Ricci almost solitons

By S. Pigola, M. Rigoli, Michele Rimoldi, A. G. Setti. 2010

336 citations recorded by the source corpus.

We introduce a natural extension of the concept of gradient Ricci soliton: the Ricci almost soliton. We provide existence and rigidity results, we deduce a-priori curvature estimates and isolation phenomena, and we investigate some topological properties. A number of differential identities involving the relevant geometric quantities are derived. Some basic tools from the weighted manifold theory such as general weighted volume comparisons and maximum principles at infinity for diffusion operators are discussed.

Subfields: Applied Mathematics, Astronomy and Astrophysics

View Paper: https://www.semanticscholar.org/paper/ad6d5cc8806ff0bff6c1f25a6e360800c35abdd4

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. A. Barros
2. E. Ribeiro
3. Amalendu Ghosh
4. G. Catino
5. Michele Rimoldi
6. M. Rigoli
7. A. G. Setti
8. S. Pigola

In Astronomy and Astrophysics

1. A. Barros
2. E. Ribeiro
3. Amalendu Ghosh
4. G. Catino
5. Ramesh Sharma
6. J. Gomes
7. M. Rigoli
8. D. Monticelli

Citations in the corpus

• A note on rigidity of the almost Ricci soliton (2013, background): “In fact, we refer the reader to [10] to see some of these changes.”
• On the geometry of gradient einstein-type manifolds (2014, background): “Other examples are, for instance, Ricci almost solitons ([41]), Yamabe solitons ([19], [31],[21], [9]), Yamabe quasi-solitons ([27], [45]), conformal gradient solitons ([43], [15]), quasi-Einstein manifolds ([30], [12], [16], [26], [35]), ρ-Einstein solitons ([17], [18]).”
• Generalized quasi-Einstein manifolds with harmonic Weyl tensor (2010, positive): “In particular, we can prove a local characterization for locally conformally flat Ricci almost solitons (which have been introduced in [11]), similar to the one for Ricci solitons [2,4].”
• Compact almost Ricci solitons with constant scalar curvature are gradient (2012, methodology): “ng; see e.g. Theorem6 in [10]. Therefore, the fourth assumption is relevant when the scalar curvature is positive. Actually, Corollary 1 is used to give an answer to the following problem proposed in [11]: Problem1. Under which conditions a compact almost Ricci soliton is necessarily gradient? As a ”
• Characterizations and integral formulae for generalized m-quasi-Einstein metrics (2012, methodology): “2) we obtain the almost Ricci soliton equation, for more details see [4, 16].”
• Remarks on non-compact gradient Ricci solitons (2009, background): “It is easy to show that Einstein, gradient Ricci solitons are either trivial or Ricci flat; see, e.g., the proof of Theorem 1.3 in [ 18 ].”
• Certain Contact Metrics as Ricci Almost Solitons (2014, background): “[18] introduced and studied the notion of Ricci almost solitons on a Riemannian manifold, defined by the same Eq.”
• Almost Ricci solitons and $$K$$K-contact geometry (2014, background): “[11] defined an almost Ricci soliton as a Riemannian manifold (M, g) satisfying the condition: £X gi j + 2Ri j = 2λgi j .”