Yamabe flow and metrics of constant scalar curvature on a complete manifold

2019 ◽  
Vol 58 (1) ◽  
Author(s):  
Li Ma
Keyword(s):  
Scalar Curvature ◽  
Constant Scalar Curvature ◽  
Complete Manifold ◽  
Yamabe Flow

Conformal Geometry Solitons

2012 ◽  
Vol 472-475 ◽  
pp. 123-126
Author(s):  
Rong Rong Cao ◽  
Xiang Gao
Keyword(s):  
Scalar Curvature ◽  
Compact Manifold ◽  
Hilbert Function ◽  
Conformal Geometry ◽  
Constant Scalar Curvature ◽  
Yamabe Flow ◽  
Noncompact Manifolds ◽  
Monotone Formula ◽  
Yamabe Solitons ◽  
Yamabe Soliton

In this paper, we deal with a generalization of the Yamabe flow named conformal geometry flow. Firstly we derive a monotone formula of the Einstein-Hilbert functional under the conformal geometry flow. Then we prove the properties that the conformal geometry solitons and conformal geometry breather both have constant scalar curvature at each time by using the modified Einstein-Hilbert function. Finally we present some properties of Yamabe solitons in compact manifold and noncompact manifolds through the equation of Yamabe soliton.

A 3-DIMENSIONAL SASAKIAN METRIC AS A YAMABE SOLITON

2012 ◽  
Vol 09 (04) ◽  
pp. 1220003 ◽  
Author(s):  
RAMESH SHARMA
Keyword(s):  
Vector Field ◽  
Scalar Curvature ◽  
Constant Curvature ◽  
Constant Scalar Curvature ◽  
Complete Manifold ◽  
3 Dimensional ◽  
Infinitesimal Automorphism ◽  
Contact Metric Structure ◽  
Sasakian Metric ◽  
Yamabe Soliton

If a 3-dimensional Sasakian metric on a complete manifold (M, g) is a Yamabe soliton, then we show that g has constant scalar curvature, and the flow vector field V is Killing. We further show that, either M has constant curvature 1, or V is an infinitesimal automorphism of the contact metric structure on M.

Yamabe solitons on 3-dimensional contact metric manifolds with Qφ = φQ

2019 ◽  
Vol 16 (03) ◽  
pp. 1950039 ◽  
Author(s):  
V. Venkatesha ◽  
Devaraja Mallesha Naik
Keyword(s):  
Vector Field ◽  
Scalar Curvature ◽  
Constant Scalar Curvature ◽  
Sasakian Manifold ◽  
Contact Metric Manifold ◽  
Reeb Vector Field ◽  
3 Dimensional ◽  
Flow Vector ◽  
Contact Metric Manifolds ◽  
Yamabe Soliton

If [Formula: see text] is a 3-dimensional contact metric manifold such that [Formula: see text] which admits a Yamabe soliton [Formula: see text] with the flow vector field [Formula: see text] pointwise collinear with the Reeb vector field [Formula: see text], then we show that the scalar curvature is constant and the manifold is Sasakian. Moreover, we prove that if [Formula: see text] is endowed with a Yamabe soliton [Formula: see text], then either [Formula: see text] is flat or it has constant scalar curvature and the flow vector field [Formula: see text] is Killing. Furthermore, we show that if [Formula: see text] is non-flat, then either [Formula: see text] is a Sasakian manifold of constant curvature [Formula: see text] or [Formula: see text] is an infinitesimal automorphism of the contact metric structure on [Formula: see text].

Generalized -Einstein Real Hypersurfaces in and

2020 ◽  
Vol 63 (4) ◽  
pp. 909-920
Author(s):  
Yaning Wang
Keyword(s):  
Scalar Curvature ◽  
Mean Curvature ◽  
Constant Coefficient ◽  
Real Hypersurface ◽  
Constant Mean Curvature ◽  
Constant Scalar Curvature ◽  
Real Hypersurfaces

AbstractIn this paper we obtain some new characterizations of pseudo-Einstein real hypersurfaces in $\mathbb{C}P^{2}$ and $\mathbb{C}H^{2}$. More precisely, we prove that a real hypersurface in $\mathbb{C}P^{2}$ or $\mathbb{C}H^{2}$ with constant mean curvature is generalized ${\mathcal{D}}$-Einstein with constant coefficient if and only if it is pseudo-Einstein. We prove that a real hypersurface in $\mathbb{C}P^{2}$ with constant scalar curvature is generalized ${\mathcal{D}}$-Einstein with constant coefficient if and only if it is pseudo-Einstein.

A Note on Randers Metrics of Scalar Flag Curvature

2012 ◽  
Vol 55 (3) ◽  
pp. 474-486 ◽  
Author(s):  
Bin Chen ◽  
Lili Zhao
Keyword(s):  
Scalar Curvature ◽  
Three Dimensional ◽  
Constant Scalar Curvature ◽  
Flag Curvature ◽  
Projectively Flat ◽  
Randers Metrics

AbstractSome families of Randers metrics of scalar flag curvature are studied in this paper. Explicit examples that are neither locally projectively flat nor of isotropic S-curvature are given. Certain Randers metrics with Einstein α are considered and proved to be complex. Three dimensional Randers manifolds, with α having constant scalar curvature, are studied.

Spacelike hypersurfaces with constant scalar curvature

1998 ◽  
Vol 95 (4) ◽  
pp. 499-505 ◽  
Author(s):  
Qing-Ming Cheng ◽  
Susumu Ishikawa
Keyword(s):  
Scalar Curvature ◽  
Constant Scalar Curvature ◽  
Spacelike Hypersurfaces
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