Bi-conformal vector fields and the local geometric characterization of conformally separable pseudo-Riemannian manifolds I

2006 ◽  
Vol 56 (7) ◽  
pp. 1069-1095 ◽  
Author(s):  
Alfonso García-Parrado Gómez-Lobo
Keyword(s):  
Riemannian Manifolds ◽  
Vector Fields ◽  
Geometric Characterization ◽  
Conformal Vector ◽  
Conformal Vector Fields

scholarly journals Closed conformal vector fields on pseudo-Riemannian manifolds

2006 ◽  
Vol 2006 ◽  
pp. 1-8 ◽  
Author(s):  
D. A. Catalano
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Riemannian Manifolds ◽  
Vector Fields ◽  
Geometric Proof ◽  
Conformal Vector Field ◽  
Conformal Vector ◽  
Conformal Vector Fields ◽  
Local Coordinates

We give here a geometric proof of the existence of certain local coordinates on a pseudo-Riemannian manifold admitting a closed conformal vector field.

Conformal vector fields on Finsler manifolds

2020 ◽  
Vol 31 (12) ◽  
pp. 2050095
Author(s):  
Qiaoling Xia
Keyword(s):  
Sectional Curvature ◽  
Vector Fields ◽  
Riemannian Metric ◽  
Finsler Manifolds ◽  
Conformal Vector ◽  
Conformal Fields ◽  
Special Cases ◽  
Conformal Vector Fields ◽  
Equivalent Characterization

In this paper, we give an equivalent characterization of conformal vector fields on a Finsler manifold [Formula: see text], whose metric [Formula: see text] is defined by a Riemannian metric [Formula: see text] and a 1-form [Formula: see text]. This characterization contains all related results in [Z. Shen and Q. Xia, On conformal vector fields on Randers manifolds, Sci. China Math. 55(9) (2012) 1869–1882; Z. Shen and M. Yuan, Conformal vector fields on some Finsler manifolds, Sci. China Math. 59(1) (2016) 107–114; X. Cheng, Y. Li and T. Li, The conformal vector fields on Kropina manifolds, Diff. Geom. Appl. 56 (2018) 344–354] as special cases. Further, we determine conformal fields on some Finsler manifolds [Formula: see text] when [Formula: see text] is of constant sectional curvature and [Formula: see text] is a conformal 1-form with respect to [Formula: see text].

scholarly journals Characterization of Almost Yamabe Solitons and Gradient Almost Yamabe Solitons with Conformal Vector Fields

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2362
Author(s):  
Ali H. Alkhaldi ◽  
Pişcoran Laurian-Ioan ◽  
Abimbola Abolarinwa ◽  
Akram Ali
Keyword(s):  
Scalar Curvature ◽  
Vector Fields ◽  
Sufficient Conditions ◽  
Constant Scalar Curvature ◽  
Topological Constraints ◽  
Conformal Vector ◽  
Conformal Vector Fields ◽  
Yamabe Solitons

In this paper, some sufficient conditions of almost Yamabe solitons are established, such that the solitons are Yamabe metrics, by which we mean metrics of constant scalar curvature. This is achieved by imposing fewer topological constraints. The properties of the conformal vector fields are exploited for the purpose of establishing various necessary criteria on the soliton vector fields of gradient almost Yamabe solitons so as to obtain Yamabe metrics.

scholarly journals On the zeros of a conformal vector field

1974 ◽  
Vol 55 ◽  
pp. 1-3 ◽  
Author(s):  
David E. Blair
Keyword(s):  
Vector Field ◽  
Riemannian Manifolds ◽  
Vector Fields ◽  
Short Note ◽  
Killing Vector ◽  
Connected Components ◽  
Conformal Vector Field ◽  
Totally Geodesic Submanifolds ◽  
Conformal Vector ◽  
Conformal Vector Fields

In [1] S. Kobayashi showed that the connected components of the set of zeros of a Killing vector field on a Riemannian manifold (Mn,g) are totally geodesic submanifolds of (Mn,g) of even codimension including the case of isolated singular points. The purpose of this short note is to give a simple proof of the corresponding result for conformal vector fields on compact Riemannian manifolds. In particular we prove the following

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