East Asian Math. J. 22 (2006), No. 2, pp. 249–257 ON CONJUGATE POINTS OF THE GROUP H(2, 1)

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. Let γ be a unit speed geodesic in a 2-step nilpotent Lie group H(2, 1) with its Lie algebra n(2, 1) and let its initial velocity γ′(0) be given by γ′(0) = Z0 +X0 ∈ z ⊕ v = n(2, 1) with its center component Z0 nonzero. Then we showed that γ(0) is conjugate to γ ( 2npiθ), where n is a nonzero intger and −θ2 is a nonzero eigenvalue of J2Z0, along γ if and only if either X0 is an eigenvector of