The curvature of the Sasaki metric of tangent sphere bundles

We study the sectional curvaturesK of the Sasaki metric of tangent sphere bundles over spaces of constant curvatureK(T ₁(M ⁿ, K)). We give precise bounds on the variation of the Ricci curvature and a bound on the scalar curvature ofT ₁ (M ⁿ, K) that is uniform onK. In an appendix we calculate and give lower bounds for the lengths of closed geodesics onT ₁ S ⁿ. titles.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 132–145.     

Notes on the Sasaki metric

We survey on the geometry of the tangent bundle of a Riemannian manifold, endowed with the classical metric established by S. Sasaki 60 years ago. Following the results of Sasaki, we try to write and deduce them by different means. Questions of vector fields, mainly those arising from the base, are related as invariants of the classical metric, contact and Hermitian structures. Attention is given to the natural notion of extension or complete lift of a vector field, from the base to the tangent manifold. Few results are original, but finally new equations of the mirror map are considered.     

Conformal vector fields with respect to the Sasaki metric tensor

The purpose of this article is to characterize conformal vector fields with respect to the Sasaki metric tensor field on the tangent bundle of a Riemannian manifold of dimension at least three. In particular, if the manifold in question is compact, it is found that the only conformal vector fields are Killing vector fields.     

Einstein metrics on even-dimensional homogeneous spaces admitting a homogeneous Riemannian metric of positive sectional curvature

Novosibirsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 32, No. 3, pp. 126–131, May–June, 1991.     

On the curvature of contact metric manifolds

Some results on Ricci-symmetric contact metric manifolds are obtained. Second order parallel tensors and vector fields keeping curvature tensor invariant are characterized on a class of contact manifolds. Conformally flat contact manifolds are studied assuming certain curvature conditions. Finally some results onk-nullity distribution of contact manifolds are obtained.     

K-spaces of constant holomorphic sectional curvature

In this note we prove the equivalence of the pointwise constancy and the global constancy of the holomorphic sectional curvature of a K-space. A criterion for the constancy of the holomorphic sectional curvature of a K-space is found. It is proved that every proper K-space of constant holomorphic sectional curvature is a six-dimensional orientable Riemannian manifold of constant positive curvature, which is isometric with the six-dimensional sphere in the case of completeness and connectedness.Translated from Matematicheskie Zametki, Vol. 19, No. 5, pp. 805–814, May, 1976.In conclusion I take this opportunity to express my gratitude to A. M. Vasil'ev for help in the preparation of this note.     

The Jung Theorem in metric spaces of curvature bounded above

The classical Jung Theorem states in essence that the diameter of a compact set in satisfies where is the circumradius of . The theorem was extended recently to the hyperbolic and the spherical -spaces. Here, the estimate above is extended to a class of metric spaces of curvature introduced by A. D. Alexandrov. The class includes the Riemannian spaces. The extended estimate is of the form where is a positive integer suitably defined for the set and its circumcenter. It can be that is not unique or does not exist. In the latter case, no estimate is derived. In case of a Riemannian -dimensional space, an integer always exists and satisfies . Then . In case of , one has .

     

The dual foliation in open manifolds with nonnegative sectional curvature

We study several properties of the Sharafutdinov dual foliation in open manifolds with nonnegative sectional curvature.

     

Dirac-Lu space with pseudo-Riemannian metric of constant curvature

In this paper, we will discuss the geometries of the Dirac-Lu space whose boundary is the third conformal space N defined by Dirac and Lu. We firstly give the SO(3, 3) invariant pseudo-Riemannian metric with constant curvature on this space, and then discuss the timelike geodesics. Finally we get a solution of the Yang-Mills equation on it by using the reduction theorem of connections.     

On the curvature of a generalization of contact metric manifolds

Summary We consider a genaralization of contact metric manifolds given by assignment of 1-formsη¹, . . . ,η^sand a compatible metric gon a manifold. With some integrability conditions they are called almost<span style='font-size:10.0pt;font-family:"Monotype Corsiva"; mso-bidi-font-family:"Monotype Corsiva"'>S-manifolds. We give a sufficient condition regarding the curvature of an almost<span style='font-size:10.0pt;font-family:"Monotype Corsiva";mso-bidi-font-family: "Monotype Corsiva"'>S-manifold to be locally isometric to a product of a Euclidean space and a sphere.     

Ricci curvature of Markov chains on metric spaces

We define the coarse Ricci curvature of metric spaces in terms of how much small balls are closer (in Wasserstein transportation distance) than their centers are. This definition naturally extends to any Markov chain on a metric space. For a Riemannian manifold this gives back, after scaling, the value of Ricci curvature of a tangent vector. Examples of positively curved spaces for this definition include the discrete cube and discrete versions of the Ornstein-Uhlenbeck process. Moreover this generalization is consistent with the Bakry-Émery Ricci curvature for Brownian motion with a drift on a Riemannian manifold.Positive Ricci curvature is shown to imply a spectral gap, a Lévy-Gromov-like Gaussian concentration theorem and a kind of modified logarithmic Sobolev inequality. The bounds obtained are sharp in a variety of examples.     

Extension of a Riemannian metric with vanishing curvature

Let $\text{Ω}$ be a connected and simply-connected open subset of $\text{R}{}^{\mathrm{n}}$ such that the geodesic distance in $\text{Ω}$ is equivalent to the Euclidean distance. Let there be given a Riemannian metric (g_ij) of class $\text{C}{}^2$ and of vanishing curvature in $\text{Ω}$, such that the functions g_ij and their partial derivatives of order $\text{?}\text{2}$ have continuous extensions to $\text{Ω}$. Then there exists a connected open subset $\text{Ω}$ of $\text{R}{}^{\mathrm{n}}$ containing $\text{Ω}$ and a Riemannian metric $\text{(}\text{g}\text{?}{}_{\mathrm{ij}}\text{)}$ of class $\text{C}{}^2$ and of vanishing curvature in $\text{Ω}$ that extends the metric (g_ij). To cite this article: P.G. Ciarlet, C. Mardare, C. R. Acad. Sci. Paris, Ser. I 338 (2004).