On the Pointwise Slant Submanifolds
In this survey paper, we consider several kinds of submanifolds in Riemannian manifolds, which are obtained by many authors. (i.e., slant submanifolds, pointwise slant submanifolds, semi-slant submanifolds, pointwise semi-slant submanifolds, pointwise alm
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Abstract In this survey paper, we consider several kinds of submanifolds in Riemannian manifolds, which are obtained by many authors. (i.e., slant submanifolds, pointwise slant submanifolds, semi-slant submanifolds, pointwise semi-slant submanifolds, pointwise almost h-slant submanifolds, pointwise almost h-semi-slant submanifolds, etc.) And we deal with some results, which are obtained by many authors at this area. Finally, we give some open problems at this area.
1 Introduction Given a Riemannian manifold (M, g) with some additional structures, there are several kinds of submanifolds: (Almost) complex submanifolds, totally real submanifolds, slant submanifolds, pointwise slant submanifolds, semi-slant submanifolds, pointwise semi-slant submanifolds, etc. In 1990, Chen [3] defined the notion of slant submanifolds of an almost Hermitian manifold as a generalization of almost complex submanifolds and totally real submanifolds. In 1994, Papaghiuc [7] introduced a semi-slant submanifold of an almost Hermitian manifold as a generalization of CR-submanifolds and slant submanifolds. In 1996, Lotta [6] introduced a slant submanifold of an almost contact metric manifold. In 1998, Etayo [5] defined the notion of pointwise slant submanifolds of an almost Hermitian manifold under the name of quasi-slant submanifolds as a generalization of slant submanifolds. In 1999, Cabrerizo, Carriazo, Fernandez, Fernandez [2] defined the notion of semi-slant submanifolds of an almost contact metric manifold. K.-S. Park (B) Division of General Mathematics, University of Seoul, 163 Seoulsiripdaero, Dongdaemun-gu, Seoul 02504, Korea e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2017 Y.J. Suh et al. (eds.), Hermitian–Grassmannian Submanifolds, Springer Proceedings in Mathematics & Statistics 203, DOI 10.1007/978-981-10-5556-0_21
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In 2012, Chen and Garay [4] studied deeply pointwise slant submanifolds of an almost Hermitian manifold. In 2013, Sahin [10] introduced pointwise semi-slant submanifolds of an almost Hermitian manifold. In 2014, Park [8] defined the notion of pointwise almost h-slant submanifolds and pointwise almost h-semi-slant submanifolds of an almost quaternionic Hermitian manifold. In 2015, Park [9] introduced pointwise slant and pointwise semi-slant submanifolds of an almost contact metric manifold. In this paper, we consider some results, which are obtained by many authors at this area. And we give some open problems at this area.
2 Preliminaries Let (M, g, J ) be an almost Hermitian manifold, where M is a C ∞ -manifold, g is a Riemannian metric on M, and J is an almost complex structure on M which is compatible with g. I.e., J ∈ End(T M), J 2 = −id, g(JX, J Y ) = g(X, Y ) for X, Y ∈ Γ (T M). Let M be a submanifold of M = (M, g, J ). We have the following notions. We call M an almost complex submanifold of M if J (Tx M) ⊂ Tx M for x ∈ M. The submanifold M is said to be a totally real submanifold if J (Tx M) ⊂ Tx M ⊥ for x ∈ M. The submanifold M is called a CR-submanifold if there exist
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