Nonsymmetric Osserman pseudo-Riemannian manifolds

Examples of Osserman pseudo-Riemannian manifolds with metric of any signature , , which are not locally symmetric are exhibited.

     

Osserman manifolds of dimension 8

For a Riemannian manifold Mⁿ with the curvature tensor R, the Jacobi operator R_X is defined by R_XY=R(X,Y)X. The manifold Mⁿ is called pointwise Osserman if, for every pMⁿ, the eigenvalues of the Jacobi operator R_X do not depend of a unit vector XT_pMⁿ, and is called globally Osserman if they do not depend of the point p either. R. Osserman conjectured that globally Osserman manifolds are flat or locally rank-one symmetric. This Conjecture is true for manifolds of dimension n8,16[14]. Here we prove the Osserman Conjecture and its pointwise version for 8-dimensional manifolds.Mathematics Subject Classification (2000): 53B20     

Nonsymmetric Osserman indefinite Kähler manifolds

The authors prove the existence of Osserman manifolds with indefinite Kähler metric of nonnegative or nonpositive holomorphic sectional curvature which are not locally symmetric.

     

Almost s-tangent manifolds

Geometriae Dedicata -     

Spectral geometry for almost isospectral hermitian manifolds

The following question was posed by M. Berger: Is it possible to determine from the spectrum of the real Laplacian whether or not a manifold is Kähler? The Kähler condition for Hermitian manifolds is found out from the invariants of the spectrum of some differential operators acting on forms of type (p, q). P. Gilkey and H. Donnelly proved the Berger conjecture for the complex Laplacian and the reduced complex Laplacian respectively. In this paper we consider the Berger conjecture of almost isospectral Hermitian manifolds about the complex Laplacian acting on forms of type (p, q). Then we can show that a closed complexm(≥ 3)-dimensional Hermitian manifold which is strongly (?2/m)-isospectral to the complex projective space CP ^m with the Fubini-Study metric is holomorphically isometric to CP ^m .     

Almost complex manifolds and Cartan's uniqueness theorem

We present a generalization of Cartan's uniqueness theorem to the almost complex manifolds.

     

Almost Kenmotsu manifolds with a condition of η-parallelism

We consider almost Kenmotsu manifolds (M²ⁿ⁺¹,φ,ξ,η,g) with η-parallel tensor h^′=h○φ, 2h being the Lie derivative of the structure tensor φ with respect to the Reeb vector field ξ. We describe the Riemannian geometry of an integral submanifold of the distribution orthogonal to ξ, characterizing the CR-integrability of the structure. Under the additional condition _ξh^′=0, the almost Kenmotsu manifold is locally a warped product. Finally, some lightlike structures on M²ⁿ⁺¹ are introduced and studied.     

Almost Schur lemma for manifolds with boundary

In this paper, we prove the almost Schur theorem, introduced by De Lellis and Topping, for the Riemannian manifold M of nonnegative Ricci curvature with totally geodesic boundary. Examples are given to show that it is optimal when the dimension of M is at least 5. We also prove that the almost Schur theorem is true when M is a 4-dimensional manifold of nonnegative scalar curvature with totally geodesic boundary. Finally we obtain a generalization of the almost Schur theorem in all dimensions only by assuming the Ricci curvature is bounded below.     

Almost complex and complex structures on real projective Stiefel manifolds

We give a complete list of real projective Stiefel manifolds which admit almost complex structures and show that many of them are in fact complex manifolds. The first named author was supported in part by Grants 1/1486/94 and 2/1225/96 of VEGA (Slovakia) during the preparation of this work.     

Almost Hermitian manifolds admitting holomorphically planar conformal vector fields

We classify and characterize an almost Hermitian manifold M admitting a holomorphically planar conformal vector (HPCV) field (a generalization of a closed conformal vector field) V . We show that if V is nowhere vanishing and strictly non-geodesic, then it is homothetic and almost analytic. If, in addition,M satisfies Gray’s first condition, then M is Kaehler. For a semi-Kaehler manifold M admitting an HPCV field V we show that either V is closed, or M becomes almost Kaehler and V is homothetic and almost analytic. Part of this work was done by the second author while he was visiting Sri Sathya Sai Institute Of Higher Learning, Prasanthinilayam, India.     

Contact metric manifolds satisfying a nullity condition

This paper presents a study of contact metric manifolds for which the characteristic vector field of the contact structure satisfies a nullity type condition, condition (*) below. There are a number of reasons for studying this condition and results concerning it given in the paper: There exist examples in all dimensions; the condition is invariant underD-homothetic deformations; in dimensions>5 the condition determines the curvature completely; and in dimension 3 a complete, classification is given, in particular these include the 3-dimensional unimodular Lie groups with a left invariant metric. Dedicated to Professor Chorng-Shi Houh on his 65th birthday