Classification of some geometric structures on 4-dimensional Riemannian Lie group

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 [12]. We consider these classes and study all of them. Finally, we study the space form problem and spectral properties of Szabo operator on G.

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