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Semiparallel isometric immersions of 3-dimensional semisymmetric Riemannian manifolds

Ülo Lumiste (2003)

Czechoslovak Mathematical Journal

A Riemannian manifold is said to be semisymmetric if R ( X , Y ) · R = 0 . A submanifold of Euclidean space which satisfies R ¯ ( X , Y ) · h = 0 is called semiparallel. It is known that semiparallel submanifolds are intrinsically semisymmetric. But can every semisymmetric manifold be immersed isometrically as a semiparallel submanifold? This problem has been solved up to now only for the dimension 2, when the answer is affirmative for the positive Gaussian curvature. Among semisymmetric manifolds a special role is played by the foliated...

Semi-symmetric four dimensional neutral Lie groups

Ali Haji-Badali, Amirhesam Zaeim (2020)

Czechoslovak Mathematical Journal

The present paper is concerned with obtaining a classification regarding to four-dimensional semi-symmetric neutral Lie groups. Moreover, we discuss some geometric properties of these spaces. We exhibit a rich class of non-Einstein Ricci soliton examples.

Singer-Thorpe bases for special Einstein curvature tensors in dimension 4

Zdeněk Dušek (2015)

Czechoslovak Mathematical Journal

Let ( M , g ) be a 4-dimensional Einstein Riemannian manifold. At each point p of M , the tangent space admits a so-called Singer-Thorpe basis (ST basis) with respect to the curvature tensor R at p . In this basis, up to standard symmetries and antisymmetries, just 5 components of the curvature tensor R are nonzero. For the space of constant curvature, the group O ( 4 ) acts as a transformation group between ST bases at T p M and for the so-called 2-stein curvature tensors, the group Sp ( 1 ) ⊂ SO ( 4 ) acts as a transformation group...

Slant and Legendre curves in Bianchi-Cartan-Vranceanu geometry

Constantin Călin, Mircea Crasmareanu (2014)

Czechoslovak Mathematical Journal

We study Legendre and slant curves for Bianchi-Cartan-Vranceanu metrics. These curves are characterized through the scalar product between the normal at the curve and the vertical vector field and in the helix case they have a proper (non-harmonic) mean curvature vector field. The general expression of the curvature and torsion of these curves and the associated Lancret invariant (for the slant case) are computed as well as the corresponding variant for some particular cases. The slant (particularly...

Slant and pseudo-slant submanifolds in LCS -manifolds

Mehmet Atçeken, Shyamal Kumar Hui (2013)

Czechoslovak Mathematical Journal

We show new results on when a pseudo-slant submanifold is a LCS-manifold. Necessary and sufficient conditions for a submanifold to be pseudo-slant are given. We obtain necessary and sufficient conditions for the integrability of distributions which are involved in the definition of the pseudo-slant submanifold. We characterize the pseudo-slant product and give necessary and sufficient conditions for a pseudo-slant submanifold to be the pseudo-slant product. Also we give an example of a slant submanifold...

Slant submanifolds in cosymplectic manifolds

Ram Shankar Gupta, S. M. Khursheed Haider, A. Sharfuddin (2006)

Colloquium Mathematicae

We give some examples of slant submanifolds of cosymplectic manifolds. Also, we study some special slant submanifolds, called austere submanifolds, and establish a relation between minimal and anti-invariant submanifolds which is based on properties of the second fundamental form. Moreover, we give an example to illustrate our result.

Smooth metric measure spaces, quasi-Einstein metrics, and tractors

Jeffrey Case (2012)

Open Mathematics

We introduce the tractor formalism from conformal geometry to the study of smooth metric measure spaces. In particular, this gives rise to a correspondence between quasi-Einstein metrics and parallel sections of certain tractor bundles. We use this formulation to give a sharp upper bound on the dimension of the vector space of quasi-Einstein metrics, providing a different perspective on some recent results of He, Petersen and Wylie.

Some Classes of Lorentzian α -Sasakian Manifolds Admitting a Quarter-symmetric Metric Connection

Santu DEY, Buddhadev Pal, Arindam BHATTACHARYYA (2016)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

The object of the present paper is to study a quarter-symmetric metric connection in an Lorentzian α -Sasakian manifold. We study some curvature properties of an Lorentzian α -Sasakian manifold with respect to the quarter-symmetric metric connection. We study locally φ -symmetric, φ -symmetric, locally projective φ -symmetric, ξ -projectively flat Lorentzian α -Sasakian manifold with respect to the quarter-symmetric metric connection.

Some constructions of biharmonic maps on the warped product manifolds

Abdelmadjid Bennouar, Seddik Ouakkas (2017)

Commentationes Mathematicae Universitatis Carolinae

In this paper, we characterize a class of biharmonic maps from and between product manifolds in terms of the warping function. Examples are constructed when one of the factors is either Euclidean space or sphere.

Some Properties of Lorentzian α -Sasakian Manifolds with Respect to Quarter-symmetric Metric Connection

Santu DEY, Arindam BHATTACHARYYA (2015)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

The aim of this paper is to study generalized recurrent, generalized Ricci-recurrent, weakly symmetric and weakly Ricci-symmetric, semi-generalized recurrent, semi-generalized Ricci-recurrent Lorentzian α -Sasakian manifold with respect to quarter-symmetric metric connection. Finally, we give an example of 3-dimensional Lorentzian α -Sasakian manifold with respect to quarter-symmetric metric connection.

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