact transformation and that [Formula: see text] is [Formula: see text]-paracontact if [Formula: see text] is collinear with Reeb vector field. Second, we proved that a [Formula: see text]-paracontact manifold admitting a Yamabe gradient soliton is of constant curvature [Formula: see text] when [Formula: see text] and for [Formula: see text], the soliton is trivial and the manifold has constant scalar curvature. Moreover, for a paraSasakian manifold admitting a Yamabe soliton, we show that it has constant scalar curvature and [Formula: see text] is Killing when [Formula: see text]. Finally, we consider a paracontact metric [Formula: see text]-manifold with a non-trivial almost Yamabe gradient soliton. In the end, we construct two examples of paracontact metric manifolds with an almost Yamabe soliton.

Conformal vector fields on f-cosymplectic manifolds

In this paper, at first we characterize f-cosymplectic manifolds admitting conformal vector fields. Next, we establish that if a 3-dimensional f -cosymplectic manifold admits a homothetic vector field V, then either the manifold is of constant sectional curvature ?f?r, V is an infinitesimal contact transformation. Furthermore, we also investigate Ricci-Yamabe solitons with conformal vector fields on f-cosymplectic manifolds. At last, two examples are constructed to validate our outcomes

A study of conformal almost Ricci solitons on Kenmotsu manifolds

The goal of this paper is to study conformal almost Ricci solitons within the framework of Kenmotsu manifolds. First, we demonstrate that if the potential vector field is Jacobi along the Reeb vector field, then the soliton reduces to a conformal Ricci soliton. If the manifold is [Formula: see text]-Einstein Kenmotsu manifold, we show that either the manifold is of constant scalar curvature or the potential vector field is an infinitesimal contact transformation. In addition, if we consider the soliton vector field as a contact vector field, then either the gradient of [Formula: see text] is pointwise collinear with the Reeb vector field or the manifold becomes [Formula: see text]-Einstein. Lastly, we develop an example of a conformal almost Ricci soliton on the Kenmotsu manifold.

Conformal Ricci soliton and almost conformal Ricci soliton in paracontact geometry

In this paper, we study conformal Ricci soliton and almost conformal Ricci soliton within the framework of paracontact manifolds. Here, we have shown the characteristics of the soliton vector field and the nature of the manifold if para-Sasakian metric satisfies conformal Ricci soliton. We also demonstrate the feature of the soliton vector field V and scalar curvature when the para-Sasakian manifold admitting conformal Ricci soliton and vector field is pointwise collinear with the characteristic vector field [Formula: see text]. Next, we prove that if a K-paracontact manifold confesses a gradient conformal Ricci soliton, then it is Einstein. Next, we show that a para-Sasakian metric reveals with an almost conformal Ricci soliton that is either Einstein or [Formula: see text]-Einstein metric if the soliton vector field V is an infinitesimal contact transformation. Lastly, we decorate an example of conformal Ricci soliton on para-Sasakian manifold.

Almost *-η-Ricci solitons on Kenmotsu pseudo-Riemannian manifolds

Abstract In this paper, we aim to study a special type of metric called almost * {*} -η-Ricci soliton on the special class of contact pseudo-Riemannian manifold. First, we prove that a Kenmotsu pseudo-Riemannian metric as an * {*} -η-Ricci soliton is Einstein if either it is η-Einstein or the potential vector field V is an infinitesimal contact transformation. Further, we prove that if a Kenmotsu pseudo-Riemannian manifold admits an almost * {*} -η-Ricci soliton with a Reeb vector field leaving the scalar curvature invariant, then it is an Einstein manifold. Finally, we present an example of * {*} -η-Ricci solitons which illustrate our results.

Conformal η-Ricci almost solitons of Kenmotsu manifolds

The aim of this paper is to find some important classes of Einstein manifolds using conformal [Formula: see text]-Ricci solitons and conformal [Formula: see text]-Ricci almost solitons. We prove that a Kenmotsu metric as conformal [Formula: see text]-Ricci soliton is Einstein if it is [Formula: see text]-Einstein or the potential vector field [Formula: see text] is infinitesimal contact transformation or collinear with the Reeb vector field [Formula: see text]. Next, we prove that a Kenmotsu metric as gradient conformal [Formula: see text]-Ricci almost soliton is Einstein if the Reeb vector field leaves the scalar curvature invariants. Finally, we construct some examples to illustrate the existence of conformal [Formula: see text]-Ricci soliton, gradient almost conformal [Formula: see text]-Ricci soliton on Kenmotsu manifold.

K-cosymplectic metric as an η-Ricci soliton

Let M be an almost cosymplectic manifold such that the Reeb vector field is Killing. In this paper, it is proved that if the metric of M satisfying the [Formula: see text]-Einstein condition is an [Formula: see text]-Ricci soliton, then either M is Ricci flat or the potential vector field is an infinitesimal contact transformation. Also, a concrete example of cosymplectic [Formula: see text]-manifold admitting non-trivial Ricci and [Formula: see text]-Ricci solitons is constructed.

Some characterizations of -Einstein solitons on Sasakian manifolds

AbstractThe $\rho $ -Einstein soliton is a self-similar solution of the Ricci–Bourguignon flow, which includes or relates to some famous geometric solitons, for example, the Ricci soliton and the Yamabe soliton, and so on. This paper deals with the study of $\rho $ -Einstein solitons on Sasakian manifolds. First, we prove that if a Sasakian manifold M admits a nontrivial $\rho $ -Einstein soliton $(M,g,V,\lambda )$ , then M is $\mathcal {D}$ -homothetically fixed null $\eta $ -Einstein and the soliton vector field V is Jacobi field along trajectories of the Reeb vector field $\xi $ , nonstrict infinitesimal contact transformation and leaves $\varphi $ invariant. Next, we find two sufficient conditions for a compact $\rho $ -Einstein almost soliton to be trivial (Einstein) under the assumption that the soliton vector field is an infinitesimal contact transformation or is parallel to the Reeb vector field $\xi $ .

Almost Cosymplectic (k,mu )-metrics as eta-Ricci Solitons

In this paper, we study eta-Ricci solitons on almost cosymplectic (k,mu )-manifolds. As an application, it is proved that if an almost cosymplectic (k,mu )-metric with k<0 represents a Ricci soliton, then the potential vector field of the Ricci soliton is a strict infinitesimal contact transformation, and the corresponding almost cosymplectic manifold is locally isometric to a Lie group whose local structure is determined completely by k<0. In addition, a concrete example is constructed to illustrate the above result.

Almost $$\eta $$-Ricci solitons on Kenmotsu manifolds

We characterize the Einstein metrics in such broad classes of metrics as almost \(\eta \)-Ricci solitons and \(\eta \)-Ricci solitons on Kenmotsu manifolds, and generalize some known results. First, we prove that a Kenmotsu metric as an \(\eta \)-Ricci soliton is Einstein metric if either it is \(\eta \)-Einstein or the potential vector field V is an infinitesimal contact transformation or V is collinear to the Reeb vector field. Further, we prove that if a Kenmotsu manifold admits a gradient almost \(\eta \)-Ricci soliton with a Reeb vector field leaving the scalar curvature invariant, then it is an Einstein manifold. Finally, we present new examples of \(\eta \)-Ricci solitons and gradient \(\eta \)-Ricci solitons, which illustrate our results.

Sasakian 3-metric as a generalized Ricci-Yamabe soliton

In the present paper, we first investigate a Sasakian 3-metric as a quasi-Yamabe gradient soliton. In the sequel, extending the notions of quasi-Yamabe soliton and Ricci-Yamabe soliton, the notion of generalized Ricci-Yamabe soliton is introduced. It is shown that if (g, V, λ, α, β, γ) is a generalized gradient Ricci-Yamabe soliton on a complete Sasakian 3-manifold M with potential function f , then M is compact Einstein and locally isometric to a unit sphere. Moreover, the potential vector field V is an infinitesimal contact transformation and pointwise collinear with the characteristic vector field ξ. Further, if h is the Hodge-de Rham potential for V, then, upto a constant, f = h.

Almost Kenmotsu $$(k,\mu )'$$-manifolds with Yamabe solitons

Let $$(M^{2n+1},\phi ,\xi ,\eta ,g)$$ be a non-Kenmotsu almost Kenmotsu $$(k,\mu )'$$ -manifold. If the metric g represents a Yamabe soliton, then either $$M^{2n+1}$$ is locally isometric to the product space $$\mathbb {H}^{n+1}(-4)\times \mathbb {R}^n$$ or $$\eta $$ is a strict infinitesimal contact transformation. The later case can not occur if a Yamabe soliton is replaced by a gradient Yamabe soliton. Some corollaries of this theorem are given and an example illustrating this theorem is constructed.

η-Ricci solitons and almost η-Ricci solitons on para-Sasakian manifolds

In this paper, we study para-Sasakian manifold [Formula: see text] whose metric [Formula: see text] is an [Formula: see text]-Ricci soliton [Formula: see text] and almost [Formula: see text]-Ricci soliton. We prove that, if [Formula: see text] is an [Formula: see text]-Ricci soliton, then either [Formula: see text] is Einstein and in such a case the soliton is expanding with [Formula: see text] or it is [Formula: see text]-homothetically fixed [Formula: see text]-Einstein manifold and in such a case the soliton is shrinking with [Formula: see text]. We show the same conclusion when the para-Sasakian manifold [Formula: see text] is of [Formula: see text] and [Formula: see text] is an almost [Formula: see text]-Ricci soliton with [Formula: see text] as infinitesimal contact transformation. Finally, we prove that, if the para-Sasakian manifold [Formula: see text] of [Formula: see text] admits a gradient almost [Formula: see text]-Ricci soliton with [Formula: see text], then [Formula: see text] is Einstein. Suitable examples are constructed to justify our results.

*-Ricci soliton on (κ, μ)′-almost Kenmotsu manifolds

Abstract Let (M, g) be a non-Kenmotsu (κ, μ)′-almost Kenmotsu manifold of dimension 2n + 1. In this paper, we prove that if the metric g of M is a *-Ricci soliton, then either M is locally isometric to the product ℍn+1(−4)×ℝn or the potential vector field is strict infinitesimal contact transformation. Moreover, two concrete examples of (κ, μ)′-almost Kenmotsu 3-manifolds admitting a Killing vector field and strict infinitesimal contact transformation are given.

Ricci solitons on 3-dimensional cosymplectic manifolds

Abstract In this paper, we prove that if a 3-dimensional cosymplectic manifold M 3 admits a Ricci soliton, then either M 3 is locally flat or the potential vector field is an infinitesimal contact transformation.

First-order analytic propagation of satellites in the exponential atmosphere of an oblate planet

The paper offers the fully analytic solution to the motion of a satellite orbiting under the influence of the two major perturbations, due to the oblateness and the atmospheric drag. The solution is presented in a time-explicit form, and takes into account an exponential distribution of the atmospheric density, an assumption that is reasonably close to reality. The approach involves two essential steps. The first one concerns a new approximate mathematical model that admits a closed-form solution with respect to a set of new variables. The second step is the determination of an infinitesimal contact transformation that allows to navigate between the new and the original variables. This contact transformation is obtained in exact form, and afterwards a Taylor series approximation is proposed in order to make all the computations explicit. The aforementioned transformation accommodates both perturbations, improving the accuracy of the orbit predictions by one order of magnitude with respect to the case when the atmospheric drag is absent from the transformation. Numerical simulations are performed for a low Earth orbit starting at an altitude of 350 km, and they show that the incorporation of drag terms into the contact transformation generates an error reduction by a factor of 7 in the position vector. The proposed method aims at improving the accuracy of analytic orbit propagation and transforming it into a viable alternative to the computationally intensive numerical methods.

Airy, Beltrami, Maxwell, Einstein and Lanczos Potentials Revisited

When ${\cal{D}}$ is a linear partial differential operator of any order, a direct problem is to look for an operator ${\cal{D}}_1$ generating the compatibility conditions (CC) ${\cal{D}}_1\eta=0$ of ${\cal{D}}\xi=\eta$. We may thus construct a differential sequence with successive operators ${\cal{D}},{\cal{D}}_1,{\cal{D}}_2, ...$, where each operator is generating the CC of the previous one. Introducing the formal adjoint $ad( )$, we have ${\cal{D}}_i\circ {\cal{D}}_{i-1}=0 \Rightarrow ad({\cal{D}}_{i-1}) \circ ad({\cal{D}}_i)=0$ but $ad({\cal{D}}_{i-1})$ may not generate all the CC of $ad({\cal{D}}_i)$. When $D=K[d_1,...,d_n]=K[d]$ is the (non-commutative) ring of differential operators with coefficients in a differential field $K$, it gives rise by residue to a differential module $M$ over $D$. The homological extension modules $ext^i(M)=ext^i_D(M,D)$ with $ext^0(M)=hom_D(M,D)$ only depend on $M$ and are measuring the above gaps, independently of the previous differential sequence.The purpose of this rather technical paper is to compute them for certain Lie operators involved in the formal theory of Lie pseudogroups in arbitrary dimension $n$. In particular, we prove that the extension modules highly depend on the Vessiot structure constants $c$. When one is dealing with a Lie group of transformations or, equivalently, when ${\cal{D}}$ is a Lie operator of finite type, then we shall prove that $ext^i(M)=0, \forall 0\leq i \leq n-1$. It will follow that the Riemann-Lanczos and Weyl-Lanczos problems just amount to prove such a result for $i=2$ and arbitrary $n$ when ${\cal{D}}$ is the Killing or conformal Killing operator. We finally prove that ${ext}^i(M)=0, \forall i\geq 1$ for the Lie operator of infinitesimal contact transformations with arbitrary $n=2p+1$. Most of these new results have been checked by means of computer algebra.