## $$r$$-almost Newton-Ricci solitons immersed in a Lorentzian manifold: examples, nonexistence and rigidity.(English)Zbl 1437.53033

Summary: We establish the concept of $$r$$-almost Newton-Ricci soliton immersed into a Lorentzian manifold, which extends in a natural way the almost Ricci solitons introduced in [S. Pigola et al., Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 10, No. 4, 757–799 (2011; Zbl 1239.53057)]. In this setting, under suitable hypothesis on the potential and soliton functions, we obtain nonexistence and rigidity results. Some interesting examples of these new geometric objects are also given.

### MSC:

 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics 53C24 Rigidity results

Zbl 1239.53057
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