CONJUGATE LOCI OF 2-STEP NILPOTENT LIE GROUPS SATISFYING J²_z= <Sz, z>A

Abstract

Abstract. Let n be a 2-step nilpotent Lie algebra which has an inner product ⟨ , ⟩ and has an orthogonal decomposition n = z ⊕ v for its center z and the orthogonal complement v of z. Then Each element z of z defines a skew symmetric linear map Jz : v −→ v given by ⟨Jzx, y⟩ = ⟨z, [x, y]⟩ for all x, y ∈ v. In this paper we characterize Jacobi fields and calculate all conjugate points of a simply connected 2-step nilpotent Lie group N with its Lie algebra n satisfying J 2 z = ⟨Sz, z⟩A for all z ∈ z, where S is a positive definite symmetric operator on z and A is a negative definite symmetric operator on v.

A different perspective on H-like Lie algebras

“…For pseudo H-type or generalized H-type nilmanifolds, the inner product is assumed to be nondegenerate instead of positive definite, and even possibly sub-semi-Riemannian ( [Cia00,GMKM13]). Lauret generalized the notion of H-type to modified H-type in [Lau99], and this was further generalized in [JLP08].…”