Narrow Results

The main purpose of the paper is to study an almost gradient Ricci–Bourguignon soliton (RB soliton) within the framework of $K$-contact manifolds and $(\kappa ,\psi )$-contact manifolds. First, we prove that if complete $K$-contact manifold endows a gradient RB soliton, then the manifold is compact Sasakian and isometric to unit sphere $S^{2n+1}$. Next, we show that if a complete contact metric satisfies an almost RB soliton with a non-zero potential vector field is collinear with the Reeb vector field $\xi$ and the Reeb vector field $\xi$ acting as an eigenvector of the Ricci operator, then it is compact Einstein Sasakian and the potential vector field is a constant multiple of the Reeb vector field $\xi$. Lastly, we prove that if the metric of a non-Sasakian $(\kappa ,\psi )$-contact manifold is an almost gradient RB soliton, then it is flat in dimension 3 and in higher dimensions it is locally isometric to $E^{n+1}\times S^n(4)$.

In this paper, we consider the heat flow for $p$-pseudoharmonic maps from a closed Sasakian manifold $(M^{2n+1},J,𝜃)$ into a compact Riemannian manifold $(N^m,g_{ij})$. We prove global existence and asymptotic convergence of the solution for the $p$-pseudoharmonic map heat flow, provided that the sectional curvature of the target manifold $N$ is non-positive. Moreover, without the curvature assumption on the target manifold, we obtain global existence and asymptotic convergence of the $p$-pseudoharmonic map heat flow as well when its initial $p$-energy is sufficiently small.

In this paper, we proved that a compact Sasakian manifold $(M,\xi ,\eta ,\mathrm{\Phi },g)$ with negative transverse holomorphic sectional curvature must have a Sasakian structure $(\xi ,{\eta }^{\prime },{\mathrm{\Phi }}^{\prime },g^{\prime })$ with negative transverse Ricci curvature. Similarly, a compact Sasakian manifold with nonpositive transverse holomorphic sectional curvature, then the negative first basic Chern class $-c_1^B(M,{{ℱ}}_{\xi })$ is transverse nef and we have the Miyaoka–Yau-type inequality. When transverse holomorphic sectional curvature is quasi-negative, we obtain a Chern number inequality.

Let be a compact orientable smooth manifold admitting an almost complex structure and for (λ, μ) ∈ ℝ² - (0, 0) be the functional defined on the space of the almost Hermitian structure . We discuss the first variational problem of the functional on the space and its subspace in the case where is a product manifold of Sasakian manifolds. Further this paper provides examples of critical Hermitian structures of the functional for various (λ, μ).

The existence of compact simply-connected K-contact, but not Sasakian, manifolds has been unknown only for dimension 5. The aim of this paper is to show that the Kollár's simply-connected example which is a Seifert bundle over the complex projective space ℂℙ² and does not carry any Sasakian structure is actually a K-contact manifold. As a consequence, we affirmatively answer the above existence problem in dimension 5, establishing that there are infinitely many compact simply-connected K-contact manifolds of dimension 5 which do not carry a Sasakian structure.

As a generalization of anti-invariant ${\xi }^{\perp }$-Riemannian submersions, we introduce semi-invariant ${\xi }^{\perp }$-Riemannian submersions from Sasakian manifolds onto Riemannian manifolds. We give examples, investigating the geometry of foliations which arise from the definition of a Riemannian submersion and proving a necessary and sufficient condition for a semi-invariant ${\xi }^{\perp }$-Riemannian submersion to be totally geodesic. Moreover, we study semi-invariant ${\xi }^{\perp }$-Riemannian submersions with totally umbilical fibers.

In this paper, we study ruled surface in 3-dimensional almost contact metric manifolds by using surface theory defined by Gök [Surfaces theory in contact geometry, PhD thesis (2010)]. We also studied the theory of curves using cross product defined by Camcı. In this study, we obtain the distribution parameters of the ruled surface and then some results and theorems are presented with special cases. Moreover, some relationships among asymptotic curve and striction line of the base curve of the ruled surface have been found.

We prove that if a Sasakian metric is a ∗-Ricci Soliton, then it is either positive Sasakian, or null-Sasakian. Next, we prove that if a complete Sasakian metric is an almost gradient ∗-Ricci Soliton, then it is positive-Sasakian and isometric to a unit sphere $S^{2n+1}$. Finally, we classify nontrivial ∗-Ricci Solitons on non-Sasakian $(\kappa ,\mu )$-contact manifolds.

The purpose of this paper is to study anti-invariant Riemannian submersions from Sasakian manifolds onto Riemannian manifolds such that characteristic vector field is vertical or horizontal vector field. We first show that any anti-invariant Riemannian submersions from Sasakian manifold is not a Riemannian submersion with totally umbilical fiber. Then we introduce anti-invariant Riemannian submersions from Sasakian manifolds with totally contact umbilical fibers. We investigate the totally contact geodesicity of fibers of such submersions. Moreover, under this condition, we investigate Ricci curvature of anti-invariant Riemannian submersions from Sasakian manifolds onto Riemannian manifolds.

In this paper, we initiate the study of critical point equation and *-critical point equation within the substructure of Sasakian manifolds. We prove that if a compact Sasakian manifold admits CPE, then either the manifold is Einstein or the potential function is harmonic in an open subset. Later, we demonstrate that if the manifold satisfies *-CPE, it is $\eta$-Einstein and obtain an expression for the *-Ricci tensor. Finally, we construct an example to illustrate the existence of CPE and *-CPE on the Sasakian manifold.

In this paper, we study ∗-Conformal $\eta$-Ricci soliton on Sasakian manifolds. Here, we discuss some curvature properties on Sasakian manifold admitting ∗-Conformal $\eta$-Ricci soliton. We obtain some significant results on ∗-Conformal $\eta$-Ricci soliton in Sasakian manifolds satisfying $R(\xi ,X)\cdot S=0$, $S(\xi ,X)\cdot R=0$, $\overline{P}(\xi ,X)\cdot S$$=0$, where $\overline{P}$ is Pseudo-projective curvature tensor. The conditions for ∗-Conformal $\eta$-Ricci soliton on $\varphi$-conharmonically flat and $\varphi$-projectively flat Sasakian manifolds have been obtained in this paper. Lastly we give an example of five-dimensional Sasakian manifolds satisfying ∗-Conformal $\eta$-Ricci soliton.

A warped product submanifold, whose tangent bundle can be decomposed to two orthogonal distributions; invariant and pointwise slant, is called a warped product pointwise semi-slant submanifold. The objective of this paper is to classify warped product pointwise semi-slant submanifolds, isometrically immersed into a Sasakian manifold. Park [25] provided the (non)-existence of a warped product pointwise semi-slant submanifold in a Sasakian manifold such that the structure vector field is tangential to fibers. In contrast, we provide intriguing theorems on warped product pointwise semi-slant submanifolds in a Sasakian manifold in terms of the shape operator and tensor fields such that the structure vector field is tangential to a base manifold and each fiber is a pointwise slant submanifold.