On shrinking gradient Ricci solitons with nonnegative sectional curvature

We introduce a study of Riemannian manifold M = ℝ2 endowed with a metric of diagonal type of the form [Formula: see text], where g is a positive function, of C∞-class, depending on the variable x2 only. We emphasize the role of metric [Formula: see text] in determining manifolds having negative, null or positive sectional curvature. Within this framework, we find a wide class of gradient Ricci solitons (see, Theorems 4 and 7) and specialize these results to discuss some 2D and 4D case studies. The present study can be thought as a natural continuation of those included in monograph [22] by Constantin Udrişte, and to those in the research article [12] by Richard S. Hamilton (the result in Proposition 8 is precisely the famous "Hamilton cigar" in polar coordinates).

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ON LOCALLY CONFORMALLY FLAT GRADIENT SHRINKING RICCI SOLITONS

Communications in Contemporary Mathematics ◽

10.1142/s0219199711004191 ◽

2011 ◽

Vol 13 (02) ◽

pp. 269-282 ◽

Cited By ~ 22

Author(s):

XIAODONG CAO ◽

BIAO WANG ◽

ZHOU ZHANG

Keyword(s):

Sectional Curvature ◽

Ricci Curvature ◽

Exponential Growth ◽

Integral Identity ◽

Ricci Solitons ◽

Riemannian Curvature ◽

Conformally Flat ◽

Gradient Ricci Solitons ◽

Riemannian Curvature Tensor ◽

Locally Conformally Flat

In this paper, we first apply an integral identity on Ricci solitons to prove that closed locally conformally flat gradient Ricci solitons are of constant sectional curvature. We then generalize this integral identity to complete noncompact gradient shrinking Ricci solitons, under the conditions that the Ricci curvature is bounded from below and the Riemannian curvature tensor has at most exponential growth. As a consequence, we classify complete locally conformally flat gradient shrinking Ricci solitons with Ricci curvature bounded from below.

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Stein manifolds of nonnegative curvature

Advances in Geometry ◽

10.1515/advgeom-2016-0025 ◽

2018 ◽

Vol 18 (3) ◽

pp. 285-287

Author(s):

Xiaoyang Chen

Keyword(s):

Fixed Point ◽

Sectional Curvature ◽

Normal Bundle ◽

Stein Manifold ◽

Riemannian Metric ◽

Nonnegative Curvature ◽

Fixed Point Set ◽

Nonnegative Sectional Curvature ◽

Point Set ◽

Nonempty Compact

AbstractLet X bea Stein manifold with an anti-holomorphic involution τ and nonempty compact fixed point set Xτ. We show that X is diffeomorphic to the normal bundle of Xτ provided that X admits a complete Riemannian metric g of nonnegative sectional curvature such that τ*g = g.

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GRADIENT RICCI SOLITONS WITH SEMI-SYMMETRY

Bulletin of the Korean Mathematical Society ◽

10.4134/bkms.2014.51.1.213 ◽

2014 ◽

Vol 51 (1) ◽

pp. 213-219

Author(s):

Jong Taek Cho ◽

Jiyeon Park

Keyword(s):

Ricci Solitons ◽

Gradient Ricci Solitons

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CLASSIFICATION OF STEADY GRADIENT RICCI SOLITONS ON TWO-MANIFOLDS

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887812500491 ◽

2012 ◽

Vol 09 (05) ◽

pp. 1250049 ◽

Cited By ~ 2

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.

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