Compact hyperbolic Ricci solitons with constant scalar curvature and 2-conformal vector fields

A Yang-Mills theory linear in the scalar curvature for two-dimensional gravity with symmetry generated by the semidirect product formed with the Lie derivative of the algebra of diffeomorphisms with the two-dimensional Abelian algebra is formulated. As compared with dilaton models, the role of the dilaton is played by the dual field strength of a $U(1)$ gauge field. All vacuum solutions are found. They are either black holes or have constant scalar curvature. Those with constant scalar curvature have constant dual field strength. In particular, solutions with vanishing cosmological constant but nonzero scalar curvature exist. In the conformal-Lorenz gauge, the model has a conformal field theory interpretation whose residual symmetry combines holomorphic diffeomorphisms with a subclass of $U(1)$ gauge transformations while preserving two-dimensional de Sitter and anti-de Sitter boundary conditions. This is the same symmetry as in Jackiw-Teitelboim-Maxwell gravity considered by Hartman and Strominger. It is argued that this is the only nontrivial Yang-Mills model linear in the scalar curvature that exists for real Lie algebras of dimension four.

Generalizing a theorem of Huang, Cheng and Wan classified the complete hypersurfaces of $\mathbb R^4$ with non-zero constant mean curvature and constant scalar curvature. In our work, we obtain results of this nature in higher dimensions. In particular, we prove that if a complete hypersurface of $\mathbb R^5$ has constant mean curvature $H\neq 0$ and constant scalar curvature $R\geq\frac{2}{3}H^2$, then $R=H^2$, $R=\frac{8}{9}H^2$ or $R=\frac{2}{3}H^2$. Moreover, we characterize the hypersurface in the cases $R=H^2$ and $R=\frac{8}{9}H^2$, and provide an example in the case $R=\frac{2}{3}H^2$. The proofs are based on the principal curvature theorem of Smyth-Xavier and a well known formula for the Laplacian of the squared norm of the second fundamental form of a hypersurface in a space form.

Complete, constant positive scalar curvature metrics on S n {p 0 ,...,p k } admit a definite asymptotic structure, i.e., the metric is asymptotically S n-1 -invariant near each puncture. This allows one to glue together two such metrics near their punctures, provided the asyrnptotics match, and create a metric whose scalar curvature is nearly constant. In this paper we perturb this new metric to obtain a new metric with constant scalar curvature.

Submanifolds with constant scalar curvature in a space form

Memory effects in the exact Kundt wave spacetimes are shown to arise in the behaviour of geodesics in such spacetimes. The types of Kundt spacetimes we consider here are direct products of the form H2×M(1,1) and S2×M(1,1). Both geometries have constant scalar curvature. We consider a scenario in which initial velocities of the transverse geodesic coordinates are set to zero (before the arrival of the pulse) in a spacetime with non-vanishing background curvature. We look for changes in the separation between pairs of geodesics caused by the pulse. Any relative change observed in the position and velocity profiles of geodesics, after the burst, can be solely attributed to the wave (hence, a memory effect). For constant negative curvature, we find there is permanent change in the separation of geodesics after the pulse has departed. Thus, there is displacement memory, though no velocity memory is found. In the case of constant positive scalar curvature (Plebański–Hacyan spacetimes), we find both displacement and velocity memory along one direction. In the other direction, a new kind of memory (which we term as frequency memory effect) is observed where the separation between the geodesics shows periodic oscillations once the pulse has left. We also carry out similar analyses for spacetimes with a non-constant scalar curvature, which may be positive or negative. The results here seem to qualitatively agree with those for constant scalar curvature, thereby suggesting a link between the nature of memory and curvature.

The aim of this note is to prove that any compact non-trivial almost Ricci soliton $$\big (M^n,\,g,\,X,\,\lambda \big )$$ with constant scalar curvature is isometric to a Euclidean sphere $$\mathbb {S}^{n}$$ . As a consequence we obtain that every compact non-trivial almost Ricci soliton with constant scalar curvature is gradient. Moreover, the vector field $$X$$ decomposes as the sum of a Killing vector field $$Y$$ and the gradient of a suitable function.

The Yamabe problem (proved in 1984) guarantees the existence of a metric of constant scalar curvature in each conformal class of Riemannian metrics on a compact manifold of dimension n ≥ 3, which minimizes the total scalar curvature on this conformal class. Let (M′, g′) and (M″, g″) be compact Riemannian n‐manifolds. We form their connected sumM′#M″ by removing small balls of radius ϵ from M′, M″ and gluing together the 𝒮n−1 boundaries, and make a metric g on M′#M″ by joining together g′, g″ with a partition of unity. In this paper, we use analysis to study metrics with constant scalar curvature on M′#M″ in the conformal class of g. By the Yamabe problem, we may rescale g′ and g″ to have constant scalar curvature 1, 0, or −1. Thus, there are 9 cases, which we handle separately. We show that the constant scalar curvature metrics either develop small “necks” separating M′ and M″, or one of M′, M″ is crushed small by the conformal factor. When both sides have positive scalar curvature, we find three metrics with scalar curvature 1 in the same conformal class.

We prove existence in the Minkowski space of entire spacelike hypersurfaces with constant negative scalar curvature and given set of lightlike directions at infinity; we also construct the entire scalar curvature flow with prescribed set of lightlike directions at infinity, and prove that the flow converges to a spacelike hypersurface with constant scalar curvature. The proofs rely on barriers construction and a priori estimates.

The purpose of the paper is to study of Para-Kenmotsu metric as a $\eta$-Ricci soliton. The paper is organized as follows: * If an $\eta$-Einstein para-Kenmotsu metric represents an $\eta$-Ricci soliton with flow vector field $V$, then it is Einstein with constant scalar curvature $r = -2n(2n+1)$. * If a para-Kenmotsu metric $g$ represents an $\eta$-Ricci soliton with the flow vector field $V$ being an infinitesimal paracontact transformation, then $V$ is strict and the manifold is an Einstein manifold with constant scalar curvature $r = -2n(2n+1)$. * If a para-Kenmotsu metric $g$ represents an $\eta$-Ricci soliton with non-zero flow vector field $V$ being collinear with $\xi$, then the manifold is an Einstein manifold with constant scalar curvature $r = -2n(2n+1)$. Finally, we cited few examples to illustrate the results obtained.

In K\"ahler geometry, Fujiki--Donaldson show that the scalar curvature arises as the moment map for Hamiltonian diffeomorphisms. In generalized K\"ahler geometry, one does not have suitable notions of Levi-Civita connection and curvature, however there still exists a precise framework for a moment map and the scalar curvature is defined as the moment map. Then a fundamental question is to understand the existence or non-existence of generalized K\"ahler structures with constant scalar curvature. In the paper, we study the Lie algebra of automorphisms of a generalized complex manifold. We assume that $H^{1}(M)=0$. Then we show that the Lie algebra of the automorphisms is a reductive Lie algebra if a generalized complex manifold admits a generalized K\"ahler structure of symplectic type with constant scalar curvature. This is a generalization of Matsushima and Lichnerowicz theorem in K\"ahler geometry. We explicitly calculate the Lie algebra of the automorphisms of a generalized complex structure given by a cubic curve on $\Bbb C P^2$. Cubic curves are classified into nine cases (see Figure.$1 -- 9$). In the three cases as in Figures. 7, 8 and 9, the Lie algebra of the automorphisms is not reductive and there is an obstruction to the existence of generalized K\"ahler structures of symplectic type with constant scalar curvature in the three cases. We also discuss deformations starting from an ordinary K\"ahler manifold $(X,\omega)$ with constant scalar curvature and show that nontrivial generalized K\"ahler structures of symplectic type with constant scalar curvature arise as deformations if the Lie algebra of automorphisms of $X$ is trivial.

Three special classes of Wintgen ideal submanifolds

We give a general procedure for gluing together possibly noncompact manifolds of constant scalar curvature which satisfy an extra nondegeneracy hypothesis. Our aim is to provide a simple paradigm for making "analytic" connected sums. In particular, we can easily construct complete metrics of constant positive scalar curvature on the complement of certain configurations of an even number of points on the sphere, which is a special case of Schoen's [S1] well-known, difficult construction. Applications of this construction produces metrics with prescribed asymptotics. In particular, we produce metrics with cylindrical ends, the simplest type of asymptotic behaviour. Solutions on the complement of an infinite number of points are also constructed by an iteration of our construction.

In this paper, we study n-dimensional complete submanifolds with constant scalar curvature in the Euclidean space En+p and n-dimensional compact submanifolds with constant scalar curvature in the unit sphere Sn+p(1). We prove that the totally umbilical sphere Sn(r), totally geodesic Euclidean space En and generalized cylinder Sn−1(c) × E1 are the only n-dimensional (n > 2) complete submanifolds Mn with constant scalar curvature n(n − 1)r in the Euclidean space En+p, which satisfy the following condition: where S denotes the squared norm of the second fundamental form of Mn. For compact submanifolds with constant scalar curvature in the unit sphere Sn+p(1), we also obtain a corresponding result (see theorem 1.3).

In this paper, we study n-dimensional complete submanifolds with constant scalar curvature in the Euclidean space En+p and n-dimensional compact submanifolds with constant scalar curvature in the unit sphere Sn+p(1). We prove that the totally umbilical sphere Sn(r), totally geodesic Euclidean space En and generalized cylinder Sn−1(c) × E1 are the only n-dimensional (n > 2) complete submanifolds Mn with constant scalar curvature n(n − 1)r in the Euclidean space En+p, which satisfy the following condition: where S denotes the squared norm of the second fundamental form of Mn. For compact submanifolds with constant scalar curvature in the unit sphere Sn+p(1), we also obtain a corresponding result (see theorem 1.3).

Let Mm be a m-dimensional submanifold in the n-dimensional unit sphere Sn without umbilic point. Two basic invariants of Mm under the Mobius transformation group of Sn are a 1-form Φ called Mobius form and a symmetric (0,2) tensor A called Blaschke tensor. In this paper, we prove the following rigidity theorem: Let Mm be a m-dimensional (m≥3) submanifold with vanishing Mobius form and with constant Mobius scalar curvature R in Sn, denote the trace-free Blaschke tensor by \(\). If \(\), then either ||A||≡0 and Mm is Mobius equivalent to a minimal submanifold with constant scalar curvature in Sn; or \(\) and Mm is Mobius equivalent to \(\) in \(\) for some c≥0 and \(\).