An anti-Kählerian Einstein structure on the tangent bundle of a space form

Vasile Oproiu¹ ; Neculai Papaghiuc²

¹ V. Oproiu Faculty of Mathematics University “Al. I. Cuza” Iaşi, Romania
² N. Papaghiuc Department of Mathematics Technical University Iaşi, Romania

Colloquium Mathematicum, Tome 103 (2005) no. 1, pp. 41-46.

Résumé

In [11] we have considered a family of almost anti-Hermitian structures $(G,J)$ on the tangent bundle $TM$ of a Riemannian manifold $(M,g)$, where the almost complex structure $J$ is a natural lift of $g$ to $TM$ interchanging the vertical and horizontal distributions $VTM$ and $HTM$ and the metric $G$ is a natural lift of $g$ of Sasaki type, with the property of being anti-Hermitian with respect to $J$. Next, we have studied the conditions under which $(TM, G, J)$ belongs to one of the eight classes of anti-Hermitian structures obtained in the classification in [2]. In this paper, we study some geometric properties of the anti-Kählerian structure obtained in [11]. In fact we prove that it is Einstein. This result offers nice examples of anti-Kählerian Einstein manifolds studied in [1].

Détail

DOI : 10.4064/cm103-1-6

Mots-clés : have considered family almost anti hermitian structures tangent bundle riemannian manifold where almost complex structure natural lift interchanging vertical horizontal distributions vtm htm metric natural lift sasaki type property being anti hermitian respect have studied conditions under which belongs eight classes anti hermitian structures obtained classification paper study geometric properties anti k hlerian structure obtained prove einstein result offers nice examples anti k hlerian einstein manifolds studied