Scalar curvature

About: Scalar curvature is a research topic. Over the lifetime, 12701 publications have been published within this topic receiving 296040 citations. The topic is also known as: Ricci scalar.

Compact self-dual manifolds with positive scalar curvature

Abstract: On etudie l'existence d'une metrique auto-duale a courbure scalaire positive sur une variete compacte simplement connexe qui n'est pas conformement equivalente a ces metriques standard sur S 4 ou CP 2

Modified gravity with R?matter couplings and (non-)geodesic motion

Abstract: We consider alternative theories of gravity with a direct coupling between matter and the Ricci scalar. We study the relation between these theories and ordinary scalar?tensor gravity, or scalar?tensor theories which include non-standard couplings between the scalar and matter. We then analyze the motion of matter in such theories, its implications for the equivalence principle, and the recent claim that they can alleviate the dark matter problem in galaxies.

Canonical Sasakian Metrics

Abstract: Let M be a closed manifold of Sasaki type. A polarization of M is defined by a Reeb vector field, and for any such polarization, we consider the set of all Sasakian metrics compatible with it. On this space we study the functional given by the square of the L 2-norm of the scalar curvature. We prove that its critical points, or canonical representatives of the polarization, are Sasakian metrics that are transversally extremal. We define a Sasaki-Futaki invariant of the polarization, and show that it obstructs the existence of constant scalar curvature representatives. For a fixed CR structure of Sasaki type, we define the Sasaki cone of structures compatible with this underlying CR structure, and prove that the set of polarizations in it that admit a canonical representative is open. We use our results to describe fully the case of the sphere with its standard CR structure, showing that each element of its Sasaki cone can be represented by a canonical metric; we compute their Sasaki-Futaki invariant, and use it to describe the canonical metrics that have constant scalar curvature, and to prove that only the standard polarization can be represented by a Sasaki-Einstein metric.

Homogeneous Manifolds With Negative Curvature

Abstract: This paper solves the problem of determining which Lie groups act simply transitively on a Riemannian manifold with negative curvature. The results obtained extend those of Heintze for the case of strictly negative curvature. Using results of Wolf and Heintze, it is established that every connected, simply connected, homogeneous manifold M with negative curvature admits a Lie group S acting simply transitively by isometries and every group with this property must be solvable. Formulas for the curvature tensor on M are established and used to show that the Lie algebra of any such group S must satisfy a number of structural conditions. Conversely, given a Lie algebra < satisfying these conditions and any member of an easily constructed family of inner products on i, a metric deforma- tion argument is used to obtain a modified inner product which gives rise to a left invariant Riemannian structure with negative curvature on the associated simply connected Lie group. 1. Introduction. This paper was motivated by the following problem: Which connected Lie groups admit a left invariant Riemannian metric with nega- tive (sectional) curvature? We emphasize that throughout the paper, we under- stand "negative" to mean "less than or equal to zero". Since the property in question is not sensitive to groups linked by a local isomorphism, we deal primar- ily with simply connected groups. Results of J. A. Wolf (13) and E. Heintze (4) show that the above problem is closely linked with the classification of connected, homogeneous Riemannian manifolds with negative curvature. Indeed, if M is such a manifold and if M is simply connected, then M is isometric to a solvable Lie group endowed with a left-invariant metric. In this paper, we give a complete solution to our original problem by show- ing that a necessary and sufficient condition for a group to have the property in question is that its Lie algebra be what we call an "NC algebra". Roughly speak- ing, the crucial properties of an NC algebra $ are that in addition to being solv- able, e must contain an abelian subalgebra a complementary to the derived

On some hereditary properties of Riemannian g-natural metrics on tangent bundles of Riemannian manifolds

Abstract: It is well known that if the tangent bundle TM of a Riemannian manifold (M,g) is endowed with the Sasaki metric gs, then the flatness property on TM is inherited by the base manifold [Kowalski, J. Reine Angew. Math. 250 (1971) 124–129]. This motivates us to the general question if the flatness and also other simple geometrical properties remain “hereditary” if we replace gs by the most general Riemannian “g-natural metric” on TM (see [Kowalski and Sekizawa, Bull. Tokyo Gakugei Univ. (4) 40 (1988) 1–29; Abbassi and Sarih, Arch. Math. (Brno), submitted for publication]). In this direction, we prove that if (TM,G) is flat, or locally symmetric, or of constant sectional curvature, or of constant scalar curvature, or an Einstein manifold, respectively, then (M,g) possesses the same property, respectively. We also give explicit examples of g-natural metrics of arbitrary constant scalar curvature on TM.