Abstract:
In this paper, we introduce the notion of Jacobi-type
vector fields on Riemannian manifolds, which is a generalization of the
Jacobi field along a geodesic. We study Ricci solitons with positive Ricci
curvature whose potential vector field is a Jacobi-type vector field and
show that if the metric on Ricci soliton is replaced by the Ricci tensor,
then we get a Riemannian manifold that is an Einstein manifold. As a
by-product, we get a criterion for compactness of a complete Ricci soliton
using a Jacobi-type vector field. Finally it is shown that a Ricci soliton
of positive Ricci curvature whose potential field is Jacobi-type vector
field is necessarily an Einstein manifold.
Key Words: Jacobi-type vector fields, Killing vector fields,
Ricci soliton, Einstein manifolds.
2000 Mathematics Subject Classification: Primary: 53C20
Secondary: 53C25, 53B21.
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