Some Classes of Almost Anti-Hermitian Structures on the Tangent Bundle

Abstract

In [11] we have considered a family of natural almost anti-Hermitian structures (G, J) on the tangent bundle T M of a Riemannian manifold (M, g), where the semi-Riemannian metric G is a lift of natural type of g to T M, such that the vertical and horizontal distributions V T M, HT M are maximally isotropic and the almost complex structure J is a usual natural lift of g of diagonal type interchanging V T M, HT M (see [5], [15]). We have obtained the conditions under which this almost anti-Hermitian structure belongs to one of the eight classes of anti-Hermitian manifolds obtained in the classification given in [1]. In this paper we consider another semi-Riemannian metric G on T M such that the vertical and horizontal distributions are orthogonal to each other. We study the conditions under which the above almost complex structure J defines, together with G, an almost anti-Hermitian structure on T M. Next, we obtain the conditions under which this structure belongs to one of the eight classes of anti-Hermitian manifolds obtained in the classification in [1]. Mathematics Subject Classification (2000). Primary 53C55, 53C15, 53C05.

Totally Real Statistical Submanifolds

“…In this section we will provide a method to construct holomorphic statistical structures on R 4 using a g-natural metric G, defined by Oproiu in 1999 (see [5]). It is known that the tangent bundle TM of an n-dimensional Riemannian manifold ðM; gÞ has a structure of 2n-dimensional manifold induced from the manifold structure of M. A local chart ðU; x 1 ; .…”

“…In §5 we give the proof of Theorem 1.1. In §6 we construct four dimensional holomorphic statistical structures, using a g-natural metric constructed by Oproiu in 1999 ( [5]). These structures depend on nine functions.…”

Однородные Пара-Кэлеровы Многообразия Эйнштейна

“…2.6). Свойства почти ги-перкэлеровых (или почти пара-гиперкэлеровых) структур были изучены в [33], а свойства пара-эрмитовых структур изучаются в [34]. В [35] авторы изучают почти пара-эрмитовы многообразия с поточечно постоянной пара-голоморфной секционной кривизной, определяемой как в эрмитовом случае.…”

Some Aspects on the Geometry of the Tangent Bundles and Tangent Sphere Bundles of a Riemannian Manifold

“…V.Oproiu and his collaborators constructed a family of Riemannian metrics on the tangent bundles of Riemannian manifolds which possess interesting geometric properties (cf. [17,18,19,20]). In particular, the scalar curvature of T (M ) can be constant also for a non-flat base manifold with constant sectional curvature.…”