Semi-Slant Submersions from Almost Product Riemannian Manifolds

Abstract

Abstract. In this paper, we introduce semi-slant submersions from almost product Riemannian manifolds onto Riemannian manifolds. We give some examples, investigate the geometry of foliations which are arisen from the definition of a Riemannian submersion. We also find necessary and sufficient conditions for a semi-slant submersion to be totally geodesic. IntroductionGiven a C 8 -submersion π from a Riemannian manifold pM, gq onto a Riemannian manifold pB, g 1 q, there are several kinds of submersions according to the conditions on it: e.g. Riemannian submersion ([5] [15], the author studied the slant and semi-slant submanifolds of an almost product Riemannian manifold. Let pM, g, F q be an almost product Riemannian manifold. A Riemannian submersion π : pM, g, F q Ñ pN, g 1 q is called a slant submersion if the angle θpXq between F X and the space kerpπ˚q p is constant for any nonzero X P T p M and p P M [6]. We call θpXq a slant angle. The paper is organized as follows. In Section 2 we recall some notions needed for this paper. In Section 3 we give definition of semi-slant submersions and provide examples. We also investigate the geometry of leaves of the distributions. Finally, we give necessary and sufficient conditions for such submersions to be totally geodesic.2010 Mathematics Subject Classification: 53C15, 53B20, 53C43.

Slant submersions in paracontact geometry

“…Given a C ∞ −submersion ψ from a (semi)-Riemannian manifold (N, g N ) onto a (semi)-Riemannian manifold (B, g B ), according to the circumstances on the map ψ : (N, g N ) → (B, g B ), we get the following: a (semi)-Riemannian submersion ( [3,8,14,20]), an almost Hermitian submersion ( [27]), a paracontact submersion ( [9]), a paracontact paracomplex submersion ( [10]), a (para) quaternionic submersion ( [6,17]), a slant submersion ( [12,19,22,23]), an anti-invariant submersion ( [11,24]), a conformal semi-slant submersion ( [1,13]), a conformal anti-invariant submersion ( [2]), a hemi-slant submersion ( [25]), etc. As we know, Riemannian submersions were severally introduced by B. O'Neill ( [20]) and A.…”

Isotropic Riemannian submersions

“…Recently, B. Şahin [27] introduced the notion of anti-invariant Riemannian submersions which are Riemannian submersions from almost Hermitian manifolds such that the vertical distribution is anti-invariant under the almost complex structure of the total manifold. Later this notion has been extended for several cases, see: [1,2,4,8,9,16,19,20,23,26,29,30,32].…”

Conformal slant submersions from cosymplectic manifolds

“…Afterwards, he also defined slant submersions from almost Hermitian manifolds in [41]. After that, many geometers studied this area and obtained lots of results on the new topic (see [3,4,6,13,21,25,26,36,38,[44][45][46]). Recent developments on the notion of Riemannian submersion can be found in [43].…”