[Paper Review] Generalized Kenmotsu Manifolds

Aysel Turgut Vanlı, Ramazan Sarı|arXiv (Cornell University)|Jun 4, 2014

Geometric Analysis and Curvature Flows12 references3 citations

TL;DR

This paper introduces generalized Kenmotsu manifolds as a higher-dimensional extension of Kenmotsu manifolds, defined on $(2n+s)$-dimensional $s$-contact metric manifolds. It establishes a necessary and sufficient condition for an almost $s$-contact metric manifold to be generalized Kenmotsu, proves such manifolds are locally warped product spaces, and shows their $\varphi$-sectional curvature is $-s$ under semi-symmetric or projective semi-symmetric curvature conditions.

ABSTRACT

In 1972, K. Kenmotsu studied a class of almost contact Riemannian manifolds. Later, such a manifold was called a Kenmotsu manifold. This paper, we studied Kenmotsu manifolds with $(2n+s)$-dimensional $s-$contact metric manifold and this manifold, we have called generalized Kenmotsu manifolds. Necessary and sufficient condition is given for an almost $s-$contact metric manifold to be a generalized Kenmotsu manifold.We show that a generalized Kenmotsu manifold is a locally warped product space. In addition, we study some curvature properties of generalized Kenmotsu manifolds. Moreover, we show that the $φ$% -sectional curvature of any semi-symmetric and projective semi-symmetric $% (2n+s)$-dimensional generalized Kenmotsu manifold is $-s$.

Motivation & Objective

• To generalize Kenmotsu manifolds from the $(2n+1)$-dimensional case to higher-dimensional $s$-contact metric manifolds with $s > 1$.
• To identify the necessary and sufficient condition for an almost $s$-contact metric manifold to be a generalized Kenmotsu manifold.
• To investigate curvature properties, particularly Ricci and sectional curvature, in the generalized setting.
• To establish that generalized Kenmotsu manifolds are locally warped product spaces.
• To determine the $\varphi$-sectional curvature under semi-symmetric or projective semi-symmetric curvature conditions.

Proposed method

• Define generalized Kenmotsu manifolds as $(2n+s)$-dimensional $s$-contact metric manifolds satisfying modified curvature conditions analogous to Kenmotsu's original equations.
• Use the Riemannian connection $\nabla$ and curvature tensor $R$ to derive structural identities involving $\varphi$, $\xi_i$, $\eta^i$, and $g$.
• Apply the Nijenhuis torsion and tensor fields $N^1$, $N^2$ to characterize normality and integrability conditions.
• Utilize the warped product structure $L^s \times_f V^{2n}$ as a model example to verify the generalized construction.
• Analyze semi-symmetric and projective semi-symmetric curvature conditions via the Riemann curvature tensor and Weyl projective tensor.
• Derive and manipulate tensorial identities involving the Ricci tensor $S$, $\nabla S$, and $\eta$-parallelism to determine curvature constraints.

Experimental results

Research questions

• RQ1 What is the necessary and sufficient condition for an almost $s$-contact metric manifold to be a generalized Kenmotsu manifold?
• RQ2 How does the curvature structure of generalized Kenmotsu manifolds differ from that of standard Kenmotsu manifolds?
• RQ3 Under what conditions is a generalized Kenmotsu manifold locally a warped product space?
• RQ4 What is the value of the $\varphi$-sectional curvature in semi-symmetric and projective semi-symmetric generalized Kenmotsu manifolds?
• RQ5 When is the Ricci tensor $\eta$-parallel in a generalized Kenmotsu manifold, and what does this imply for curvature?

Key findings

• A necessary and sufficient condition for an almost $s$-contact metric manifold to be a generalized Kenmotsu manifold is given by a specific curvature identity involving $\nabla\varphi$, $d\Phi$, and the Nijenhuis torsion.
• Generalized Kenmotsu manifolds are locally isometric to warped product spaces $L^s \times_f V^{2n}$, with $f(t) = ce^t$.
• The $\varphi$-sectional curvature of any semi-symmetric $(2n+s)$-dimensional generalized Kenmotsu manifold is exactly $-s$.
• The $\varphi$-sectional curvature of any projective semi-symmetric $(2n+s)$-dimensional generalized Kenmotsu manifold is also $-s$.
• A Ricci semi-symmetric generalized Kenmotsu manifold satisfies $S(X,Y) = -2n\{s g(\varphi X, \varphi Y) + \sum_{i,j=1}^s \eta^i(X)\eta^j(Y)\}$, implying Einstein structure when $s=1$.
• For $s=1$, a projectively semi-symmetric Kenmotsu manifold is Einstein if and only if its $\varphi$-sectional curvature is $-1$.