Some Result on Contact Pseudo-Slant Submanifolds of a Sasakian Manifold

Slant Submanifold ◽

Slant Submanifolds ◽

Integrability Conditions ◽

Definition Of ◽

In this paper, we study the geometry of the contact pseudo-slant submanifolds of a Sasakian manifold. We derive the integrability conditions of distributions in the definition of a contact pseudo-slant submanifold. The notions contact pseudo-slant product is defined, and the necessary and sufficient conditions for a submanifold to contact pseudo-slant product is given. Also, a non-trivial example is used to demonstrate that the method presented in this paper is effective.

On warped product semi-slant submanifolds of nearly Trans-Sasakian manifolds

Filomat ◽

10.2298/fil1817845a ◽

2018 ◽

Vol 32 (17) ◽

pp. 5845-5856

Author(s):

Akram Ali ◽

Ali Alkhaldi ◽

Rifaqat Ali

Keyword(s):

Warped Product ◽

Sufficient Conditions ◽

Characterization Theorem ◽

Sasakian Manifold ◽

Sasakian Manifolds ◽

Slant Submanifold ◽

Slant Submanifolds ◽

In this paper, we study warped product semi-slant submanifold of type M = NT xf N? with slant fiber, isometrically immersed in a nearly Trans-Sasakian manifold by finding necessary and sufficient conditions in terms of Weingarten map. A characterization theorem is proved as main result.

The geometry of contact pseudo-slant submanifolds of a Sasakian manifold

Filomat ◽

10.2298/fil1917551y ◽

2019 ◽

Vol 33 (17) ◽

pp. 5551-5559

Author(s):

Ümit Yıldırım

Keyword(s):

Tensor Field ◽

Sufficient Conditions ◽

Sasakian Manifold ◽

Slant Submanifold ◽

Slant Submanifolds ◽

In this paper, we study the geometry of the contact pseudo-slant submanifolds of a Sasakian manifold. We verify some properties of the components of the tensor field acting on that kind of submanifold and find out the necessary and sufficient conditions for them to be parallel. Also, necessary and sufficient conditions are given for a submanifold to be a pseudo-slant submanifold, contact pseudo-slant product, D?, D? and mixed-geodesic in Sasakian manifold.

Radical transversal SCR-lightlike submanifolds of indefinite Sasakian manifolds

Filomat ◽

10.2298/fil2108585s ◽

2021 ◽

Vol 35 (8) ◽

pp. 2585-2594

Author(s):

S.S. Shukla ◽

Akhilesh Yadav

Keyword(s):

Sufficient Conditions ◽

Characterization Theorem ◽

Sasakian Manifold ◽

Sasakian Manifolds ◽

Totally Geodesic ◽

Integrability Conditions ◽

Cauchy Riemann ◽

Lightlike Submanifolds

In this paper, we introduce the notion of radical transversal screen Cauchy-Riemann (SCR)- lightlike submanifolds of indefinite Sasakian manifolds giving characterization theorem with some nontrivial examples of such submanifolds. Integrability conditions of distributions D1, D2, D and D? on radical transversal SCR-lightlike submanifolds of an indefinite Sasakian manifold have been obtained. Further, we obtain necessary and sufficient conditions for foliations determined by above distributions to be totally geodesic.

Generalized Timelike Mannheim Curves in Minkowski Space-Time

Mathematical Problems in Engineering ◽

10.1155/2011/539378 ◽

2011 ◽

Vol 2011 ◽

pp. 1-19 ◽

Cited By ~ 3

Author(s):

M. Akyig~it ◽

S. Ersoy ◽

İ. Özgür ◽

M. Tosun

Keyword(s):

Minkowski Space ◽

Sufficient Conditions ◽

Space Time ◽

Minkowski Space Time ◽

Mannheim Curve ◽

Definition Of ◽

We give the definition of generalized timelike Mannheim curve in Minkowski space-time . The necessary and sufficient conditions for the generalized timelike Mannheim curve are obtained. We show some characterizations of generalized Mannheim curve.

Fair bets and inductive probabilities

Journal of Symbolic Logic ◽

10.2307/2268222 ◽

1955 ◽

Vol 20 (3) ◽

pp. 263-273 ◽

Cited By ~ 110

Author(s):

John G. Kemeny

Keyword(s):

Sufficient Conditions ◽

The Other ◽

Inductive Probability ◽

Other Hand ◽

Frequency Interpretation ◽

Definition Of ◽

Degree Of Confirmation

The question of what constitutes fairness in betting quotients has been studied by Ramsey, deFinetti, and Shimony. Thanks to their combined efforts we now have a satisfactory definition of fairness.On the other hand, the explication of the concept of degree of confirmation (inductive probability) has progressed rapidly in recent years, thanks primarily to Carnap. This explication has usually proceeded by laying down the axioms for frequency-probabilities, and elaborating on these. While in the case where a frequency interpretation is intended these axioms are clearly justified, in our case they have been laid down without any justification. Carnap's presentation has been criticized for just this reason.The purpose of this paper is to show that the probability axioms are necessary and sufficient conditions to assure that the degrees of confirmation form a set of fair betting quotients. In addition it will be shown that one additional, highly controversial, axiom is precisely the condition needed to assure that not only deFinetti's weaker criterion but Shimony's criterion of fairness is also satisfied.

Para-CR structures on almost paracontact metric manifolds

Journal of Applied Analysis ◽

10.1515/jaa-2014-0012 ◽

2014 ◽

Vol 20 (2) ◽

Cited By ~ 6

Author(s):

Joanna Wełyczko

Keyword(s):

Sectional Curvature ◽

Sufficient Conditions ◽

Sasakian Manifold ◽

Constant Sectional Curvature ◽

Conformally Flat ◽

Cr Manifolds ◽

Cosymplectic Manifolds ◽

Cr Structures ◽

AbstractAlmost paracontact metric manifolds are the famous examples of almost para-CR manifolds. We find necessary and sufficient conditions for such manifolds to be para-CR. Next we examine these conditions in certain subclasses of almost paracontact metric manifolds. Especially, it is shown that normal almost paracontact metric manifolds are para-CR. We establish necessary and sufficient conditions for paracontact metric manifolds as well as for almost para-cosymplectic manifolds to be para-CR. We find also basic curvature identities for para-CR paracontact metric manifolds and study their consequences. Among others, we prove that any para-CR paracontact metric manifold of constant sectional curvature and of dimension greater than 3 must be para-Sasakian and its curvature equal to -1. The last assertion does not hold in dimension 3. We show that a conformally flat para-Sasakian manifold is of constant sectional curvature equal to -1. New classes of examples of para-CR manifolds are established.

Conceptualising strategic litigation

Oñati Socio-legal Series ◽

10.35295/osls.iisl/0000-0000-0000-1226 ◽

2021 ◽

Author(s):

Kris van der Pas

Keyword(s):

Research Method ◽

Sufficient Conditions ◽

Database Research ◽

Conceptual Relations ◽

Definition Of ◽

Exact Definition ◽

Strategic Litigation

This article conceptualises the term “strategic litigation” in order to provide for a definition of it. Strategic litigation is a tool increasingly used in Europe by individuals and organisations to attain different objectives. Next to that, there is increasing academic attention for the topic. Nevertheless, the exact definition of “strategic litigation” remains unclear. Therefore, this article uses the research method of conceptualisation as well as a database research and additional literature to provide for a definition. It looks firstly at the background concept, involving the range of meanings associated with the term “strategic litigation”, after which a systematised concept is formed. Thereby, use is made of the “necessary and sufficient conditions” (NSC) approach, to develop the conditions necessary and/or sufficient for a case to fit within the category of strategic litigation. Moreover, the external conceptual relations of the term are explored.

Isomorphism of Binary Operations in Differential Geometry

Symmetry ◽

10.3390/sym12101634 ◽

2020 ◽

Vol 12 (10) ◽

pp. 1634

Author(s):

Nikita E. Barabanov

Keyword(s):

Sufficient Conditions ◽

Canonical Representation ◽

Function Parameter ◽

Binary Operations ◽

Möbius Addition ◽

Algebraic Automorphism ◽

Definition Of ◽

Metric Tensors

We consider smooth binary operations invariant with respect to unitary transformations that generalize the operations of the Beltrami–Klein and Beltrami–Poincare ball models of hyperbolic geometry, known as Einstein addition and Möbius addition. It is shown that all such operations may be recovered from associated metric tensors that have a canonical form. Necessary and sufficient conditions for canonical metric tensors to generate binary operations are found. A definition of algebraic isomorphism of binary operations is given. Necessary and sufficient conditions for binary operations to be isomorphic are provided. It is proved that every algebraic automorphism gives rise to isomorphism of corresponding gyrogroups. Necessary and sufficient conditions in terms of metric tensors for binary operations to be isomorphic to Euclidean addition are given. The problem of binary operations to be isomorphic to Einstein addition is also solved in terms of necessary and sufficient conditions. We also obtain necessary and sufficient conditions for binary operations having the same function-parameter in the canonical representation of metric tensors to be isomorphic.

On extensions of the Riemann and Lebesgue Integrals by Nets

Canadian Mathematical Bulletin ◽

10.4153/cmb-1975-002-1 ◽

1975 ◽

Vol 18 (1) ◽

pp. 7-17 ◽

Cited By ~ 1

Author(s):

O. S. Bellamy ◽

H. W. Ellis

Keyword(s):

General Theory ◽

Sufficient Conditions ◽

Closed Set ◽

Denjoy Integral ◽

Definition Of ◽

Open Extension

In this note our principal interest is in using nets to give spaces of non-absolutely convergent integrals as extensions of the spaces of absolutely convergent Riemann and Lebesgue integrals. For this purpose we develop a general theory of extensions, by nets, of functions defined on the open intervals with closures in the complement of a fixed closed set, the nets being directed by inclusion for finite disjoint collections of such intervals. Two cases are considered leading to open extension (OE-) and conditional open extension (COE-) nets, the latter being subnets of the former. Necessary and sufficient conditions for the convergence of the OE- and COE-nets are given, those for the COE-nets being similar to conditions that arise in the definition of the restricted Denjoy integral. Properties of inner continuity, weak additivity and the existence of a continuous integral are defined and studied. These relate to the more specialized nets that are suitable for the extension of integrals.

Additivity of the Pn-Integral

Canadian Journal of Mathematics ◽

10.4153/cjm-1978-067-2 ◽

1978 ◽

Vol 30 (4) ◽

pp. 783-796 ◽

Cited By ~ 1

Author(s):

George Cross

Keyword(s):

Sufficient Conditions ◽

Slight Modification ◽

Definition Of ◽

It is known that the Pn-tegral as originally defined is not additive on abutting intervals. This paper offers a slight modification in the definition of the integral and develops necessary and sufficient conditions for the integral to be additive.