Keun Park

Conjugate Points on 2-Step Nilpotent Groups

Let N be a simply connected 2-step nilpotent Lie group equipped with a left-invariant metric. We consider the characterizations of Jacobi fields and conjugate points along geodesics emanating from the identity element in N. We obtain a partial result for N and the complete result for N with a one-dimensional center.

CONJUGATE POINTS ON THE QUATERNIONIC HEISENBERG GROUP

Let N be the quaternionic Heisenberg group equipped with a left-invariant metric. We characterize all the conjugate points along the geodesics on N:

CONJUGATE LOCI OF 2-STEP NILPOTENT LIE GROUPS SATISFYING J 2 z = A

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.

GEODESIC SPHERES AND BALLS OF THE HEISENBERG GROUPS

Let H 2n+1 be the (2n + 1)-dimensional Heisenberg group equipped with a left-invariant metric. In this paper we study the Gaussian curvatures of the geodesic spheres and the volumes of geodesic balls in H 2n+1 .