Semi-Invariant Riemannian Submersions with Semi-Symmetric Non-Metric Connection

Abstract

A conventional way to compare two manifolds is by defining smooth maps from one manifold to another. One such map is submersion, whose rank equals to the dimension of the target manifold. Riemannian submersion between Riemannian submanifolds were first introduced by O’ Neill and Gray [1, 2]. Later many authors studied different geometric properties of the Riemannian submersions [3], semi-slant submersions [4–6], hemi-slant submersions [7–9], semi-invariant submersions [10–12], antiinvariant submersions [13–15]. On the other hand, Friedmann et al. defined the concept of the semi-symmetric non-metric connection in a differential manifold [16]. Hayden studied metric connection with torsion a Riemannian manifold [17]. Later, Yano investigated a Riemannian manifold with new connection, which is called a semi-symmetric metric connection [18]. Afterwards, Agashe et al. studied semi-symmetric non-metric connection (SSNMC) on a Riemannian manifold [19]. Many author have studied semi-symmetric connection [20–26]. Let M be differentiable manifold with linear connection ∇. Therefore, for all K,L ∈ Γ(TN), we get T (K,L) = ∇KL−∇LK − [K,L], where T is torsion tensor of ∇. If the torsion tensor T = 0, then the connection ∇ is said to be symmetric, otherwise it is called non-symmetric. Moreover, for all K,L ∈ Γ(TN), the connection ∇ is said to be semi-symmetric if T (K,L) = η(L)K − η(K)L where η is a 1-form on N . However, ∇ is called metric connection if ∇g = 0 with Riemannian metric g, otherwise it is said to be non-metric.