[Paper Review] Almost cosymplectic and almost Kenmotsu (\kappa,\mu, u)-spaces

Alfonso Carriazo, Verónica Martín-Molina|arXiv (Cornell University)|Jan 26, 2012

Advanced Differential Geometry Research3 citations

TL;DR

This paper investigates the Riemann curvature tensor in (κ,µ, u)-spaces endowed with almost cosymplectic or almost Kenmotsu structures, providing an explicit expression for the curvature. It introduces and studies a natural generalization of contact metric (κ,µ, u)-spaces, offering examples and obstruction results across all possible cases.

ABSTRACT

We study the Riemann curvature tensor of (\kappa,\mu, u)-spaces when they have almost cosymplectic and almost Kenmotsu structures, giving its writing explicitly. This leads to the definition and study of a natural generalisation of the contact metric (\kappa,\mu, u)-spaces. We present examples or obstruction results of these spaces in all possible cases.

Motivation & Objective

• To analyze the Riemann curvature tensor in (κ,µ, u)-spaces equipped with almost cosymplectic or almost Kenmotsu structures.
• To define and study a natural generalization of contact metric (κ,µ, u)-spaces based on these structures.
• To provide explicit constructions or obstruction results for such spaces in all possible geometric configurations.
• To clarify the geometric and curvature properties of these generalized (κ,µ, u)-spaces under the given structures.

Proposed method

• Explicit computation of the Riemann curvature tensor for (κ,µ, u)-spaces with almost cosymplectic or almost Kenmotsu structures.
• Use of tensorial and differential geometric techniques to derive curvature expressions in terms of the structure parameters (κ, µ, u).
• Application of the integrability conditions for almost cosymplectic and almost Kenmotsu structures to constrain curvature behavior.
• Construction of examples and identification of obstruction conditions for the existence of such structures on (κ,µ, u)-spaces.
• Comparison of the derived curvature forms with known results in contact metric geometry to highlight generalizations.
• Systematic analysis of all possible cases for the parameters (κ, µ, u) to determine geometric realizability.

Experimental results

Research questions

• RQ1 How does the Riemann curvature tensor of a (κ,µ, u)-space decompose when equipped with an almost cosymplectic structure?
• RQ2 What are the curvature implications of endowing a (κ,µ, u)-space with an almost Kenmotsu structure?
• RQ3 What is the nature of the generalization of contact metric (κ,µ, u)-spaces arising from these structures?
• RQ4 Under what conditions do almost cosymplectic or almost Kenmotsu structures exist on (κ,µ, u)-spaces?
• RQ5 What examples or obstructions arise in the classification of such structures across all (κ,µ, u) parameter regimes?

Key findings

• The Riemann curvature tensor of (κ,µ, u)-spaces with almost cosymplectic or almost Kenmotsu structures is explicitly computed and expressed in terms of the parameters (κ, µ, u).
• A natural generalization of contact metric (κ,µ, u)-spaces is defined and studied, extending the class of structures beyond the standard contact metric setting.
• Examples of such spaces are constructed in various cases, demonstrating the geometric realizability of the structures under specific parameter constraints.
• Obstruction results are derived, showing that not all (κ,µ, u)-spaces can carry these structures, particularly depending on the values of κ, µ, and u.
• The curvature expressions reveal structural differences between almost cosymplectic and almost Kenmotsu cases, highlighting distinct geometric behaviors.
• The analysis confirms the existence of non-trivial geometric structures on (κ,µ, u)-spaces beyond the classical contact metric framework.