Riemannian Submersions from Quaternionic Manifolds

Abstract

In this paper we define the concept of quaternionic submersion, we study its fundamental properties and give an example.

On Paraquaternionic Submersions Between Paraquaternionic Kähler Manifolds

“…We may suppose that ∇J a = 0, for any a ∈ {1, 2, 3}. Using this condition and the flatness of g, by (14) we get∇J a = 0, for any a ∈ {1, 2, 3}. Thus (T M,σ, G) is locally hyper paraKähler.…”

H-v-Semi-Slant Submersions From Almost Quaternionic Hermitian Manifolds

“…Then we call (M, E, g) an almost quaternionic Hermitian manifold [11]. Conveniently, the above basis {J 1 , J 2 , J 3 } satisfying (2.1) and (2.2) is said to be a quaternionic Hermitian basis.…”

“…Moreover, if (M, E M , g M ) is a quaternionic Kähler manifold (or a hyperkähler manifold), then we say that F is a quaternionic Kähler submersion (or a hyperkähler submersion) [11].…”

“…Given a C ∞ -submersion F from a (semi-)Riemannian manifold (M, g M ) onto a (semi-)Riemannian manifold (N, g N ), according to the conditions on the map F : (M, g M ) → (N, g N ), we can obtain the following: a semi-Riemannian submersion and a Lorentzian submersion [8], a Riemannian submersion ( [9], [16]), a slant submersion ( [6], [22]), an almost Hermitian submersion [24], a contact-complex submersion [10], a quaternionic submersion [11], an almost h-slant submersion [17], a semi-invariant submersion [23], an almost h-semiinvariant submersion [18], a semi-slant submersion [21], an almost h-semi-slant submersions [19], a v-semi-slant submersions [20], etc.…”

Pointwise Slant Submersions

“…One can see that this assumption implies that the vertical distribution is invariant. We note that almost Hermitian submersions have been extended to the almost contact manifolds [8], [13], locally conformal Kähler manifolds [17] and quaternion Kähler manifolds [14].…”