On the spectral geometry of 4-dimensional Lorentzian Lie group

The main focus of this paper is concern to the study on the point-wise Osserman structure on 4-dimensional Lorentzian Lie group. In this paper we study on the spectrum of the Jacobi operator and spectrum of the skew-symmetric curvature operator on the non-abelian 4-dimensional Lie group G, whenever G equipped with an orthonormal left invariant pseudo-Riemannian metric g of signature (-;+;+; +), i.e, Lorentzian metric, where e1 is a unit time-like vector. The Lie algebra structure in dimension four has key role in our investigation, also in this case we study on the classification of 1-Stein and mixed IP spaces. At the end we show that G does not admit any space form and Einstein structures.