Slant Submanifolds of Semi-Riemannian Manifolds

Abstract

In this study, we search slant submanifolds of almost product semi-Riemannian manifold. Accordingly, we investigate slant submanifolds of semi-Riemannian manifold by making the classifications as slant Riemannian, slant semi-Riemannian and slant lightlike submanifold. Moreover, we add the theorems which characterize existence of slant submanifold.

Some Results for Bislant and Semi-Slant Submanifolds of Semi-Riemannian Manifolds

“…Şahin (2008) also stated the slant submanifold for lightlike submanifolds of indefinite Hermitian manifolds. Moreover, Aydın and Çöken (2013) worked the almost product structure on semi-Riemannian manifold similarly to Yano and Kon (1984)'s study. According to that, let be a -dimensional manifold.…”

“…Let semi-Riemannian manifold be an immersed submanifold of almost product semi-Riemannian manifold . For any ∈ , ∈ � ′ �, is a slant semi-Riemannian submanifold of if and only if the angle between and F is constant where F is an almost product structure (Aydın and Çöken, 2013). Now, we can give the slant submanifolds for lightlike submanifolds.…”

“…where � � is a lightlike transversal bundle, ′ is a complementary distribution to � � on � � and is a semi-Riemannian distribution. Thus, we describe slant lightlike submanifold as following (Duggal and Bejancu, 1996;Aydın and Çöken, 2013 (Aydın and Çöken, 2013). (see Şahin (2008) for indefinite Hermitian manifolds)…”

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.…”

Nonexistence of $$\mathcal {P}\mathcal {R}$$PR-semi-slant warped product submanifolds in paracosymplectic manifolds

“…In light of that, many authors discussed the existence and nonexistence of such warped product submanifolds in Lorentzian settings [33,42]. Recently, Chen and Munteanu [13], the authors of [30,31] and Aydin and Cöken [2] initiated the study of the geometry of pseudo-Riemannian warped products submanifolds in para-Kähler manifolds, paracontact manifolds and slant submanifold in semi-Riemannian manifolds, respectively. Motivated by the above studies, in the present paper, we investigate the existence or nonexistence of PR-semi-slant warped product submanifolds in paracosymplectic manifolds.…”