RIGIDITY OF GRADIENT RICCI SOLITONS

By Peter Petersen, W. Wylie. 2007

316 citations recorded by the source corpus.

We define a gradient Ricci soliton to be rigid if it is a flat bundle N ×GRk where N is Einstein. It is known that not all gradient solitons are rigid. Here we offer several natural conditions on the curvature that characterize rigid gradient solitons. Other related results on rigidity of Ricci solitons are also explained in the last section.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/fae3d925d74f3eba291f6b49bf460b4ae71ab53a

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In Applied Mathematics

In Astronomy and Astrophysics

In Geometry and Topology

Citations in the corpus

Rigidity of shrinking Ricci solitons (2011, background): “Generalizing the trivial Ricci solitons, Petersen and Wylie [16] introduced the notion of rigidity of gradient Ricci solitons.”
On the classification of gradient Ricci solitons (2007, background): “Proof The proofs of the first three identities can be found in [35] (also see [8]).”
Analysis of weighted Laplacian and applications to Ricci solitons (2011, background): “The rigidity of gradient Ricci solitons with constant scalar curvature has been studied in [22].”
Remarks on non-compact gradient Ricci solitons (2009, methodology): “In this note we will focus our attention on geodesically complete, gradient Ricci Solitons. Here are some typical examples, [ 11 ].”
Rigidity of Complete Gradient Steady Ricci Solitons with Harmonic Weyl Curvature (2021, unknown): “immediately follows that complete gradient Ricci solitons with harmonic curvature are rigid, which was first proved by Petersen and Wylie in [38].”
A remark on Einstein warped products (2010, methodology): “n−1)λ. The proof of Theorem 3 will require the following formula, which generalizes to the case m < +∞ similar formulas for Ricci solitons (m = +∞) obtained previously by P. Petersen and W. Wylie, [11]. Following the terminology introduced in [10], the f-Laplacian on the weighted manifold (M,gM,e”
Characterizations and integral formulae for generalized m-quasi-Einstein metrics (2012, methodology): “Before to announce the next results we point out that they are generalizations of ones found in [15] and [3] for Ricci solitons, [4] for almost Ricci solitons and [9] for quasi-Einstein metrics.”
Ricci solitons on Lorentzian manifolds with large isometry groups (2010, background): “Rigid solitons have been systematically studied [15] and further investigated in [8, 10].”
Noncompact Shrinking 4-Solitons with Nonnegative Curvature (2007, result): “ons with positive soliton hessian The results of this section are similar to those proved by the author in the unpublished work by the author [10]. For other interesting results in this direction see [15]. Theorem 7.1. Let (M,g,f) be complete shrinking soliton. Assume Rc≥ 0 and that ∇2f>0, then (”
Ricci almost solitons (2010, background): “Note that, contrary to [23], we do not assume that the scalar curvature is either constant or bounded.”