In this paper we study the spectral geometry of a $4$-dimensional Lie group. The main focus of this paper is to study the $2$-Stein and $2$-Osserman structures on a $4$-dimensional Riemannian Lie group. In this paper, we study the spectrum and trace of Jacobi operator and also we study the characteristic polynomial of generalized Jacobi operator on the non-abelian $4$-dimensional Lie group $G$, whenever $G$ is equipped with an orthonormal left invariant Riemannian metric $g$. The Lie algebra structures in dimension four have key role in this paper. It is known that in the classification of $4$-dimensional non-abelian Lie algebras there are nineteen classes of Lie algebras up to isomorphism . We consider these classes and study all of them. Finally, we study the space form problem and spectral properties of Szabo operator on $G$.
Lie group, pointwise Osserman, sectional curvature, 2-Stein
PEYGHAN, ESMAEIL and SEIFIPOUR, DAVOOD
"Classification of some geometric structures on 4-dimensional Riemannian Lie group,"
Turkish Journal of Mathematics: Vol. 46:
1, Article 18.
Available at: https://journals.tubitak.gov.tr/math/vol46/iss1/18