[Paper Review] On warped product manifolds satisfying some pseudosymmetric type conditions

Absos Ali Shaikh, Haradhan Kundu|arXiv (Cornell University)|Dec 10, 2016

Geometric Analysis and Curvature Flows21 references19 citations

TL;DR

This paper investigates warped product semi-Riemannian manifolds satisfying pseudosymmetric type curvature conditions, particularly involving the projective curvature tensor. It derives a characterization theorem for manifolds satisfying $ R\cdot R = L_1 Q(g,R) + L_2 Q(S,R) $, showing that if $ L_2 \neq 0 $, then either the base is flat or the fiber is Einstein, with special cases recovering semisymmetry, pseudosymmetry, and projective curvature-based conditions.

ABSTRACT

The object of the present paper is to study the characterization of warped product manifolds satisfying some pseudosymmetric type conditions, especially, due to projective curvature tensor. For this purpose we consider a warped product manifold satisfying the pseudosymmetric type condition $R\cdot R = L_1 Q(g,R) + L_2 Q(S,R)$ and evaluate its characterization theorem. As special cases of $L_1$ and $L_2$ we find out the necessary and sufficient condition for a warped product manifold to satisfy various pseudosymmetric type, such as pseudosymmetry, Ricci generalized pseudosymmetry, semisymmetry due to projective curvature tensor ($P\cdot R = 0$), pseudosymmetry due to projective curvature tensor ($P\cdot R = L Q(g,R)$) etc. Finally we present some suitable examples of warped product manifolds satisfying such pseudosymmetric type conditions.

Motivation & Objective

• To characterize warped product semi-Riemannian manifolds satisfying the pseudosymmetric type condition $ R\cdot R = L_1 Q(g,R) + L_2 Q(S,R) $.
• To determine necessary and sufficient conditions under which such warped product manifolds satisfy specific curvature restrictions, including semisymmetry, pseudosymmetry, and projective curvature-based conditions.
• To explore the geometric implications on the base and fiber manifolds, especially when $ L_2 \neq 0 $, showing that either the base is flat or the fiber is Einstein.
• To construct explicit examples of warped product manifolds satisfying various pseudosymmetric type conditions, demonstrating the theoretical results.
• To establish that a pseudosymmetric manifold can be a totally umbilical hypersurface of a semisymmetric manifold, using warped product structure.

Proposed method

• Derives the curvature condition $ R\cdot R = L_1 Q(g,R) + L_2 Q(S,R) $ for warped product manifolds $ M = \overline{M} \times_f \widetilde{M} $, using the Levi-Civita connection and curvature tensors.
• Applies the warped product structure to decompose the Riemann curvature tensor $ R $, Ricci tensor $ S $, and projective curvature tensor $ P $ into components of the base $ \overline{M} $ and fiber $ \widetilde{M} $.
• Uses the condition $ R\cdot R = L_1 Q(g,R) + L_2 Q(S,R) $ to derive a system of equations involving the warping function $ f $, scalar curvature $ \kappa $, and curvature tensors of base and fiber.
• Analyzes the resulting equations to show that if $ L_2 \neq 0 $, then either $ \overline{R} = 0 $ (base is flat) or $ \widetilde{S} = \frac{\widetilde{\kappa}}{n-p} \widetilde{g} $ (fiber is Einstein).
• Applies the general condition to special cases: $ L_1 = 0 $, $ L_2 = 0 $, $ L_1 = L_2 = 1 $, and $ L_2 = 0 $ with $ L_1 $ scalar, to recover semisymmetry, pseudosymmetry, and projective curvature-based conditions.
• Constructs explicit examples using specific warping functions and metrics to verify that the curvature conditions are satisfied, including one with $ R\cdot R = a Q(g,R) $ and another with $ R\cdot R = Q(g,R) + \text{non-constant term} \cdot Q(S,R) $.

Experimental results

Research questions

• RQ1 Under what conditions does a warped product manifold $ M = \overline{M} \times_f \widetilde{M} $ satisfy the pseudosymmetric type condition $ R\cdot R = L_1 Q(g,R) + L_2 Q(S,R) $?
• RQ2 What geometric constraints are imposed on the base $ \overline{M} $ and fiber $ \widetilde{M} $ when $ L_2 \neq 0 $ in the curvature condition?
• RQ3 How do special cases of $ L_1 $ and $ L_2 $, such as $ L_1 = 1, L_2 = 0 $ or $ L_1 = 0, L_2 = 1 $, correspond to known curvature conditions like semisymmetry or Ricci generalized pseudosymmetry?
• RQ4 Can a pseudosymmetric manifold arise as a totally umbilical hypersurface in a semisymmetric warped product manifold?
• RQ5 What explicit examples of warped product manifolds satisfy $ R\cdot R = L_1 Q(g,R) + L_2 Q(S,R) $, and how do they verify the derived conditions?

Key findings

• If a warped product manifold $ M = \overline{M} \times_f \widetilde{M} $ satisfies $ R\cdot R = L_1 Q(g,R) + L_2 Q(S,R) $ with $ L_2 $ nowhere zero, then either the base $ \overline{M} $ is flat or the fiber $ \widetilde{M} $ is Einstein.
• For the special case $ R\cdot R = 0 $, the base $ \overline{M} $ is semisymmetric and the fiber $ \widetilde{M} $ is pseudosymmetric.
• In the case $ R\cdot R = L_1 Q(g,R) $, both the base and fiber are pseudosymmetric.
• For $ P\cdot R = 0 $, the manifold is semisymmetric due to the projective curvature tensor, and this holds if and only if either the base is flat or the fiber is Einstein.
• For $ P\cdot R = L Q(g,R) $, the condition reduces to a scalar multiple of $ Q(g,R) $, and again, either the base is flat or the fiber is Einstein.
• An explicit example is constructed where $ R\cdot R = a Q(g,R) $, showing that $ M $ is a warped product pseudosymmetric manifold of constant type, and if $ a = 0 $, it becomes semisymmetric; in this case, $ W\cdot R = 0 $, so it is semisymmetric type due to the concircular curvature tensor.