SOME RESULTS ON CONCIRCULAR VECTOR FIELDS AND THEIR APPLICATIONS TO RICCI SOLITONS
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초록
A vector field on a Riemannian manifold (M, g) is called concircular if it satisfies ∇Xv = μX for any vector X tangent to M, where ∇ is the Levi-Civita connection and μ is a non-trivial function on M. A smooth vector field ξ on a Riemannian manifold (M, g) is said to define a Ricci soliton if it satisfies the following Ricci soliton equation: 1/2 Lξg + Ric = λg, where Lξg is the Lie-derivative of the metric tensor g with respect to ξ, Ric is the Ricci tensor of (M, g) and λ is a constant. A Ricci soliton (M, g, ξ,λ) on a Riemannian manifold (M, g) is said to have concircular potential field if its potential field ξ 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.
Abstract
A vector field on a Riemannian manifold (M, g) is called concircular if it satisfies ∇Xv = μX for any vector X tangent to M, where ∇ is the Levi-Civita connection and μ is a non-trivial function on M. A smooth vector field ξ on a Riemannian manifold (M, g) is said to define a Ricci soliton if it satisfies the following Ricci soliton equation: 1/2 Lξg + Ric = λg, where Lξg is the Lie-derivative of the metric tensor g with respect to ξ, Ric is the Ricci tensor of (M, g) and λ is a constant. A Ricci soliton (M, g, ξ,λ) on a Riemannian manifold (M, g) is said to have concircular potential field if its potential field ξ 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.
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