Curvature of indefinite almost contact manifolds
type: Item
instance of: scholarly article
title: Curvature of indefinite almost contact manifolds
zbMATH Open document ID: 1008.53032
DOI: 10.1007/BF01222928
author: A. Bonome
author: R. Castro
author: Eduardo García-Río
author: L. M. Hervella
published in: Journal of Geometry
publication date: 1997-08-18T00:00:00Z
review text: The authors investigate the curvature properties of indefinite almost contact manifolds \((M,\varphi,\xi,\) \(\eta,g).\) Formally, \((\varphi,\xi,\eta,g)\) is an almost contact metric structure on the differentiable manifold \(M\) [cf., e.g., \textit{D. E. Blair} [Contact manifolds in Riemannian geometry. Lect. Notes Math. 509 (1976; Zbl 0319.53026)] but the metric \(g\) is not assumed to be positive definite. The investigation focuses on the so-called \(C(\alpha)\)-manifolds introduced by \textit{D. Janssens} and \textit{L. Vanhecke} [Kodai Math. J. 4, 1-27 (1981; Zbl 0472.53043)], i.e., those satisfying the condition \[ R(X,Y,Z,W-R(X,Y,\varphi Z,\varphi W)= \] \[ \alpha\bigl(g(Y,Z)g(X,W)-g(X,Z)g(Y,W)-g(Y,\varphi Z)g(X,\varphi W) +g(X,\varphi Z)g(Y,\varphi W)\bigr) \] for any vector fields, \(\alpha\) being a constant. By means of the study of the Jacobi operator along spacelike, timelike and null geodesics, spaces of constant curvature are characterized as well as spaces of pointwise constant \(\varphi\)-sectional curvature. There is an essential difference on the behaviour of the Jacobi operator along null and non-null geodesics. This motivates the definition of the \(\varphi\)-isotropic \(C(\alpha)\)-manifolds as those satisfying the condition \(R(U,\varphi U)\varphi U=c_UU\) (\(c_U=\text{const.}\)) for any null vector \(U\) tangent to the contact distribution. A local classification of \(\varphi\)-isotropic \(C(\alpha)\)-manifolds with parallel and diagonalizable Ricci tensor is obtained.
reviewed by: Zbigniew Olszak
Mathematics Subject Classification ID: 53C15
zbMATH DE Number: 1000346
zbMATH Keywords: indefinite almost contact manifold
zbMATH Keywords: \(\phi\)-sectional curvature
zbMATH Keywords: \(C(\alpha)\)-manifold
MaRDI profile type: Publication
cites work: Q4033561
cites work: Null holomorphically flat indefinite almost Hermitian manifolds
cites work: On the boundedness of Ricci curvature of an indefinite metric
cites work: On sectional curvature of indefinite metrics
cites work: Almost contact structures and curvature tensors
cites work: A class of almost contact Riemannian manifolds
cites work: Ein Beitrag zur Whitney-Regularität im unendlichdimensionalen Fall
cites work: Remarks on Sectional Curvature of an Indefinite Metric
cites work: On the existence of generalized complex space forms
cites work: Sasakian manifold with pseudo-Riemannian metric
cites work: Constancy of holomorphic sectional curvature in almost Hermitian manifolds
cites work: Foliations on Riemannian manifolds
cites work: Curvature Tensors on Almost Hermitian Manifolds
cites work: Q5678143
cites work: Q3222791
full work available at URL: https://doi.org/10.1007/bf01222928
OpenAlex ID: W1989738581
Recommended article: Q5740910
Recommended article: Almost \(C(\lambda)\)-manifolds
Recommended article: Q5506924
Recommended article: Projective invariants of almost \(C(\lambda)\)-manifolds
Recommended article: Q3787084
Recommended article: Q4918681
Recommended article: On conformally flat almost contact metric manifolds
Recommended article: Q4012272
Recommended article: On some classes of almost contact metric manifolds
Recommended article: Q5506990
description: scientific article; zbMATH DE number 1000346