scholarly journals A NATURAL CONNECTION ON A BASIC CLASS OF RIEMANNIAN PRODUCT MANIFOLDS

2012 ◽  
Vol 09 (07) ◽  
pp. 1250057 ◽  
Author(s):  
DOBRINKA GRIBACHEVA
Keyword(s):  
Curvature Tensor ◽  
Sufficient Conditions ◽  
Flat Connection ◽  
Riemannian Product ◽  
Natural Connection ◽  
Hermitian Geometry ◽  
Locally Conformal Kähler ◽  
Basic Class ◽  
Necessary And Sufficient ◽  
Locally Conformal

A Riemannian manifold M with an integrable almost product structure P is called a Riemannian product manifold. Our investigations are on the manifolds (M, P, g) of the largest class of Riemannian product manifolds, which is closed with respect to the group of conformal transformations of the metric g. This class is an analogue of the class of locally conformal Kähler manifolds in almost Hermitian geometry. In the present paper we study a natural connection D on (M, P, g) (i.e. DP = Dg = 0). We find necessary and sufficient conditions, the curvature tensor of D to have properties similar to the Kähler tensor in Hermitian geometry. We pay attention to the case when D has a parallel torsion. We establish that the Weyl tensors for the connection D and the Levi-Civita connection coincide as well as the invariance of the curvature tensor of D with respect to the usual conformal transformation. We consider the case when D is a flat connection. We construct an example of the considered manifold by a Lie group where D is a flat connection with non-parallel torsion.

scholarly journals Infinitesimal Transformations of Locally Conformal Kähler Manifolds

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 658
Author(s):  
Yevhen Cherevko ◽  
Volodymyr Berezovski ◽  
Irena Hinterleitner ◽  
Dana Smetanová
Keyword(s):  
Sufficient Conditions ◽  
Lie Derivative ◽  
Conformal Transformations ◽  
Projective Transformations ◽  
Integrability Conditions ◽  
Locally Conformal Kähler ◽  
Necessary And Sufficient ◽  
Locally Conformal ◽  
Lee Form ◽  
Infinitesimal Transformations

The article is devoted to infinitesimal transformations. We have obtained that LCK-manifolds do not admit nontrivial infinitesimal projective transformations. Then we study infinitesimal conformal transformations of LCK-manifolds. We have found the expression for the Lie derivative of a Lee form. We have also obtained the system of partial differential equations for the transformations, and explored its integrability conditions. Hence we have got the necessary and sufficient conditions in order that the an LCK-manifold admits a group of conformal motions. We have also calculated the number of parameters which the group depends on. We have proved that a group of conformal motions admitted by an LCK-manifold is isomorphic to a homothetic group admitted by corresponding Kählerian metric. We also established that an isometric group of an LCK-manifold is isomorphic to some subgroup of the homothetic group of the coresponding local Kählerian metric.

scholarly journals Infinitesimal Transformations of Locally Conformal Kähler Manifolds

2019 ◽  
Author(s):  
Yevhen Cherevko ◽  
Volodymyr Berezovski ◽  
Irena Hinterleitner ◽  
Dana Smetanová
Keyword(s):  
Partial Differential Equations ◽  
Sufficient Conditions ◽  
Lie Derivative ◽  
Conformal Transformations ◽  
Projective Transformations ◽  
Integrability Conditions ◽  
Necessary And Sufficient ◽  
Locally Conformal ◽  
Lee Form ◽  
Infinitesimal Transformations

The article is devoted to infinitesimal transformations. We have obtained that LCK-manifolds do not admit nontrivial infinitesimal projective transformations. Then we study infinitesimal conformal transformations of LCK-manifolds. We have found the expression for the Lie derivative of a Lee form. Also we have obtained the system of partial differential equations for the transformations, and explored its integrability conditions. Hence we have got the necessary and sufficient conditions in order that the an LCK-manifold admits a group of conformal motions. Also we have calculated the number of parameters which the group depends on. We have proved that a group of conformal motions admitted by an LCK-manifold is isomorphic to a homothetic group admitted by corresponding K\"{a}hlerian metric. We also established that an isometric group of an LCK-manifold is isomorphic to a some subgroup of homothetic group of the coresponding local K\"{a}hlerian metric.

scholarly journals On locally conformal Kähler space forms

1985 ◽  
Vol 8 (1) ◽  
pp. 69-74 ◽  
Author(s):  
Koji Matsumoto
Keyword(s):  
Space Form ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Covariant Differentiation ◽  
Space Forms ◽  
Riemannian Curvature Tensor ◽  
Locally Conformal Kähler ◽  
Necessary And Sufficient ◽  
Locally Conformal ◽  
Lee Form

Anm-dimensional locally conformal Kähler manifold (l.c.K-manifold) is characterized as a Hermitian manifold admitting a global closedl-formαλ(called the Lee form) whose structure(Fμλ,gμλ)satisfies∇νFμλ=−βμgνλ+βλgνμ−αμFνλ+αλFνμ, where∇λdenotes the covariant differentiation with respect to the Hermitian metricgμλ,βλ=−Fλϵαϵ,Fμλ=Fμϵgϵλand the indicesν,μ,…,λrun over the range1,2,…,m.For l. c. K-manifolds, I. Vaisman [4] gave a typical example and T. Kashiwada ([1], [2],[3]) gave a lot of interesting properties about such manifolds.In this paper, we shall study certain properties of l. c. K-space forms. In§2, we shall mainly get the necessary and sufficient condition that an l. c. K-space form is an Einstein one and the Riemannian curvature tensor with respect togμλwill be expressed without the tensor fieldPμλ. In§3, we shall get the necessary and sufficient condition that the length of the Lee form is constant and the sufficient condition that a compact l. c. K-space form becomes a complex space form. In the last§4, we shall prove that there does not exist a non-trivial recurrent l. c. K-space form.

scholarly journals NATURAL CONNECTION WITH TOTALLY SKEW-SYMMETRIC TORSION ON RIEMANNIAN ALMOST PRODUCT MANIFOLDS

2012 ◽  
Vol 09 (01) ◽  
pp. 1250003 ◽  
Author(s):  
DIMITAR MEKEROV ◽  
MANCHO MANEV
Keyword(s):  
Linear Connection ◽  
Riemannian Metric ◽  
Sufficient Condition ◽  
Product Manifold ◽  
Necessary And Sufficient Condition ◽  
Natural Connection ◽  
Almost Product Manifold ◽  
Hermitian Geometry ◽  
Levi Civita Connection ◽  
Necessary And Sufficient

On a Riemannian almost product manifold (M, P, g), we consider a linear connection preserving the almost product structure P and the Riemannian metric g and having a totally skew-symmetric torsion. We determine the class of the manifolds (M, P, g) admitting such a connection and prove that this connection is unique in terms of the covariant derivative of P with respect to the Levi-Civita connection. We find a necessary and sufficient condition the curvature tensor of the considered connection to have similar properties like the ones of the Kähler tensor in Hermitian geometry. We pay attention to the case when the torsion of the connection is parallel. We consider this connection on a Riemannian almost product manifold (G, P, g) constructed by a Lie group G.

scholarly journals Curvature properties of almost Ricci-like solitons with torse-forming vertical potential on almost contact b-metric manifolds

Filomat ◽  
2021 ◽  
Vol 35 (8) ◽  
pp. 2679-2691
Author(s):  
Mancho Manev
Keyword(s):  
Curvature Tensor ◽  
Ricci Tensor ◽  
Einstein Manifold ◽  
Arbitrary Dimension ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Ricci Solitons ◽  
Contact Distribution ◽  
Necessary And Sufficient

A generalization of ?-Ricci solitons is considered involving an additional metric and functions as soliton coefficients. The soliton potential is torse-forming and orthogonal to the contact distribution of the almost contact B-metric manifold. Then such a manifold can also be considered as an almost Einstein like manifold, a generalization of an ?-Einstein manifold with respect to both B-metrics and functions as coefficients. Necessary and sufficient conditions are found for a number of properties of the curvature tensor and its Ricci tensor of the studied manifolds. Finally, an explicit example of an arbitrary dimension is given and some of the results are illustrated.

scholarly journals The Concircular Curvature Tensor Of The Locally Conformal Kahler Manifold

2019 ◽  
Vol 24 (7) ◽  
pp. 110
Author(s):  
Ali Abdalmajed. Shihab1 ◽  
Dheyaa Nathim Ahmed2
Keyword(s):  
Curvature Tensor ◽  
Kähler Manifold ◽  
Riemannian Curvature ◽  
Classical Symmetry ◽  
Kahler Manifold ◽  
Symmetry Properties ◽  
Conharmonic Curvature Tensor ◽  
Locally Conformal Kähler ◽  
Concircular Curvature Tensor ◽  
Locally Conformal

In this research, we are calculated components conharmonic curvature tensor in some aspects Hermeation manifolding in particular of the Locally Conformal Kahler manifold. And we prove that this tensor possesses the classical symmetry properties of the Riemannian curvature. They also, establish relationships between the components of the tensor in this manifold   http://dx.doi.org/10.25130/tjps.24.2019.137

scholarly journals Generalized Helicoidal Surfaces in Euclidean 5-space

2021 ◽  
Vol 29 (3) ◽  
pp. 269-283
Author(s):  
Ali Uçum ◽  
Makoto Sakaki
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Curvature Tensor ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Normal Curvature ◽  
Helicoidal Surfaces ◽  
Necessary And Sufficient

Abstract In this paper, we study generalized helicoidal surfaces in Euclidean 5-space. We obtain the necessary and sufficient conditions for generalized helicoidal surfaces in Euclidean 5-space to be minimal, flat or of zero normal curvature tensor, which are ordinary differential equations. We solve those equations and discuss the completeness of the surfaces.

scholarly journals Warped product semi-slant submanifolds in locally conformal Kaehler manifolds

2017 ◽  
Vol 10 (2) ◽  
Author(s):  
Koji Matsumoto
Keyword(s):  
Warped Product ◽  
Sufficient Conditions ◽  
Hermitian Manifold ◽  
Kaehler Manifold ◽  
Slant Submanifold ◽  
Kaehler Manifolds ◽  
Slant Submanifolds ◽  
Necessary And Sufficient ◽  
Locally Conformal ◽  
Nearly Kaehler Manifold

In 1994, in [13], N. Papaghiuc introduced the notion of semi-slant submanifold in a Hermitian manifold which is a generalization of CR- and slant-submanifolds. In particular, he considered this submanifold in Kaehlerian manifolds, [13]. Then, in 2007, V. A. Khan and M. A. Khan considered this submanifold in a nearly Kaehler manifold and obtained interesting results, [11]. Recently, we considered semi-slant submanifolds in a locally conformal Kaehler manifold and gave a necessary and sufficient conditions for two distributions (holomorphic and slant) to be integrable. Moreover, we considered these submanifolds in a locally conformal Kaehler space form, [4]. In this paper, we define 2-kind warped product semi-slant submanifolds in a locally conformal Kaehler manifold and consider some properties of these submanifolds.

scholarly journals Curvature Properties of Almost Ricci-Like Solitons with Torse-Forming Vertical Potential on Almost Contact B-Metric Manifolds

2021 ◽  
Author(s):  
MANCHO MANEV
Keyword(s):  
Curvature Tensor ◽  
Ricci Tensor ◽  
Einstein Manifold ◽  
Arbitrary Dimension ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Ricci Solitons ◽  
Contact Distribution ◽  
Necessary And Sufficient

A generalization of $\eta$-Ricci solitons is considered involving an additional metric and functions as soliton coefficients. The soliton potential is torse-forming and orthogonal to the contact distribution of the almost contact B-metric manifold. Then such a manifold can also be considered as an almost Einstein-like manifold, a generalization of an $\eta$-Einstein manifold with respect to both B-metrics and functions as coefficients. Necessary and sufficient conditions are found for a number of properties of the curvature tensor and its Ricci tensor of the studied manifolds. Finally, an explicit example of an arbitrary dimension is given and some of the results are illustrated.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

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