Clairaut anti-invariant submersions from locally product Riemannian manifolds
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Clairaut anti-invariant submersions from locally product Riemannian manifolds Yılmaz Gündüzalp1 Received: 29 August 2019 / Accepted: 7 February 2020 © The Managing Editors 2020
Abstract In this paper, we investigate some geometric properties of Clairaut submersions whose total space is a locally product Riemannian manifold. Keywords Locally product Riemannian manifold · Riemannian submersion · Clairaut anti-invariant submersion Mathematics Subject Classification Primary 53C15; Secondary 53C40
1 Introduction Given a C ∞ -submersion F from a (semi)-Riemannian manifold (N , g N ) onto a (semi)-Riemannian manifold (B, g B ), according to the circumstances on the map F : (N , g N ) → (B, g B ), we get the following: Riemannian submersion (Falcitelli et al. 2004; O’Neill 1966; Gray 1967), almost Hermitian submersion (Watson 1976), paracontact paracomplex submersios (Gündüzalp and S.ahin 2014), quaternionic submersion (Ianus et al. 2008), slant submersion (Akyol and Gündüzalp 2016; Gündüzalp 2013b; Gündüzalp and Akyol 2018; S.ahin 2011), anti-invariant submersion (Beri et al. 2016; S.ahin 2010), Clairaut submersion (Bishop 1972; Gündüzalp 2019; Tas.tan and Gerdan 2017; Lee et al. 2015; Allison 1996), conformal anti-invariant submersion (Akyol 2017; Akyol and S.ahin 2016), etc. In the present paper, we take into account Clairaut anti-invariant submersions from a locally product Riemannian manifold onto a Riemannian manifold. In Sect. 2, we recall some concepts, which are needed in the following section. In Sect. 3, we first obtain necessary and sufficient conditions for a curve on the manifold N of anti-
B 1
Yılmaz Gündüzalp [email protected] Department of Mathematics, Dicle University, 21280 Diyarbakir, Turkey
123
Beitr Algebra Geom
invariant submersions to be geodesic. Then we present a new characterization for Clairaut anti-invariant submersions. Also, we present an example.
2 Preliminaries 2.1 Riemannian submersions A C ∞ -submersion F : N → B between two Riemannian manifolds (N , g N ) and (B, g B ) is called a Riemannian submersion if it satisfies conditions: (i) the fibres F −1 (b), b ∈ B, are r -dimensional Riemannian submanifolds of N , where r = dim(N ) − dim(B). (ii) F∗ preserves the lengths of horizontal vectors. The vectors tangent to the fibres are called vertical and those normal to the fibres are called horizontal. We denote by (ker F∗ ) the vertical distribution, by (ker F∗ )⊥ the horizontal distribution and by v and h the vertical and horizontal projection. A horizontal vector field X 1 on N is said to be fundamental if X 1 is F-related to a vector field X ∗1 on B. A Riemannian submersion F : N → B defines two (1, 2) tensor fields T and A on N , by the formulas: T X 1 X 2 = h∇v X 1 v X 2 + v∇v X 1 h X 2
(1)
A X 1 X 2 = v∇h X 1 h X 2 + h∇h X 1 v X 2
(2)
and
for any X 1 , X 2 ∈ χ (N ) (see Falcitelli et al. 2004). Using (1) and (2), one can get ∇U1 U2 = TU1 U2 + ∇ˆ U1 U2 ; ∇U1 X 1 = TU1 X 1 + h(∇U1 X 1 );
(3) (4)
∇ X 1 U1 = A X 1 U1 + v(∇ X 1 U1 ), ∇ X 1 X 2 = A X 1 X 2 + h(∇ X 1 X 2 )
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