On geometry of weakly cosymplectic manifolds

We consider classes of weakly cosymplectic manifolds whose Riemannian curvature tensors satisfy contact analogs of the Riemannian–Christoffel identities. Additional properties of the Riemannian curvature tensor symmetry are found and a classification of weakly cosymplectic manifolds is obtained.     

Riemannian manifolds with a generalized axiom of planes

This paper introduces a class of Riemannian manifolds satisfying the axiom of (l, s)-planes that is a generalization of Cartan's axiom of p-planes. The structure of the curvature and Riemannian metric of a manifold with the axiom of (n – 2, n – 1)-planes is established.Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 103–110, 1992.     

Upper eigenvalue estimates for Dirac operators

We derive upper eigenvalue estimates for generalized Dirac operators on closed Riemannian manifolds. In the case of the classical Dirac operator the estimates on the first eigenvalues are sharp for spheres of constant curvature.     

Total curvatures of geodesic spheres associated to quadratic curvature invariants

Total scalar curvatures of geodesic spheres obtained by integrating the second-order scalar invariants of the curvature tensor are investigated. The first terms in their power-series expansions are derived and these results are used to characterize the two-point homogeneous spaces among Riemannian manifolds with adapted holonomy. Dedicated to Professor L. VanheckeMathematics Subject Classification (2000) 53C25, 53C30     

Local embeddability of CR manifolds into spheres

In this paper we solve local CR embeddability problem of smooth CR manifolds into spheres under a certain nondegeneracy condition on the Chern–Moser’s curvature tensor. We state necessary and sufficient conditions for the existence of CR embeddings as finite number of equations and rank conditions on the Chern–Moser’s curvature tensors and their derivatives. We also discuss the rigidity of those embeddings. J.-W. Oh was partially supported by BK21-Yonsei University.     

A Generalized Mass Involving Higher Order Symmetric Functions of the Curvature Tensor

We define a generalized mass for asymptotically flat manifolds using some higher order symmetric function of the curvature tensor. This mass is non-negative when the manifold is locally conformally flat and the σ _k curvature vanishes at infinity. In addition, with the above assumptions, if the mass is zero, then, near infinity, the manifold is isometric to a Euclidean end.     

Rigidity Theorems for Complete Sasakian Manifolds with Constant Pseudo-Hermitian Scalar Curvature

The orthogonal decomposition of the Webster curvature provides us a way to characterize some canonical metrics on a pseudo-Hermitian manifold. We derive some subelliptic differential inequalities from the Weitzenböck formulas for the traceless pseudo-Hermitian Ricci tensor of Sasakian manifolds with constant pseudo-Hermitian scalar curvature and the Chern–Moser tensor of the Sasakian pseudo-Einstein manifolds, respectively. By means of either subelliptic estimates or maximum principle, some rigidity theorems are established to characterize Sasakian pseudo-Einstein manifolds among Sasakian manifolds with constant pseudo-Hermitian scalar curvature and Sasakian space forms among Sasakian pseudo-Einstein manifolds, respectively.     

李奇曲率平行的黎曼流形的曲率张量模长

李安民和赵国松［１］提出了下面的问题：找出李奇曲率平行的黎曼流形的曲率张量模长的最佳拼挤常数并确定达到该值的流形．本文确定了非爱因斯坦流形的最佳拼挤常数和达到该值的黎曼流形．在ｎ１２时，回答了［１］中提出的问题．     

Generalized quasi-Einstein manifolds with harmonic Weyl tensor

In this paper we introduce the notion of generalized quasi-Einstein manifold that generalizes the concepts of Ricci soliton, Ricci almost soliton and quasi-Einstein manifolds. We prove that a complete generalized quasi-Einstein manifold with harmonic Weyl tensor and with zero radial Weyl curvature is locally a warped product with (n ? 1)-dimensional Einstein fibers. In particular, this implies a local characterization for locally conformally flat gradient Ricci almost solitons, similar to that proved for gradient Ricci solitons.     

On some Kähler manifolds with Norden metric

We investigate the Kähler manifolds with Norden metric whose curvature tensor can be expressed in the terms of Ricci tensors only and whose holomorphically conformal curvature tensor vanishes.     

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.     

Curvature identities on almost Hermitian manifolds and applications

We systematically derive the Bianchi identities for the canonical connection on an almost Hermitian manifold.Moreover,we also compute the curvature tensor of the Levi-Civita connection on almost Hermitian manifolds in terms of curvature and torsion of the canonical connection.As applications of the curvature identities,we obtain some results about the integrability of quasi K¨ahler manifolds and nearly K¨ahler manifolds.     

Critical metrics for quadratic functionals in the curvature on 4-dimensional manifolds

We study critical metrics for the squared L²-norm functionals of the curvature tensor, the Ricci tensor and the scalar curvature by making use of a curvature identity on 4-dimensional Riemannian manifolds.     

On the Curvature of Kähler Manifolds with Zero Ricci Tensor

Mathematical Notes - The behavior of the modulus of the curvature tensor and of the holomorphic sectional curvature on Ricci-flat Kähler manifolds is investigated.     

Homogeneity and Curvatures of Geodesic Spheres

We study some scalar curvature invariants on geodesic spheres and use them to characterize several kinds of Riemannian manifolds such as homogenous manifolds and in particular, the two-point homogeneous spaces and the Damek-Ricci spaces.     

On the class of generalized Landsberg manifolds

In 2000, Bejancu–Farran introduced the class of generalized Landsberg manifolds which contains the class of Landsberg manifolds. In this paper, we prove three global results for generalized Landsberg manifolds. First, we show that every compact generalized Landsberg manifold is a Landsberg manifold. Then we prove that every complete generalized Landsberg manifold with relatively isotropic Landsberg curvature reduces to a Landsberg manifold. Finally, we show that every generalized Landsberg manifold with vanishing Douglas curvature satisfies \(\mathbf{H}=0\).     

Homogeneity and Curvatures of Geodesic Spheres

We study some scalar curvature invariants on geodesic spheres and use them to characterize several kinds of Riemannian manifolds such as homogenous manifolds and in particular, the two-point homogeneous spaces and the Damek-Ricci spaces.     

Rigidity of complete manifolds with parallel Cotton tensor

The aim of this paper is to show some rigidity results for complete Riemannian manifolds with parallel Cotton tensor. In particular, we prove that any compact manifold of dimension \(n\ge 3\) with parallel Cotton tensor and positive constant scalar curvature is isometric to a finite quotient of \({\mathbb {S}}^n\) under a pointwise or integral pinching condition. Moreover, a rigidity theorem for stochastically complete manifolds with parallel Cotton tensor is also given. The proofs rely mainly on curvature elliptic estimates and the weak maximum principle.     

Covariant derivative of the curvature tensor of pseudo-Kählerian manifolds

It is well known that the curvature tensor of a pseudo-Riemannian manifold can be decomposed with respect to the pseudo-orthogonal group into the sum of the Weyl conformal curvature tensor, the traceless part of the Ricci tensor and of the scalar curvature. A similar decomposition with respect to the pseudo-unitary group exists on a pseudo-Kählerian manifold; instead of the Weyl tensor one obtains the Bochner tensor. In the present paper, the known decomposition with respect to the pseudo-orthogonal group of the covariant derivative of the curvature tensor of a pseudo-Riemannian manifold is refined. A decomposition with respect to the pseudo-unitary group of the covariant derivative of the curvature tensor for pseudo-Kählerian manifolds is obtained. This defines natural classes of spaces generalizing locally symmetric spaces and Einstein spaces. It is shown that the values of the covariant derivative of the curvature tensor for a non-locally symmetric pseudo-Riemannian manifold with an irreducible connected holonomy group different from the pseudo-orthogonal and pseudo-unitary groups belong to an irreducible module of the holonomy group.     

Kähler manifolds of quasi-constant holomorphic sectional curvatures

The Kähler manifolds of quasi-constant holomorphic sectional curvatures are introduced as Kähler manifolds with complex distribution of codimension two, whose holomorphic sectional curvature only depends on the corresponding point and the geometric angle, associated with the section. A curvature identity characterizing such manifolds is found. The biconformal group of transformations whose elements transform Kähler metrics into Kähler ones is introduced and biconformal tensor invariants are obtained. This makes it possible to classify the manifolds under consideration locally. The class of locally biconformal flat Kähler metrics is shown to be exactly the class of Kähler metrics whose potential function is only a function of the distance from the origin in ? ⁿ . Finally we show that any rotational even dimensional hypersurface carries locally a natural Kähler structure which is of quasi-constant holomorphic sectional curvatures.