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Semi-slant and bi-slant submanifolds of almost contact metric 3-structure manifolds

In this paper we introduce the notions of semi-slant and bi-slant submanifolds of an almost contact 3-structure manifold. We give some examples and characterization theorems about these submanifolds. Moreover, the distributions of semi-slant submanifolds of 3-cosymplectic and 3-Sasakian manifolds are studied. | Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Research Article Turk J Math (2013) 37: 1030 – 1039 ¨ ITAK ˙ c TUB ⃝ doi:10.3906/mat-1207-35 Semi-slant and bi-slant submanifolds of almost contact metric 3-structure manifolds Fereshteh MALEK,∗ Mohammad Bagher KAZEMI BALGESHIR Department of Mathematics, K. N. Toosi University of Technology, Tehran, Iran Received: 25.07.2012 • Accepted: 18.12.2012 • Published Online: 23.09.2013 • Printed: 21.10.2013 Abstract: In this paper we introduce the notions of semi-slant and bi-slant submanifolds of an almost contact 3-structure manifold. We give some examples and characterization theorems about these submanifolds. Moreover, the distributions of semi-slant submanifolds of 3-cosymplectic and 3-Sasakian manifolds are studied. Key words: Almost contact 3-structure manifold, semi-slant and bi-slant submanifold, 3-Sasakian manifold 1. Introduction After slant submanifolds of complex manifolds were introduced by Chen [6], the properties of slant submanifolds became an interesting subjects in differential geometry, both in complex geometry and in contact geometry. Lotta [9] introduced this notion in contact manifolds and Cabrerizo et al. [4] studied widely in this area and found many interesting results, especially on slant submanifolds of Sasakian manifolds. On the other hand, Papaghiuc [12] defined semi-slant submanifolds as a generalization of slant and CR-submanifolds. Carriazo [5] generalized these notions by introducing bi-slant submanifolds. Moreover, in [3], the authors investigated bi-slant and semi-slant submanifolds of Sasakian manifolds. From then on, many authors have studied these types of submanifolds when the ambient manifolds have been endowed with other structures such as trans-Sasakian and Kenmotsu [1, 14, 15, 17]. In fact, one of the important reasons for studying slant and semi-slant submanifolds is that they are a generalization of invariant, anti-invariant, semi-invariant, and .