×

Almost \(\eta\)-Ricci solitons in \((LCS)_n\)-manifolds. (English) Zbl 1414.53021

In [Kyungpook Math. J. 43, No. 2, 305–314 (2003; Zbl 1054.53056)] A. A. Shaikh introduced the notion of Lorentzian concircular structure (briefly, \((\mathrm{LCS})_n\) Structure) as follows.
Let \((M,g)\) be an \(n\)-dimensional Lorentzian manifold and \(\zeta\) a unit time-like concircular vector field, \(\nabla\zeta=\alpha(I+\eta\otimes\zeta)\), with \(\alpha\) a nowhere zero smooth function on \(M\) such that \(d\alpha=\rho\eta\), for a smooth function \(\rho\), where \(\nabla\) is the Levi-Civita connection of \(g\) and \(\eta=i_{\zeta}g\) and \(\varphi=I+\eta\otimes\zeta\).
Another key definition is an almost \(\eta\)-Ricci soliton on \(M\) which is a data \((g,\zeta,\lambda,\mu)\) satisfying the equation: \[ \mathfrak{L}_{\zeta}g+2S+2\lambda g+2\mu \eta\otimes \eta=0, \] where \(\mathfrak{L}_{\zeta}\) is the Lie derivative operator along \(\eta\), \(S\) is the Ricci curvature tensor of \(g\), and \(\lambda\) and \(\mu\) are smooth functions on \(M\). In this paper, the author investigates the question whether \(\zeta\) is of potential type, i.e., \(\zeta=\mathrm{grad}(f)\), and provides lower and upper bounds for the Ricci curvature norm and a Bochner-type formula for the gradient almost \(\eta\)-Ricci soliton case.

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)

Citations:

Zbl 1054.53056
PDFBibTeX XMLCite
Full Text: arXiv Euclid