[Paper Review] Almost contact B-metric manifolds as extensions of a 2-dimensional space-form

Hristo Manev|arXiv (Cornell University)|Jun 18, 2015

Geometric Analysis and Curvature Flows4 citations

TL;DR

This paper constructs 3-dimensional almost contact B-metric manifolds as extensions of a 2-dimensional Norden manifold via two methods: the cone construction and the $S^1$-solvable extension. It characterizes the resulting manifolds in the Ganchev-Mihova-Gribachev classification, showing they belong to the $ancyscript{F}_1 igoplus ancyscript{F}_4$ class, and establishes that they are $ancyscript{F}_4$-manifolds if and only if the base is a Kähler-Norden manifold, with explicit curvature and Ricci tensor relations derived.

ABSTRACT

The object of investigations are almost contact B-metric manifolds which are derived as a product of a real line and a 2-dimensional manifold equipped with a complex structure and a Norden metric. There are used two different methods for generation of the B-metric on the product manifold. The constructed manifolds are characterised with respect to the Ganchev-Mihova-Gribachev classification and their basic curvature properties.

Motivation & Objective

• To construct 3-dimensional almost contact B-metric manifolds by extending a 2-dimensional Norden manifold using two distinct geometric methods: the cone and the $S^1$-solvable extension.
• To classify the resulting manifolds within the Ganchev-Mihova-Gribachev framework of 11 basic classes for almost contact B-metric structures.
• To determine curvature properties, including sectional curvatures, Ricci tensors, and scalar curvatures, in relation to the geometry of the base manifold.
• To establish conditions under which the extended manifold belongs to specific classes, particularly $ancyscript{F}_4$, and to identify when it becomes $ancyscript{F}_1 igoplus ancyscript{F}_4$.
• To derive explicit formulas for the Lee forms, curvature components, and scalar curvatures in terms of the base manifold’s geometry and the extension parameter $t$.

Proposed method

• Constructs the almost contact B-metric structure on the product manifold $S^1(N) = S^1 imes N$, where $N$ is a 2-dimensional Norden manifold with complex structure $J$ and Norden metric $h$.
• Applies two distinct geometric extensions: the cone construction and the $S^1$-solvable extension, each defining a different B-metric $g$ on $S^1(N)$ via specific metric and almost contact structure formulas.
• Uses the Levi-Civita connection $ abla$ of $g$ to compute the tensor $F(x,y,z) = g(( abla_x heta)y, z)$, which classifies the manifold via the Ganchev-Mihova-Gribachev classification.
• Computes the Lee forms $ heta$, $ heta^*$, and $ ho$ using the metric components and curvature tensors, with explicit expressions in a $ancyscript{F}$-basis.
• Derives the curvature components $R_{ijkar{l}}$ and sectional curvatures $k_{ij}$ using the structure equations and the base manifold’s curvature $k'$, particularly for $k_{12}$ and $k_{13}=k_{23}=1$.
• Expresses the Ricci tensor $ ho$, its associated $ ho^*$, and scalar curvatures $ au$, $ au^*$, $ au^{**}$ in terms of $k'$ and the angle parameter $t$, enabling classification and curvature analysis.

Experimental results

Research questions

• RQ1 What is the Ganchev-Mihova-Gribachev classification of 3-dimensional almost contact B-metric manifolds constructed as $S^1$-solvable extensions of a 2-dimensional Norden manifold?
• RQ2 Under what conditions does the $S^1$-solvable extension of a Norden manifold result in a manifold belonging to the $ancyscript{F}_4$ class?
• RQ3 How do the curvature components, sectional curvatures, and Ricci tensors of the extended manifold depend on the curvature of the base 2-dimensional space-form?
• RQ4 What is the relationship between the scalar curvatures $ au$, $ au^*$, and $ au^{**}$ of the extended manifold and the base manifold’s curvature $k'$?
• RQ5 When is the extended manifold $ancyscript{F}_1 igoplus ancyscript{F}_4$-type, and when is it $ancyscript{F}_4$-type, in terms of the base geometry?

Key findings

• The $S^1$-solvable extension of a 2-dimensional Norden manifold results in a 3-dimensional almost contact B-metric manifold that belongs to the class $ancyscript{F}_1 igoplus ancyscript{F}_4$.
• The manifold is in class $ancyscript{F}_4$ if and only if the base manifold $(N,J,h)$ is a Kähler-Norden manifold, which is equivalent to the vanishing of the Lee form $ heta^*$ and specific curvature conditions.
• The sectional curvatures of the $ancyscript{F}$-sections are constant: $k_{13} = k_{23} = 1$, while $k_{12} = R_{1212}$ depends on $k'$ and the angle $t$.
• The Ricci tensor $ ho$ is $ancyscript{F}_1 igoplus ancyscript{F}_4$-type and satisfies $ ho = k' heta igotimes g + (2 - k' heta) heta igotimes heta$ when the base is Kähler-Norden, where $ heta = ext{Re}(e^{2it})$.
• The scalar curvature $ au = 2(k' heta + 1)$ and $ au^{**} = 2(k' heta - 1)$, where $ heta = ext{Re}(e^{2it}) = ext{Re}( ext{cis}(2t))$, showing explicit dependence on the base curvature $k'$ and extension parameter $t$.
• When the base is Kähler-Norden, $ au^* = 0$ and $ au^{**} = au - 4$, and the manifold is $ancyscript{F}_4$-type and $ancyscript{F}_1$-type only if $k' = 0$, which corresponds to $ au = 2$, $ au^{**} = -2$.