[Paper Review] ParaSasakian manifolds with a constant paraholomorphic section curvature

Simeon Zamkovoy|arXiv (Cornell University)|Dec 9, 2008

Geometric Analysis and Curvature Flows4 references3 citations

TL;DR

This paper classifies paraSasakian manifolds with constant paraholomorphic sectional curvature by proving that any such (2n+1)-dimensional, connected, simply connected manifold is locally isometric to a specific model space: the hyperbolic Heisenberg group when curvature k=3, or a hyperboloid in para-Kähler space when k≠3. The classification relies on curvature tensor analysis and analytic continuation of connections and curvature via isomorphism of jet spaces.

ABSTRACT

In this paper paraSasakian manifolds with a constant paraholomorphic section curvature are considered.

Motivation & Objective

• To classify paraSasakian manifolds with constant paraholomorphic sectional curvature.
• To determine the local structure of such manifolds via curvature and connection analysis.
• To establish that manifolds with constant curvature k are locally isometric to either the hyperbolic Heisenberg group (k=3) or a hyperboloid in para-Kähler space (k≠3).
• To prove that the curvature tensor and its covariant derivatives determine the local geometry via analytic continuation of the connection.

Proposed method

• Uses the PC-Bochner curvature tensor and its decomposition to analyze curvature symmetries in paraSasakian manifolds.
• Applies the curvature formula (6.58) for the Riemann curvature tensor in terms of metric, structure tensor φ, and 1-form η.
• Constructs a linear isomorphism F between tangent spaces of the manifold M and the model space H(k) using orthonormal frames and φ-compatibility.
• Uses analyticity of the connection and curvature tensors to extend the local isomorphism F to a global isomorphism f via jet space equivalence.
• Verifies that f preserves ξ, φ, and η, hence is a paraSasakian isomorphism.
• Applies Theorem 7.2 from [8] to conclude that the isomorphism f exists and satisfies fξ = ξ*.

Experimental results

Research questions

• RQ1 Under what conditions is a paraSasakian manifold with constant paraholomorphic sectional curvature locally isometric to a model space?
• RQ2 What is the local geometric structure of a paraSasakian manifold when the paraholomorphic sectional curvature k is 3?
• RQ3 What is the local geometric structure when k ≠ 3, and how does it differ from the k=3 case?
• RQ4 Can the curvature tensor and its covariant derivatives uniquely determine the local geometry of such manifolds?
• RQ5 Is there a canonical model space to which all simply connected paraSasakian manifolds with constant paraholomorphic curvature are locally isometric?

Key findings

• When k=3, the curvature tensor vanishes identically, and the manifold is locally isometric to the hyperbolic Heisenberg group H^{2n+1}.
• For k≠3, the manifold is locally isometric to the hyperboloid H^{2n+1}_{n+1}(1) in para-Kähler space R^{2n+2} with the standard para-Kähler structure.
• The curvature tensor R has a specific algebraic form (6.58) depending on g, φ, η, and the curvature k.
• The PC-Bochner curvature tensor B is shown to coincide with the W^{pc} tensor, confirming curvature rigidity.
• The connection ∇ and curvature R are analytic, enabling the extension of local isomorphisms to global ones via jet space theory.
• The isomorphism f preserves ξ, φ, η, and g, so f is a paraSasakian isomorphism, proving local rigidity.