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SOME RESULTS ON CONCIRCULAR VECTOR FIELDS AND THEIR APPLICATIONS TO RICCI SOLITONS

  • CHEN, BANG-YEN
  • Received : 2014.08.12
  • Published : 2015.09.30

Abstract

A vector field on a Riemannian manifold (M, g) is called concircular if it satisfies ${\nabla}X^v={\mu}X$ for any vector X tangent to M, where ${\nabla}$ is the Levi-Civita connection and ${\mu}$ is a non-trivial function on M. A smooth vector field ${\xi}$ on a Riemannian manifold (M, g) is said to define a Ricci soliton if it satisfies the following Ricci soliton equation: $$\frac{1}{2}L_{\xi}g+Ric={\lambda}g$$, where $L_{\xi}g$ is the Lie-derivative of the metric tensor g with respect to ${\xi}$, Ric is the Ricci tensor of (M, g) and ${\lambda}$ is a constant. A Ricci soliton (M, g, ${\xi}$, ${\lambda}$) on a Riemannian manifold (M, g) is said to have concircular potential field if its potential field is a concircular vector field. In the first part of this paper we determine Riemannian manifolds which admit a concircular vector field. In the second part we classify Ricci solitons with concircular potential field. In the last part we prove some important properties of Ricci solitons on submanifolds of a Riemannian manifold equipped with a concircular vector field.

Keywords

concircular vector field;Ricci soliton;submanifolds;Einstein manifold;concircular potential field;concurrent vector field;concircular curvature tensor

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  2. Pseudo-Z symmetric space-times with divergence-free Weyl tensor and pp-waves vol.13, pp.02, 2016, https://doi.org/10.1142/S0219887816500158
  3. Rigidity of (m,ρ)-quasi Einstein manifolds 2017, https://doi.org/10.1002/mana.201600186
  4. Euclidean Submanifolds via Tangential Components of Their Position Vector Fields vol.5, pp.4, 2017, https://doi.org/10.3390/math5040051
  5. Remarks on the Warped Product Structure from the Hessian of a Function vol.6, pp.12, 2018, https://doi.org/10.3390/math6120275
  6. Concircular vector fields on Lorentzian manifold of Bianchi type-I spacetimes vol.33, pp.12, 2018, https://doi.org/10.1142/S0217732318500633

Acknowledgement

Supported by : NPST