scholarly journals ON UNIQUENESS AND DIFFERENTIABILITY IN THE SPACE OF YAMABE METRICS

2005 ◽  
Vol 07 (03) ◽  
pp. 299-310 ◽  
Author(s):  
MICHAEL T. ANDERSON
Keyword(s):  
Scalar Curvature ◽  
Sufficient Conditions ◽  
Locally Maximal ◽  
Generic Set ◽  
Scalar Curvature Functional

It is shown that there is a unique Yamabe representative for a generic set of conformal classes in the space of metrics on any manifold. At such classes, the scalar curvature functional is shown to be differentiable on the space of Yamabe metrics. In addition, some sufficient conditions are given which imply that a Yamabe metric of locally maximal scalar curvature is necessarily Einstein.

Local behavior of Iwasawa’s invariants

2017 ◽  
Vol 13 (04) ◽  
pp. 1013-1036 ◽  
Author(s):  
Sören Kleine
Keyword(s):  
Number Field ◽  
Sufficient Conditions ◽  
Main Tool ◽  
Local Behavior ◽  
Locally Maximal

Let [Formula: see text] be a number field, let [Formula: see text] denote a fixed rational prime. We study the local behavior of Iwasawa’s invariants as functions on the set [Formula: see text] of all [Formula: see text]-extensions of [Formula: see text]. With respect to a certain topology on [Formula: see text] that takes care of ramification, we prove that for each [Formula: see text] the [Formula: see text]-invariant of [Formula: see text] is locally maximal among the [Formula: see text]-invariants, and we give sufficient conditions for the [Formula: see text]-invariant to be locally maximal (e.g., a vanishing [Formula: see text]-invariant). This concerns a question raised by R. Greenberg in 1973. Our main result also provides information about [Formula: see text]- (and even [Formula: see text]-) invariants in the case of a nonvanishing [Formula: see text]-invariant. The main tool used in the proof is a new result based on the stabilization of certain ranks, which considerably generalizes a theorem of T. Fukuda.

TENSORIAL CURVATURE AND D-DIFFERENTIATION PART II: "PRINCIPAL" KIND AND EINSTEIN–MAXWELL THEORY

2007 ◽  
Vol 04 (05) ◽  
pp. 847-860 ◽  
Author(s):  
D. J. HURLEY ◽  
M. A. VANDYCK
Keyword(s):  
Scalar Curvature ◽  
Tensor Field ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Curvature Operator ◽  
Maxwell Theory ◽  
Covariant Differentiation ◽  
Necessary And Sufficient

The class of "commutative" D-operators, which was introduced in the first part of this paper, is generalized to obtain the "principal" class. It is established that principal D-operators are expressible in terms of covariant differentiation and a tensor field. Necessary and sufficient conditions are determined for the curvature operator to be tensorial, and for the scalar curvature to exist. As an application, the Einstein–Maxwell theory is recast in a new geometrical framework.

scholarly journals Conformal vector fields and Yamabe solitons

2019 ◽  
Vol 16 (04) ◽  
pp. 1950053
Author(s):  
Nasser Bin Turki ◽  
Bang-Yen Chen ◽  
Sharief Deshmukh
Keyword(s):  
Scalar Curvature ◽  
Vector Fields ◽  
Sufficient Conditions ◽  
Constant Scalar Curvature ◽  
Conformal Vector ◽  
Conformal Vector Fields ◽  
Yamabe Solitons

In this paper, we use less topological restrictions and more geometric and analytic conditions to obtain some sufficient conditions on Yamabe solitons such that their metrics are Yamabe metrics, that is, metrics of constant scalar curvature. More precisely, we use properties of conformal vector fields to find several sufficient conditions on the soliton vector fields of Yamabe solitons under which their metrics are Yamabe metrics.

ALMOST KÄHLER 4-MANIFOLDS, L2-SCALAR CURVATURE FUNCTIONAL AND SEIBERG–WITTEN EQUATIONS

2004 ◽  
Vol 15 (06) ◽  
pp. 573-580 ◽  
Author(s):  
MITSUHIRO ITOH
Keyword(s):  
Scalar Curvature ◽  
Topological Condition ◽  
Witten Theory ◽  
Almost Kähler ◽  
Scalar Curvature Functional

We show in this paper by applying the Seiberg–Witten theory developed by Taubes and LeBrun that a compact almost Kähler–Einstein 4-manifold of negative scalar curvature s is Kähler–Einstein if and only if the L2-norm satisfies ∫Ms2dv=32π2(2χ+3τ)(M). The Einstein condition can be weakened by the topological condition (2χ+3τ)(M)>0.

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.

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