On the classification of warped product Einstein metrics

By Chenxu He, Peter Petersen, W. Wylie. 2010

156 citations recorded by the source corpus.

The question of when an Einstein metric can be written as a warped product is posed in the text Einstein metrics by Besse. Recently, there have been some interesting results about these spaces found by Kim–Kim, Case–Shu–Wei, and Case. They take the perspective of studying these metrics by studying the equation on the base of the warped product. The resulting equation is similar to the Ricci soliton equation and is a natural equation from the work of Bakry– Emery and their collaborators on curvature dimension inequalities. We will also take this perspective and prove two new results: one is an extension of the work of Kim and Kim and Case-Shu-Wei to manifolds with boundary; the other is a classi…cation result of warped products Einstein metrics over locally conformally ‡at base space.

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

View Paper: https://www.semanticscholar.org/paper/2dbd6b5be4088e8ac4ada8d5762d6f37f78d63e3

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. Qiang Chen
4. G. Catino
5. Xiuxiong Chen
6. Yuanqi Wang
7. M. Rigoli
8. P. Mastrolia

In Astronomy and Astrophysics

1. A. Barros
2. E. Ribeiro
3. Qiang Chen
4. Xiuxiong Chen
5. Yuanqi Wang
6. G. Catino
7. M. Rigoli
8. D. Monticelli

In Geometry and Topology

1. A. Barros
2. E. Ribeiro
3. Qiang Chen
4. Xiuxiong Chen
5. Yuanqi Wang
6. G. Catino
7. M. Rigoli
8. D. Monticelli

Citations in the corpus

• 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, unknown): “Natural examples of GQE manifolds are given by Einstein manifolds (when f and λ are two constants), gradient Ricci solitons (when λ is constant and μ = 0), gradient Ricci almost solitons (when μ = 0, see [11]) and quasi-Einstein manifolds (when μ and λ are two constants, see [3,5,9]).”
• On Bach flat warped product Einstein manifolds (2011, methodology): “9 in [HPW1] which is the main step for the global classification in Theorem 7.”
• Characterizations and integral formulae for generalized m-quasi-Einstein metrics (2012, methodology): “2) reduces to one associated to a gradient Ricci soliton, for a good survey in this subject we recommend the work due to Cao in [8], as well as if λ is only constant and m is a positive integer, it corresponds to m-quasi-Einstein metrics that are exactly those n-dimensional manifolds which are the b”
• On Four-Dimensional Anti-self-dual Gradient Ricci Solitons (2011, background): “Later, when Manuel Fernández López kindly brought the references [20] and [18] to our attention, we realized that all Ricci Solitons are analytic without any further assumption.”
• Rigidity of Complete Gradient Steady Ricci Solitons with Harmonic Weyl Curvature (2021, unknown): “As a gradient Ricci soliton, (M, g, f) is real analytic in harmonic coordinates; see [29, 28].”