[Paper Review] Curvature of a class of indefinite globally framed $f$-manifolds

Letizia Brunetti, Anna Maria Pastore|ArXiv.org|Mar 4, 2008

Geometric Analysis and Curvature Flows13 references15 citations

TL;DR

This paper investigates the curvature properties of indefinite globally framed $f$-manifolds, particularly focusing on indefinite almost $\mathcal{S}$-manifolds and $\mathcal{S}$-space forms. It derives an explicit expression for the Riemann curvature tensor in terms of $\varphi$-sectional curvature and proves that when the number of spacelike and timelike characteristic vector fields are equal, the manifold becomes an indefinite $\mathcal{S}$-space form with vanishing $\varphi$-sectional curvature, exemplified by a 4-dimensional Lorentzian manifold with mixed-signature characteristic fields.

ABSTRACT

We present a compared analysis of some properties of indefinite almost $\mathcal{S}$-manifolds and indefinite $\mathcal{S}$-manifolds. We give some characterizations in terms of the Levi-Civita connection and of the characteristic vector fields. We study the sectional and $ϕ$-sectional curvature of indefinite almost $\mathcal{S}$-manifolds and state an expression of the curvature tensor field for the indefinite $\mathcal{S}$-space forms. We analyse the sectional curvature of indefinite $\mathcal{S}$-manifold in which the number of the spacelike characteristic vector fields is equal to that of the timelike characteristic vector fields. Some examples are also described.

Motivation & Objective

• To analyze the geometric and curvature properties of indefinite almost $\mathcal{S}$-manifolds and $\mathcal{S}$-manifolds with indefinite metrics.
• To characterize the Levi-Civita connection and the role of characteristic vector fields in these structures.
• To derive an explicit expression for the Riemann curvature tensor of indefinite $\mathcal{S}$-space forms in terms of $\varphi$-sectional curvature.
• To investigate the sectional curvature in indefinite $\mathcal{S}$-manifolds where the count of spacelike and timelike characteristic vector fields is balanced.
• To construct and analyze explicit examples of 6-dimensional and 4-dimensional indefinite $\mathcal{S}$-manifolds with varying causal types of characteristic vector fields.

Proposed method

• Utilizes the Levi-Civita connection and compatibility conditions between the $f$-structure $\varphi$, the metric $g$, and the characteristic vector fields $\xi_\alpha$.
• Applies the fundamental identity $\varphi^3 + \varphi = 0$ to decompose the tangent bundle into $\operatorname{Im}\varphi$ and $\ker\varphi$, with $\varphi$ inducing a complex structure on $\operatorname{Im}\varphi$.
• Employs the curvature identity $g(R(\varphi X,\varphi Y,\varphi Z),\varphi W) = g(R(X,Y,Z),W)$ to relate curvatures under $\varphi$-action.
• Derives a formula for the $\varphi$-sectional curvature in terms of the tensor $H_p(X,Y)$, which depends on the metric and $\varphi$-action on vectors in $\mathfrak{D}$.
• Uses Christoffel symbols computed from the metric to evaluate curvature components, particularly $R(X,\varphi X,X)$, to determine $H(X)$.
• Establishes that the Riemann curvature tensor of an indefinite $\mathcal{S}$-space form is fully determined by the $\varphi$-sectional curvature $c$, via the explicit formula (14).

Experimental results

Research questions

• RQ1 How do the Levi-Civita connection and characteristic vector fields influence the curvature structure of indefinite almost $\mathcal{S}$-manifolds?
• RQ2 In what way do $\varphi$-sectional curvatures determine the full sectional curvature in indefinite $\mathcal{S}$-manifolds?
• RQ3 What is the explicit form of the Riemann curvature tensor for indefinite $\mathcal{S}$-space forms with constant $\varphi$-sectional curvature?
• RQ4 Under what conditions does an indefinite $\mathcal{S}$-manifold with balanced spacelike and timelike characteristic vector fields become an $\mathcal{S}$-space form with zero $\varphi$-sectional curvature?
• RQ5 Can explicit examples of indefinite $\mathcal{S}$-manifolds be constructed with mixed causal types of characteristic vector fields, and what are their curvature properties?

Key findings

• The $\varphi$-sectional curvature completely determines the sectional curvature in indefinite almost $\mathcal{S}$-manifolds.
• An explicit formula for the Riemann curvature tensor of indefinite $\mathcal{S}$-space forms is derived, showing it depends only on the $\varphi$-sectional curvature $c$ and the metric structure.
• In the 4-dimensional Lorentzian example with one spacelike and one timelike characteristic vector field ($\varepsilon_1 = 1$, $\varepsilon_2 = -1$), the $\varphi$-sectional curvature $c$ is zero.
• The curvature tensor of the 4-dimensional example is fully determined by the formula (14) with $c=0$, and the resulting tensor is non-zero, indicating non-trivial curvature in the $\ker\varphi$ directions.
• The tensor $Q$ does not vanish in general, as shown by $Q(\xi_\alpha, Y; \xi_\beta, W) = -\varepsilon_\alpha\varepsilon_\beta g(W, \varphi Y)$, indicating non-trivial curvature contributions from the characteristic fields.
• The computation of $H(X)$ for the vector $X = \partial_x - y\xi_1 - y\xi_2$ yields $H(X) = 0$, confirming that the $\varphi$-sectional curvature is zero, and thus the manifold is an indefinite $\mathcal{S}$-space form with $c=0$.