Classification of gradient shrinking Ricci solitons with bounded Ricci curvature

2020 ◽  
Vol 71 ◽  
pp. 101637
Author(s):  
Fei Yang ◽  
Zijun Wang ◽  
Liangdi Zhang
Keyword(s):  
Ricci Curvature ◽  
Ricci Solitons

CLASSIFICATION OF STEADY GRADIENT RICCI SOLITONS ON TWO-MANIFOLDS

2012 ◽  
Vol 09 (05) ◽  
pp. 1250049 ◽  
Author(s):  
GABRIEL BERCU ◽  
MIHAI POSTOLACHE
Keyword(s):  
Riemannian Manifold ◽  
Case Studies ◽  
Wide Class ◽  
Positive Function ◽  
Ricci Solitons ◽  
Gradient Ricci Solitons

In our very recent published work [Int. J. Geom. Meth. Mod. Phys.8(4) (2011) 783–796], we considered the Riemannian manifold M = ℝ2 endowed with the warped metric ḡ(x, y) = diag (g(y), 1), where g is a positive function, of C∞-class, depending on the variable y only. Within this framework, we found a wide class of 2D gradient Ricci solitons and specialized our results to discuss some case studies. This research is a natural continuation, providing classification results for the subclass of steady gradient Ricci solitons.

Classification of gradient expanding and steady Ricci solitons

2019 ◽  
Vol 301 (1) ◽  
pp. 371-384
Author(s):  
Fei Yang ◽  
Shouwen Fang ◽  
Liangdi Zhang
Keyword(s):  
Ricci Solitons

scholarly journals Classification of Ricci solitons on Euclidean hypersurfaces

2014 ◽  
Vol 25 (11) ◽  
pp. 1450104 ◽  
Author(s):  
Bang-Yen Chen ◽  
Sharief Deshmukh
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Riemannian Manifolds ◽  
Potential Field ◽  
Vector Fields ◽  
Ricci Soliton ◽  
Position Vector ◽  
Ricci Solitons ◽  
Euclidean Submanifolds

A Ricci soliton (M, g, v, λ) on a Riemannian manifold (M, g) is said to have concurrent potential field if its potential field v is a concurrent vector field. Ricci solitons arisen from concurrent vector fields on Riemannian manifolds were studied recently in [Ricci solitons and concurrent vector fields, preprint (2014), arXiv:1407.2790]. The most important concurrent vector field is the position vector field on Euclidean submanifolds. In this paper we completely classify Ricci solitons on Euclidean hypersurfaces arisen from the position vector field of the hypersurfaces.

COMPACT LOCALLY CONFORMALLY FLAT RIEMANNIAN MANIFOLDS

2001 ◽  
Vol 33 (4) ◽  
pp. 459-465 ◽  
Author(s):  
QING-MING CHENG
Keyword(s):  
Scalar Curvature ◽  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Space Form ◽  
Constant Scalar Curvature ◽  
Topological Classification ◽  
Conformally Flat ◽  
Riemannian Product ◽  
Locally Conformally Flat

First, we shall prove that a compact connected oriented locally conformally flat n-dimensional Riemannian manifold with constant scalar curvature is isometric to a space form or a Riemannian product Sn−1(c) × S1 if its Ricci curvature is nonnegative. Second, we shall give a topological classification of compact connected oriented locally conformally flat n-dimensional Riemannian manifolds with nonnegative scalar curvature r if the following inequality is satisfied: [sum ]i,jR2ij [les ] r2/(n−1), where [sum ]i,jR2ij is the squared norm of the Ricci curvature tensor.

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