Critical points of the total scalar curvature functional on the space of metrics of constant scalar curvature

2000 ◽  
Vol 103 (2) ◽  
pp. 135-142 ◽  
Author(s):  
Seungsu Hwang
Keyword(s):  
Scalar Curvature ◽  
Critical Points ◽  
Constant Scalar Curvature ◽  
Total Scalar Curvature ◽  
Scalar Curvature Functional

scholarly journals Killing fields generated by multiple solutions to the Fischer–Marsden equation

2015 ◽  
Vol 26 (04) ◽  
pp. 1540006 ◽  
Author(s):  
Paul Cernea ◽  
Daniel Guan
Keyword(s):  
Scalar Curvature ◽  
Multiple Solutions ◽  
Vector Fields ◽  
Killing Vector ◽  
Solution Space ◽  
Constant Scalar Curvature ◽  
Killing Fields ◽  
Sectional Curvatures ◽  
Almost All ◽  
Scalar Curvature Functional

In the process of finding Einstein metrics in dimension n ≥ 3, we can search critical metrics for the scalar curvature functional in the space of the fixed-volume metrics of constant scalar curvature on a closed oriented manifold. This leads to a system of PDEs (which we call the Fischer–Marsden Equation, after a conjecture concerning this system) for scalar functions, involving the linearization of the scalar curvature. The Fischer–Marsden conjecture said that if the equation admits a solution, the underlying Riemannian manifold is Einstein. Counter-examples are known by O. Kobayashi and J. Lafontaine. However, almost all the counter-examples are homogeneous. Multiple solutions to this system yield Killing vector fields. We show that the dimension of the solution space W can be at most n+1, with equality implying that (M, g) is a sphere with constant sectional curvatures. Moreover, we show that the identity component of the isometry group has a factor SO(W). We also show that geometries admitting Fischer–Marsden solutions are closed under products with Einstein manifolds after a rescaling. Therefore, we obtain a lot of non-homogeneous counter-examples to the Fischer–Marsden conjecture. We then prove that all the homogeneous manifold M with a solution are in this case. Furthermore, we also proved that a related Besse conjecture is true for the compact homogeneous manifolds.

scholarly journals On static manifolds and related critical spaces with cyclic parallel Ricci tensor

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
H. Baltazar ◽  
A. Da Silva
Keyword(s):  
Scalar Curvature ◽  
Ricci Tensor ◽  
Critical Spaces ◽  
3 Dimensional ◽  
Critical Metrics ◽  
Volume Functional ◽  
Total Scalar Curvature ◽  
Cyclic Parallel Ricci Tensor ◽  
Scalar Curvature Functional ◽  
Parallel Ricci Tensor

Abstract We classify 3-dimensional compact Riemannian manifolds (M 3, g) that admit a non-constant solution to the equation −Δfg +Hess f − f Ric = μ Ric +λg for some special constants (μ, λ), under the assumption that the manifold has cyclic parallel Ricci tensor. Namely, the structures that we study here are: positive static triples, critical metrics of the volume functional, and critical metrics of the total scalar curvature functional. We also classify n-dimensional critical metrics of the volume functional with non-positive scalar curvature and satisfying the cyclic parallel Ricci tensor condition.

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].

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