On the Riemannian geometry of the nilpotent groups 𝐻(𝑝,𝑟)

Piu, Paola; Goze, Michel

doi:10.1090/S0002-9939-1993-1189548-9

On the Riemannian geometry of the nilpotent groups $H(p,r)$
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by Paola Piu and Michel Goze PDF

Proc. Amer. Math. Soc. 119 (1993), 611-619 Request permission

Abstract:

We study some aspect of the left-invariant Riemannian geometry on a class of nilpotent Lie groups $H(p,r)$ that generalize the Heisenberg group ${H_{2p + 1}}$. Let us prove that the groups of type $H$ (or Kaplan’s spaces) and the $H(p,r)$ groups have same common Riemannian properties but they are not the same spaces.

References

J. E. D’Atri and H. K. Nickerson, Geodesic symmetries in spaces with special curvature tensors, J. Differential Geometry 9 (1974), 251–262. MR 394520

Étude locale des systèmes de Pfaff

Michel Goze, Systèmes de Pfaff associés aux algèbres de Lie de type $H$, Rend. Sem. Mat. Univ. Politec. Torino 46 (1988), no. 1, 91–110 (1990) (French). MR 1084140
Michel Goze and Yuri Haraguchi, Sur les $r$-systèmes de contact, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 2, 95–97 (French, with English summary). MR 651795
Michel Goze and Paola Piu, Classification des métriques invariantes à gauche sur le groupe de Heisenberg, Rend. Circ. Mat. Palermo (2) 39 (1990), no. 2, 299–306 (French, with English summary). MR 1106594, DOI 10.1007/BF02844764

Sur une généralisation des structures de contact

Gary R. Jensen, Homogeneous Einstein spaces of dimension four, J. Differential Geometry 3 (1969), 309–349. MR 261487
Aroldo Kaplan, On the geometry of groups of Heisenberg type, Bull. London Math. Soc. 15 (1983), no. 1, 35–42. MR 686346, DOI 10.1112/blms/15.1.35
Wilfred Reyes, A note about Killing vector fields on $H(p,r)$, Rend. Sem. Fac. Sci. Univ. Cagliari 55 (1985), no. 1, 25–29 (English, with Italian summary). MR 849723
F. Tricerri and L. Vanhecke, Homogeneous structures on Riemannian manifolds, London Mathematical Society Lecture Note Series, vol. 83, Cambridge University Press, Cambridge, 1983. MR 712664, DOI 10.1017/CBO9781107325531