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In this paper, pointwise slant submanifolds of a Kaehler manifold are studied. First of all, if the submanifold of a Kaehler manifold is totally umbilical or pseudo-umbilical, the conditions for this submanifold to be a pointwise slant are investigated. Then given a pointwise submanifold of a Kaehler manifold, we investigate the conditions for this submanifold to be holomorphic, totally real, slant or CR-submanifold and obtain conditions depending on the behavior of the projections defined on the submanifold and the sectional curvature. Certain conditions are also found for the submanifold to be totally geodesic.

Almost Hermitian manifolds have a large number of submanifolds, such as holomorphic, totally real, CR-, slant, semi-slant, hemi-slant, bi-slant etc., depending on the behavior of the almost complex structure. In this paper, a new class of submanifolds, called partially slant submanifolds (abbreviated as PS-submanifolds), is defined, which includes the abovementioned CR-submanifolds, semi-slant submanifolds, hemi-slant submanifolds and bi-slant submanifolds. Such submanifolds of a Kaehler manifold are introduced, a proper example is given, the integrability of distributions and the geometry of maximal integral manifolds of these distributions are investigated. The effects of morphisms that naturally occur on such submanifolds on the geometry of the submanifold are investigated. In addition, special cases are determined when PS-submanifolds lie in a complex space form.

We introduce sequential warped product submanifolds of Kaehler manifolds, provide examples and establish Chen’s inequality for such submanifolds. The equality case is also studied. Moreover, inspired by Lawson and Simons’s integral currrent’s theorem on a submanifold, we find a similar pinching inequality for a sequential warped product submanifold and obtain geometric results when the equality case is satisfied.

In this paper, we introduce a new class of submanifolds which are called pointwise quasi bi-slant submanifolds in almost Hermitian manifolds which extends quasi bi-slant, bi-slant, hemi-slant, semi-slant and slant submanifolds in a very natural way. Several basic results in this respect are proved in this paper. Moreover, we obtain some conditions of the distributions which is involved in the definition of the new submanifolds. We also get some results for totally geodesic and mixed totally geodesic conditions for pointwise quasi bi-slant submanifolds. Finally, we illustrate some examples in order to guarantee the new kind of submanifolds.

The notion of a slant helix in Euclidean space was defined by Izumiya and Takeuchi [5], and many authors have studied such curves in Euclidean spaces. The aim of this paper is to introduce the slant helix notion on Riemannian manifolds. The necessary conditions for a curve on a Riemannian manifold to be a slant helix are obtained in terms of differential equations. In addition, certain conditions were found for the slant helix along an immersion to be a slant helix in the ambient space. Moreover, a criterion is given for the slant helix along an immersion to be a circle in the ambient space (or vice versa).

In this paper, we introduce generalized screen generic lightlike submanifolds of indefinite Kaehler manifolds which is new class and an umbrella of invariant (complex), screen real, CR, SCR, GCR and screen generic lightlike submanifolds. We give a non-trivial example for new class of submanifolds and find new conditions for the induced connection to be a metric connection. Then we obtain a characterization of such lightlike submanifolds in a complex space form. Moreover, we find some necessary and sufficient conditions for minimal generalized screen generic lightlike submanifolds and give an example of minimal generalized screen generic lightlike submanifold.

In this article, we consider pointwise slant and pointwise bi-slant submanifolds whose ambient spaces are para-Kaehler manifolds. We prove that there exist pointwise bi-slant K = k 2 K θ 1 × k 1 K θ 2 non-trivial doubly warped product type 1-2 submanifolds whose ambient spaces are para-Kaehler manifolds by constructing examples. We get a characterization and some theorems. Then, we obtain an inequality and we get some results by using the inequality.

In this paper we study invariant lightlike submanifolds of Lorentzian para-sasakian manifolds. We investigate geodesic CR-lightlike submanifolds of Lorentzian para-sasakian manifolds. We study screen CR-lightlike submanifolds of Lorentzian para-sasakian manifolds. Some necessary and sufficient conditions for mixed geodesic, totally geodesic, D ¯ -geodesic and D ˊ -geodesic contact CR-lightlike submanifolds and SCR-lightlike submanifolds are obtained.

Let X be an abstract not necessarily compact orientable CR manifold of dimension 2n-1, n\\geqslant 2, and let L^k be the k-th tensor power of a CR complex line bundle L over X. Given q\\in \\{0,1,\\ldots ,n-1\\}, let \\Box ^{(q)}_{b,k} be the Gaffney extension of Kohn Laplacian for (0,q) forms with values in L^k. For \\lambda \\geq 0, let \\Pi ^{(q)}_{k,\\leq \\lambda} :=E((-\\infty ,\\lambda ]), where E denotes the spectral measure of \\Box ^{(q)}_{b,k}. In this work, the author proves that \\Pi ^{(q)}_{k,\\leq k^{-N_0}}F^*_k, F_k\\Pi ^{(q)}_{k,\\leq k^{-N_0}}F^*_k, N_0\\geq 1, admit asymptotic expansions with respect to k on the non-degenerate part of the characteristic manifold of \\Box ^{(q)}_{b,k}, where F_k is some kind of microlocal cut-off function. Moreover, we show that F_k\\Pi ^{(q)}_{k,\\leq 0}F^*_k admits a full asymptotic expansion with respect to k if \\Box ^{(q)}_{b,k} has small spectral gap property with respect to F_k and \\Pi^{(q)}_{k,\\leq 0} is k-negligible away the diagonal with respect to F_k. By using these asymptotics, the authors establish almost Kodaira embedding theorems on CR manifolds and Kodaira embedding theorems on CR manifolds with transversal CR S^1 action.

The aim of our paper is to focus on some properties of hemi-slant submanifolds in metallic (and Golden) Riemannian manifolds. We give some characterizations of hemi-slant submanifolds in metallic or Golden Riemannian manifolds and we obtain integrability conditions for the distributions involved. Examples of hemi-slant submanifolds in metallic and Golden Riemannian manifolds are given.