Abstract
Finding characterizations of trivial solitons is an important problem in geometry of Ricci solitons. In this paper, we find several characterizations of a trivial Ricci soliton. First, on a complete shrinking Ricci soliton, we show that the scalar curvature satisfying a certain inequality gives a characterization of a trivial Ricci soliton. Then, it is shown that the potential field having geodesic flow and length of potential field satisfying certain inequality gives another characterization of a trivial Ricci soliton. Finally, we show that the potential field of constant length satisfying an inequality gives a characterization of a trivial Ricci soliton.
Highlights
Recall that Ricci solitons, being self-similar solutions of the Ricci flow, are a topic of current interest
If the potential field u = ∇f is a gradient of a smooth function f, ðM, g,∇f, λÞ is called a gradient Ricci soliton, and in this case, equation (1) takes the form
One of the important findings on Ricci solitons is that if it is compact, the potential field u is a gradient of a smooth function f, that is, a compact Ricci soliton is a gradient Ricci soliton
Summary
Recall that Ricci solitons, being self-similar solutions of the Ricci flow (cf. [1]), are a topic of current interest. One of the important findings on Ricci solitons is that if it is compact, the potential field u is a gradient of a smooth function f , that is, a compact Ricci soliton is a gradient Ricci soliton (cf [1]). We show that for a connected Ricci soliton ðM, g, u, λÞ the flow of potential field u being geodesic flow with its length kuk satisfying certain inequality gives a characterization of a trivial Ricci soliton (cf Theorem 2). It is observed that potential field u being of constant length satisfying certain inequality on a connected Ricci soliton ðM, g, u, λÞ gives a characterization of a trivial Ricci soliton (cf Theorem 4)
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