An almost Complex Structure with Norden Metric on the Phase Space

On the total space of the cotangent bundle of a Riemannian manifold, we construct a semi-Riemannian metric $G$, with respect to which an almost complex structure $J$ introduced by Oproiu and Poro\cb{s}niuc is self-adjoint. The structure $(J,G)$ turnes out to be an almost complex structure with Norden metric (this notion is known in the literature from Norden's papers). The semi-Riemannian context is different from the Riemannian one, as it is pointed out by Duggal and Bejancu in their monograph. We study this structure and provide some necessary and sufficient conditions for it to be a K\"ahler structure with Norden metric.

Keywords:

almost complex structure Norden metric, cotangent bundle,

PDF

___

[1] Bejan, C.-L., Nakova, G.: Almost Complex and Hypercomplex Norden Structures Induced by Natural Riemann Extensions. Mathematics. Vol. 10, Issue 15 (2022). 10.3390/math10152625
[2] Bejan, C.-L., Nakova, G., Blaga, A.: On Bochner Flat Kähler B-Manifolds. Axioms. 12 (4):336 (2023).
[3] Canchev, G., Borisov A.: Note on the almost complex manifolds with a Norden metric. Compt. Rend. Acad. Bulg. Sci. 39 (5), 31-34 (1986)
[4] Druta-Romaniuc, S.-L.: Bochner curvature of cotangent bundles with natural diagonal Kähler structures. World Scientific Publishing Company, (2022). https://doi.org/10.1142/978981124810 8 − 0010
[5] Duggal, K. L. and Bejancu, A.: Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications. Kluwer Academic, 364, (1996)
[6] Norden, A. P.: On a certain class of four-dimensional A-spaces. Izv. Vuzov Mat. 4, 145-157 (1960).
[7] Oproiu, V., Poros, niuc, D.D.:A class of Kaehler Einstein structures on the cotangent bundle. Publ. Math. Debrecen. 66, 457-478 (2005).
[8] Poroşniuc, D.D.: A class of Kähler Einstein structures on the nonzero cotangent bundle of a space form. Rev. Roumaine Math. Pures Appl. 50,237-252 (2005).
[9] Yano, K., Ishihara, S.: Tangent and cotangent bundles. Differential Geometry, M. Dekker, New York (1973).