scholarly journals Biwarped product submanifolds of a Kähler manifold

Filomat ◽  
2018 ◽  
Vol 32 (7) ◽  
pp. 2349-2365 ◽  
Author(s):  
Hakan Taştan
Keyword(s):  
Fundamental Form ◽  
Warped Product ◽  
Kähler Manifold ◽  
Second Fundamental Form ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Equality Case ◽  
The Second Fundamental Form ◽  
Special Cases ◽  
Necessary And Sufficient

We study biwarped product submanifolds which are special cases of multiply warped product submanifolds in K?hler manifolds. We observe the non-existence of such submanifolds under some circumstances. We show that there exists a non-trivial biwarped product submanifold of a certain type by giving an illustrate example. We also give a necessary and sufficient condition for such submanifolds to be locally trivial. Moreover, we establish an inequality for the squared norm of the second fundamental form in terms of the warping functions for such submanifolds. The equality case is also discussed.

scholarly journals Spherical Product Surfaces in E4

2012 ◽  
Vol 20 (1) ◽  
pp. 41-54 ◽  
Author(s):  
Betül Bulca ◽  
Kadri Arslan ◽  
Bengü (Kılıç) Bayram ◽  
Günay Oztürk
Keyword(s):  
Fundamental Form ◽  
Second Fundamental Form ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Second Fundamental Form ◽  
Curvature Ellipse ◽  
Necessary And Sufficient ◽  
Rotational Surfaces

Abstract In the present study we calculate the coefficients of the second fundamental form and curvature ellipse of spherical product surfaces in E4. Otsuki rotational surfaces and Ganchev-Milousheva rotational surfaces are the special type of spherical product surfaces in E4. Further, we give necessary and sufficient condition for the origin of NpM to lie on the curvature ellipse of such surfaces. Finally we get the necessary condition for Ganchev-Milousheva rotational surfaces in E4 to become flat or Chen type. We also give some examples of the projections of these surfaces in E3

Geometric aspects of CR-warped product submanifolds of T-manifolds

2017 ◽  
Vol 10 (04) ◽  
pp. 1750067
Author(s):  
Akram Ali ◽  
Wan Ainun Mior Othman
Keyword(s):  
Warped Product ◽  
Characterization Theorem ◽  
Second Fundamental Form ◽  
Warping Function ◽  
Sufficient Condition ◽  
Warped Products ◽  
Necessary And Sufficient Condition ◽  
Space Forms ◽  
The Second Fundamental Form ◽  
Necessary And Sufficient

In this paper, we study CR-warped product submanifolds of [Formula: see text]-manifolds. We prove that the CR-warped product submanifolds with invariant fiber are trivial warped products and provide a characterization theorem of CR-warped products with anti-invariant fiber of [Formula: see text]-manifolds. Moreover, we develop an inequality of CR-warped product submanifolds for the second fundamental form in terms of warping function and the equality cases are considered. Also, we find a necessary and sufficient condition for compact oriented CR-warped products turning into CR-products of [Formula: see text]-space forms.

scholarly journals On warped product bi-slant submanifolds of Kenmotsu manifolds

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Siraj Uddin ◽  
Ion Mihai ◽  
Adela Mihai
Keyword(s):  
Fundamental Form ◽  
Warped Product ◽  
Second Fundamental Form ◽  
Warping Function ◽  
Equality Case ◽  
General Inequality ◽  
Kaehler Manifolds ◽  
Slant Submanifolds ◽  
The Second Fundamental Form ◽  
Kenmotsu Manifolds

Chen (2001) initiated the study of CR-warped product submanifolds in Kaehler manifolds and established a general inequality between an intrinsic invariant (the warping function) and an extrinsic invariant (second fundamental form).In this paper, we establish a relationship for the squared norm of the second fundamental form (an extrinsic invariant) of warped product bi-slant submanifolds of Kenmotsu manifolds in terms of the warping function (an intrinsic invariant) and bi-slant angles. The equality case is also considered. Some applications of derived inequality are given.

scholarly journals Chen’s improved inequality for pointwise hemi-slant warped products in Kaehler manifolds

Filomat ◽  
2020 ◽  
Vol 34 (3) ◽  
pp. 807-814
Author(s):  
Monia Naghi ◽  
Mica Stankovic ◽  
Fatimah Alghamdi
Keyword(s):  
Fundamental Form ◽  
Warped Product ◽  
Second Fundamental Form ◽  
Warping Function ◽  
Warped Products ◽  
Equality Case ◽  
Kaehler Manifolds ◽  
Slant Submanifolds ◽  
The Second Fundamental Form

Recently, B.-Y. Chen discovered a technique to find the relation between second fundamental form and the warping function of warped product submanifolds. In this paper, we extend our further study of [24] by giving non-trivial examples of warped product pointwise hemi-slant submanifolds. Finally, we establish a sharp estimation for the squared norm of the second fundamental form ||h||2 in terms of the warping function f. The equality case is also investigated.

scholarly journals Warped product submanifolds of Kenmotsu manifolds with slant fiber

Filomat ◽  
2018 ◽  
Vol 32 (6) ◽  
pp. 2115-2126 ◽  
Author(s):  
Monia Naghi ◽  
Siraj Uddin ◽  
Falleh Al-Solamy
Keyword(s):  
Fundamental Form ◽  
Warped Product ◽  
Second Fundamental Form ◽  
Warping Function ◽  
Gradient Vector ◽  
Equality Case ◽  
Slant Submanifolds ◽  
The Second Fundamental Form ◽  
Kenmotsu Manifolds ◽  
Two Factors

Recently, wehave discussed the warped product pseudo-slant submanifolds of the typeM?xfM? of Kenmotsu manifolds. In this paper, we study other type of warped product pseudo-slant submanifolds by reversing these two factors in Kenmotsu manifolds. The existence of such warped product immersions is proved by a characterization. Also, we provide an example of warped product pseudo-slant submanifolds. Finally, we establish a sharp estimation such as ||h||2?2pcos2?(||??(ln f)||2-1) for the squared norm of the second fundamental form khk2, in terms of the warping function f, where ??(ln f) is the gradient vector of the function ln f. The equality case is also discussed.

scholarly journals Warped product pointwise pseudo-slant submanifolds of locally product Riemannian manifolds

Filomat ◽  
2018 ◽  
Vol 32 (2) ◽  
pp. 423-438 ◽  
Author(s):  
Lamia Alqahtani ◽  
Siraj Uddina
Keyword(s):  
Riemannian Manifold ◽  
Fundamental Form ◽  
Riemannian Manifolds ◽  
Warped Product ◽  
Characterization Theorem ◽  
Second Fundamental Form ◽  
Warping Function ◽  
Equality Case ◽  
Slant Submanifolds ◽  
The Second Fundamental Form

In [3], it was shown that there are no warped product submanifolds of a locally product Riemannian manifold such that the spherical submanifold of a warped product is proper slant. In this paper, we introduce the notion of warped product submanifolds with a slant function and show that there exists a class of non-trivial warped product submanifolds of a locally product Riemannian manifold such that the spherical submanifold is pointwise slant by giving some examples. We present a characterization theorem and establish a sharp relationship between the squared norm of the second fundamental form and the warping function in terms of the slant function for such warped product submanifolds of a locally product Riemannian manifold. The equality case is also considered.

Bi-warped product submanifolds of Kenmotsu manifolds and their applications

2019 ◽  
Vol 16 (01) ◽  
pp. 1950001 ◽  
Author(s):  
Siraj Uddin ◽  
Ali H. Alkhaldi
Keyword(s):  
Lower Bound ◽  
Fundamental Form ◽  
Warped Product ◽  
Second Fundamental Form ◽  
Kenmotsu Manifold ◽  
Equality Case ◽  
The Second Fundamental Form ◽  
Kenmotsu Manifolds

In this paper, we study bi-warped product submanifolds of the form [Formula: see text] in a Kenmotsu manifold. We obtain a lower bound for the squared norm of the second fundamental form of a bi-warped product submanifold such as [Formula: see text], where [Formula: see text] and [Formula: see text] and [Formula: see text] are the warping functions on [Formula: see text]. The equality case is also considered.

FOURTH MOMENT STRUCTURE OF THE GARCH(p,q) PROCESS

1999 ◽  
Vol 15 (6) ◽  
pp. 824-846 ◽  
Author(s):  
Changli He ◽  
Timo Teräsvirta
Keyword(s):  
Statistical Theory ◽  
Autocorrelation Function ◽  
Fourth Moment ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Special Cases ◽  
The Moment ◽  
Necessary And Sufficient

In this paper, a necessary and sufficient condition for the existence of the unconditional fourth moment of the GARCH(p,q) process is given and also an expression for the moment itself. Furthermore, the autocorrelation function of the centered and squared observations of this process is derived. The statistical theory is further illustrated by a few special cases such as the GARCH(2,2) process and the ARCH(q) process.

On the general bilinear time series model

1988 ◽  
Vol 25 (3) ◽  
pp. 553-564 ◽  
Author(s):  
Jian Liu ◽  
Peter J. Brockwell
Keyword(s):  
Time Series ◽  
Stationary Solution ◽  
Time Series Model ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Special Cases ◽  
Non Linear ◽  
Necessary And Sufficient ◽  
Special Case

A sufficient condition is derived for the existence of a strictly stationary solution of the general bilinear time series equations. The condition is shown to reduce to the conditions of Pham and Tran (1981) and Bhaskara Rao et al. (1983) in the special cases which they consider. Under the condition specified, a solution is constructed which is shown to be causal, stationary and ergodic. It is moreover the unique causal solution and the unique stationary solution of the defining equations. In the special case when the defining equations contain no non-linear terms, our condition reduces to the well-known necessary and sufficient condition for existence of a causal stationary solution.

scholarly journals Einstein doubly warped product manifolds with semi-symmetric metric connection

2020 ◽  
Vol 57 ◽  
pp. 7-24
Author(s):  
Punam Gupta ◽  
Abdoul Salam Diallo
Keyword(s):  
Warped Product ◽  
Sufficient Condition ◽  
Product Manifold ◽  
Necessary And Sufficient Condition ◽  
Class A ◽  
Warped Product Manifold ◽  
Metric Connection ◽  
Necessary And Sufficient

In this paper, we study the doubly warped product manifolds with semi-symmetric metric connection. We derive the curvature formulas for doubly warped product manifold with semi-symmetric metric connection in terms of curvatures of components of doubly warped product manifolds. We also prove the necessary and sufficient condition for a doubly warped product manifold to be a warped product manifold. We obtain some results for an Einstein doubly warped product manifold and Einstein-like doubly warped product manifold of class A with respect to a semi-symmetric metric connection.

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