Stable hypersurfaces in spheres with constant scalar curvature

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].

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Perturbative solutions to the extended constant scalar curvature equations on asymptotically hyperbolic manifolds

Proceedings of the American Mathematical Society ◽

10.1090/s0002-9939-09-09703-2 ◽

2009 ◽

Vol 137 (07) ◽

pp. 2293-2298

Author(s):

Erwann Delay

Keyword(s):

Scalar Curvature ◽

Constant Scalar Curvature ◽

Hyperbolic Manifolds ◽

Asymptotically Hyperbolic Manifolds ◽

Asymptotically Hyperbolic

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Curvature bounds for the spectrum of a compact Riemannian manifold of constant scalar curvature

Journal of Geometric Analysis ◽

10.1007/bf02922246 ◽

2005 ◽

Vol 15 (4) ◽

pp. 589-606 ◽

Cited By ~ 12

Author(s):

Sharief Deshmukh ◽

Afifah Al-Eid

Keyword(s):

Riemannian Manifold ◽

Scalar Curvature ◽

Compact Riemannian Manifold ◽

Constant Scalar Curvature ◽

Curvature Bounds

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Generalized -Einstein Real Hypersurfaces in and

Canadian Mathematical Bulletin ◽

10.4153/s0008439520000156 ◽

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.

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A Note on Randers Metrics of Scalar Flag Curvature

Canadian Mathematical Bulletin ◽

10.4153/cmb-2011-092-1 ◽

2012 ◽

Vol 55 (3) ◽

pp. 474-486 ◽

Cited By ~ 7

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.

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