Certain types of metrics on almost coKähler manifolds
15 Apr 2021-pp 1-17
TL;DR: In this article, it was shown that Bach flat almost coKahler manifold admits Ricci solitons, satisfying the critical point equation (CPE) or Bach flat.
Abstract: In this paper, we study an almost coKahler manifold admitting certain metrics such as $$*$$
-Ricci solitons, satisfying the critical point equation (CPE) or Bach flat. First, we consider a coKahler 3-manifold (M, g) admitting a $$*$$
-Ricci soliton (g, X) and we show in this case that either M is locally flat or X is an infinitesimal contact transformation. Next, we study non-coKahler $$(\kappa ,\mu )$$
-almost coKahler metrics as CPE metrics and prove that such a g cannot be a solution of CPE with non-trivial function f. Finally, we prove that a $$(\kappa , \mu )$$
-almost coKahler manifold (M, g) is coKahler if either M admits a divergence free Cotton tensor or the metric g is Bach flat. In contrast to this, we show by a suitable example that there are Bach flat almost coKahler manifolds which are non-coKahler.
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TL;DR: In this article , it was shown that if the metric of an almost co-K?hler manifold is a Riemann soliton with the soliton vector field, then the manifold is flat.
Abstract: The aim of the present paper is to characterize almost co-K?hler manifolds
whose metrics are the Riemann solitons. At first we provide a necessary and
sufficient condition for the metric of a 3-dimensional manifold to be
Riemann soliton. Next it is proved that if the metric of an almost
co-K?hler manifold is a Riemann soliton with the soliton vector field ?,
then the manifold is flat. It is also shown that if the metric of a (?,
?)-almost co-K?hler manifold with ? < 0 is a Riemann soliton, then the
soliton is expanding and ?, ?, ? satisfies a relation. We also prove that
there does not exist gradient almost Riemann solitons on (?, ?)-almost
co-K?hler manifolds with ? < 0. Finally, the existence of a Riemann soliton
on a three dimensional almost co-K?hler manifold is ensured by a proper
example.
2 citations
TL;DR: In this article, the authors studied generalized Ricci soliton in the framework of paracontact metric manifolds and proved that the scalar curvature r is constant and the squared norm of Ricci operator is constant.
Abstract: In the present paper, we study generalized Ricci soliton in the framework of paracontact metric manifolds. First, we prove that if the metric of a paracontact metric manifold M with $$Q\varphi =\varphi Q$$
is a generalized Ricci soliton (g, X) and if $$X
e 0$$
is pointwise collinear to $$\xi$$
, then M is K-paracontact and $$\eta$$
-Einstein. Next, we consider closed generalized Ricci soliton on K-paracontact manifold and prove that it is Einstein provided $$\beta (\lambda +2n\alpha )
e 1$$
. Next, we study K-paracontact metric as gradient generalized almost Ricci soliton and in this case we prove that (i) the scalar curvature r is constant and is equal to $$-2n(2n+1)$$
; (ii) the squared norm of Ricci operator is constant and is equal to $$4n^2(2n+1)$$
, provided $$\alpha \beta
e -1$$
.
1 citations
TL;DR: In this article, the authors considered CPE on almost f-cosymplectic manifolds and proved that the CPE conjecture is true for almost f cosymetric manifolds.
TL;DR: In this paper , a special class of contact pseudo-Riemannian manifold, called almost * {*} -η-Ricci solitons, is studied and shown to be an Einstein manifold if the potential vector field V is an infinitesimal contact transformation.
Abstract: 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.
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TL;DR: Prof. Pavel Brunovský belonged to the most important contemporary Slovak mathematicians whose significance has crossed over the borders of Slovakia a long time ago.
Abstract: On Dec. 14, 2018, Prof. RNDr. Pavel Brunovský, DrSc. passed away after a short but hard illness. Prof. Pavel Brunovský, Dec. 5, 1934, Vienna, has studied Mathematical Analysis at the Faculty of Natural Sciences of the Comenius University, Bratislava. After his studies, 1958, he started to work at the Institute of Technical Cybernetics of the Slovak Academy of Sciences, Bratislava (1959–1970). The fathers-founders of ITC SAS have realized very well that cybernetics cannot be done without good mathematicians. In years 1970–74 he worked at the Institute of Mathematics SAS, Bratislava. Since 1974 he was a collaborator of the nowadays Faculty of Mathematics, Physics, and Informatics of the Comenius University, Bratislava. In 1964 he defended his CSc. (= PhD.), in 1978 he did DrSc. (the highest scientific degree in Czecho-Slovakia). As a university professor, he was appointed in 1991 only after the political changes in our country. Prof. Pavel Brunovský belonged to the most important contemporary Slovak mathematicians whose significance has crossed over the borders of Slovakia a long time ago. He was a holder of the Pribina Cross (estimation of the state president), an emeritus member of the Learned Society of the Slovak Academy of Sciences, a member of the Academic Society, and a honorary emeritus member of the Learned Society of the Czech republic. After political-economical changes in our country at the beginning of Nineties, he become a member of the Presidium of the Slovak Academy of Sciences. In 2015 he was awarded by the Golden medal of the Slovak Academy of Sciences for the lifelong scientific opus. At the beginning of his scientific carrier, his main area of his research was theory and applications of optimal control. He has achieved a lot of remarkable and important results like synthesis regularity of optimal control for important classes of problems, or a canonical form for linear control systems, which nowadays is said to be Brunovský’s normal form (1970). Other important results are classifications of typical bifurcations of discrete dynamical system (1971), regularity of optimal feedback binding (1976–78), an attractor structure description of scalar reactive-diffusion equation (1985–1990). His name shines on more than 100 top scientific papers with hundreds of citations. Later he was concentrated to a related area of the mathematical research and soon he become an expert also in the theory of dynamical systems. Then he has joined these results with further
3 citations
TL;DR: In this article, it was shown that the gradient of scalar curvature of any Bochner-Kahler manifold is an infinitesimal harmonic transformation, and if it is conformal, then the curvature is constant.
Abstract: We have classified Bochner-Kahler manifolds of real dimension $$> 4$$
, which are also Bach flat. In the 4-dimensional case, we have shown that if the scalar curvature is harmonic, then it is constant. Finally, we show that the gradient of scalar curvature of any Bochner-Kahler manifold is an infinitesimal harmonic transformation, and if it is conformal, then the scalar curvature is constant.
2 citations
2 citations