Bi-Slant Submanifolds of Para Hermitian Manifolds

Abstract

In this paper we introduce the notion of bi-slant submanifolds of a para Hermitian manifold. They naturally englobe CR, semi-slant and hemi-slant submanifolds. We study their first properties and present a whole gallery of examples.2000 Mathematics Subject Classification : 53C40, 53C50. PreliminariesLet M be a 2n-dimensional semi-Riemannian manifold. If it is endowed with a structure (J, g), where J is a (1, 1) tensor, and g is a semi-defined metric, satisfying (2.1) J 2 X = X, g(JX, Y ) + g(X, JY ) = 0, for any vector fields X, Y on M , it is called a para Hermitian manifold. It is said to be para Kaehler if, in addition, ∇J = 0, where ∇ is the Levi-Civita connection of g. Let now M be a submanifold of ( M , J, g). The Gauss and Weingarten formulas are given by2010 Mathematics Subject Classification. 53C15, 53C25, 53C40, 53C50.

Pointwise PR-pseudo slant submanifolds of para-Kähler manifolds

“…The behavior of the λ equals to cos 2 θ(X) or − sinh 2 θ(X) or cosh 2 θ(X) depending on the nature of vector fields (that is, angle between spacelike-spacelike or timelike-spacelike or timelike-timelike). A variety of different possibilities for λ as slant constant-coefficient have been addressed in [1,2,3], depending on the behaviour of vector fields.…”

“…• PR-pseudo slant submanifold, if d 1 .d 2 = 0 and slant function λ is globally constant [1,19], in particular if d 1 = 0 and λ = 1 then M is a PRsubmanifold [8].…”

“…Srivastava and the author continued the study for paracosymplectic manifolds [20]. Recently Alegre-Carriazo [1,2] and Srivastava-Sharma [17,18] carried out the notion for slant, semi-slant, pseudo slant submanifolds in para geometries particularly in para-Hermitian and paracosymplectic manifolds, and presented analogies and differences between structure admitted semi-Riemannian and Riemannian metrics.…”

Warped product pointwise hemi-slant submanifolds of a Para-Kaehler manifold

“…al defined slant submanifols of a neutral Kaehler manifold in [6], while Alegre studied slant submanifolds of Lorentzian Sasakian and para-Sasakian manifolds in [1]. Recently, slant, bi-slant and quasi bi-slant submanifolds of (para)-Hermitian manifolds have been defined in [2,3,5]. As an extension of slant submanifolds, Etayo [17] defined the notion of pointwise slant submanifolds under the name of quasi-slant submanifolds.…”

Bi-slant submanifolds of an S manifold

“…A submanifold M of an almost Hermitian manifold is called a bi-slant submanifold if there exist two orthogonal slant distributions, D 1 and D 2 , on tangent bundle TM of M with slant angles θ 1 and θ 2 , respectively, such that one writes In the literature, there exist very interesting works on bi-slant submanifolds of various spaces [9][10][11][12][13][14]. An important aspect of slant submanifolds is that they can be considered as a generalization of semi-slant, hemi-slant and CR submanifolds.…”