On gradient \(\eta\)-Einstein solitons. (English) Zbl 1488.53132
Summary: If the potential vector field of an \(\eta\)-Einstein soliton is of gradient type, using Bochner formula, we derive from the soliton equation a nonlinear second order PDE. Under certain conditions, the existence of an \(\eta\)-Einstein soliton forces the manifold to be of constant scalar curvature.
MSC:
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |
| 53B50 | Applications of local differential geometry to the sciences |
| 53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |
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