What question did this study set out to answer?

This work aims to identify solutions for gradient Riemann solitons that are invariant under the orthogonal group.

February 9, 2026Filomat0 citationsOpen Access

On gradient Riemann solitons invariant by rotation

IMIlton MenezesUniversidade Federal do Oeste da Bahia EBElismar Dias BatistaUniversidade Federal do Oeste da Bahia

Key Points

This work aims to identify solutions for gradient Riemann solitons that are invariant under the orthogonal group.
Considered Riemann solitons conformal to Euclidean space
Assumed invariance under the action of the orthogonal group
Analyzed properties of the gradient Riemann solitons through a system of partial differential equations
Demonstrated that a gradient Riemann soliton is geodesically complete and rotationally symmetric
Concluded that the soliton metric is the canonical product on R \times S^(n-1)
Established that the identified soliton is shrinking

Abstract

In this paper, we consider Riemann solitons that are conformal to a Euclidean space. Assuming that the solutions of the presented system of partial differential equations are invariant under the action of the orthogonal group, we provide all solutions for the gradient Riemann solitons. We show that a gradient Riemann soliton (M, g) is both geodesically complete and rotationally symmetric if and only if g is the canonical product metric on R S^n-1. Furthermore, this soliton is shrinking.

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