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In 1979, P. Wintgen obtained a basic relationship between the extrinsic normal curvature the intrinsic Gauss curvature, and squared mean curvature of any surface in a Euclidean 4-space with the equality holding if and only if the curvature ellipse is a circle. In 1999, P. J. De Smet, F. Dillen, L. Verstraelen and L. Vrancken gave a conjecture of Wintgen inequality, named as the DDVV-conjecture, for general Riemannian submanifolds in real space forms. Later on, this conjecture was proven to be true by Z. Lu and by Ge and Z. Tang independently. Since then, the study of Wintgen’s inequalities and Wintgen ideal submanifolds has attracted many researchers, and a lot of interesting results have been found during the last 15 years. The main purpose of this paper is to extend this conjecture of Wintgen inequality for bi-slant submanifold in conformal Sasakian space form endowed with a quarter symmetric metric connection.

The authors used standard technique for obtaining generalized Wintgen inequality for bi-slant submanifold in conformal Sasakian space form endowed with a quarter symmetric metric connection.

The authors establish the generalized Wintgen inequality for bi-slant submanifold in conformal Sasakian space form endowed with a quarter symmetric metric connection, and also find conditions under which the equality holds. Some particular cases are also stated.

The research may be a challenge for new developments focused on new relationships in terms of various invariants, for different types of submanifolds in that ambient space with several connections.

The authors set the goal to find the solution of the Eisenhart problem within the framework of three-dimensional trans-Sasakian manifolds. Also, they prove some results of the Ricci solitons, η-Ricci solitons and three-dimensional weakly  symmetric trans-Sasakian manifolds. Finally, they give a nontrivial example of three-dimensional proper trans-Sasakian manifold.

The authors have used the tensorial approach to achieve the goal.

A second-order parallel symmetric tensor on a three-dimensional trans-Sasakian manifold is a constant multiple of the associated Riemannian metric g.

The authors declare that the manuscript is original and it has not been submitted to any other journal for possible publication.

The central idea of this research article is to examine the characteristics of Clairaut submersions from Lorentzian trans-Sasakian manifolds of type (α, β) and also, to enhance this geometrical analysis with some specific cases, namely Clairaut submersion from Lorentzian α-Sasakian manifold, Lorentzian β-Kenmotsu manifold and Lorentzian cosymplectic manifold. Furthermore, the authors discuss some results about Clairaut Lagrangian submersions whose total space is a Lorentzian trans-Sasakian manifolds of type (α, β). Finally, the authors furnished some examples based on this study.

This research discourse based on classifications of submersion, mainly Clairaut submersions, whose total manifolds is Lorentzian trans-Sasakian manifolds and its all classes like Lorentzian Sasakian, Lorenztian Kenmotsu and Lorentzian cosymplectic manifolds. In addition, the authors have explored some axioms of Clairaut Lorentzian submersions and illustrates our findings with some non-trivial examples.

The major finding of this study is to exhibit a necessary and sufficient condition for a submersions to be a Clairaut submersions and also find a condition for Clairaut Lagrangian submersions from Lorentzian trans-Sasakian manifolds.

The results and examples of the present manuscript are original. In addition, more general results with fair value and supportive examples are provided.

The author considers an invariant lightlike submanifold M, whose transversal bundle tr(TM) is flat, in an indefinite Sasakian manifold M¯(c) of constant φ¯-sectional curvature c. Under some geometric conditions, the author demonstrates that c=1, that is, M¯ is a space of constant curvature 1. Moreover, M and any leaf M′ of its screen distribution S(TM) are, also, spaces of constant curvature 1.

The author has employed the techniques developed by K. L. Duggal and A. Bejancu of reference number 7.

The author has discovered that any totally umbilic invariant ligtlike submanifold, whose transversal bundle is flat, in an indefinite Sasakian space form is, in fact, a space of constant curvature 1 (see Theorem 4.4).

To the best of the author’s findings, at the time of submission of this paper, the results reported are new and interesting as far as lightlike geometry is concerned.

This paper aims to study almost Ricci–Yamabe soliton in the context of certain contact metric manifolds.

The paper is designed as follows: In Section 3, a complete contact metric manifold with the Reeb vector field ξ as an eigenvector of the Ricci operator admitting almost Ricci–Yamabe soliton is considered. In Section 4, a complete K-contact manifold admits gradient Ricci–Yamabe soliton is studied. Then in Section 5, gradient almost Ricci–Yamabe soliton in non-Sasakian (k, μ)-contact metric manifold is assumed. Moreover, the obtained result is verified by constructing an example.

We prove that if the metric g admits an almost (α, β)-Ricci–Yamabe soliton with α ≠ 0 and potential vector field collinear with the Reeb vector field ξ on a complete contact metric manifold with the Reeb vector field ξ as an eigenvector of the Ricci operator, then the manifold is compact Einstein Sasakian and the potential vector field is a constant multiple of the Reeb vector field ξ. For the case of complete K-contact, we found that it is isometric to unit sphere S²ⁿ⁺¹ and in the case of (k, μ)-contact metric manifold, it is flat in three-dimension and locally isometric to Eⁿ⁺¹ × Sⁿ(4) in higher dimension.

All results are novel and generalizations of previously obtained results.

Bi-slant submanifolds of S-manifolds are introduced, and some examples of these submanifolds are presented.

Some properties of D_i-geodesic and D_i-umbilical bi-slant submanifolds are examined.

The Riemannian curvature invariants of these submanifolds are computed, and some results are discussed with the help of these invariants.

The topic is original, and the manuscript has not been submitted to any other journal.

Let (M, g) be a n-dimensional smooth Riemannian manifold. In the present paper, the authors introduce a new class of natural metrics denoted by g^f and called gradient Sasaki metric on the tangent bundle TM. The authors calculate its Levi-Civita connection and Riemannian curvature tensor. The authors study the geometry of (TM, g^f) and several important results are obtained on curvature, scalar and sectional curvatures.

In this paper the authors introduce a new class of natural metrics called gradient Sasaki metric on tangent bundle.

The authors calculate its Levi-Civita connection and Riemannian curvature tensor. The authors study the geometry of (TM, gf) and several important results are obtained on curvature scalar and sectional curvatures.

The authors calculate its Levi-Civita connection and Riemannian curvature tensor. The authors study the geometry of (TM, gf) and several important results are obtained on curvature scalar and sectional curvatures.