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On the Geometry of Groups of Heisenberg Type

On the Geometry of Groups of Heisenberg Type AROLDO KAPLAN Consider the following two properties for a Riemannian manifold: (I) The local geodesic symmetries preserve the Riemannian measure. (11) Every geodesic is the orbit of a one-parameter group of isometries. These are known to hold on any naturally reductive space, but other examples have been missing. The purpose of this article is to show that the nilpotent Lie groups introduced in [6, 7] , endowed with their natural left-invariant metrics, are not naturally reductive, except for the Heisenberg group or its quaternionic analogue; that some—but not all—of the non-naturally reductive ones satisfy (II); and that they all satisfy (I). Property (I) was studied in general by D'Atri and Nickerson in [3, 4, 5], where they showed it to hold on naturally reductive spaces. T. J. Willmore pointed out to us the lack of other examples and suggested our [7] as a possible source of them, motivating this work. Property (II) was discussed by Ambrose and Singer in [1]. However, Tricceri and Vanhecke observed that the non-naturally reductive spaces which we show here to satisfy (I) provide counterexamples to Theorem 5.4 in that paper; they will discuss this in detail in a forthcoming article [12]. Other geometric properties http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

On the Geometry of Groups of Heisenberg Type

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/15.1.35
Publisher site
See Article on Publisher Site

Abstract

AROLDO KAPLAN Consider the following two properties for a Riemannian manifold: (I) The local geodesic symmetries preserve the Riemannian measure. (11) Every geodesic is the orbit of a one-parameter group of isometries. These are known to hold on any naturally reductive space, but other examples have been missing. The purpose of this article is to show that the nilpotent Lie groups introduced in [6, 7] , endowed with their natural left-invariant metrics, are not naturally reductive, except for the Heisenberg group or its quaternionic analogue; that some—but not all—of the non-naturally reductive ones satisfy (II); and that they all satisfy (I). Property (I) was studied in general by D'Atri and Nickerson in [3, 4, 5], where they showed it to hold on naturally reductive spaces. T. J. Willmore pointed out to us the lack of other examples and suggested our [7] as a possible source of them, motivating this work. Property (II) was discussed by Ambrose and Singer in [1]. However, Tricceri and Vanhecke observed that the non-naturally reductive spaces which we show here to satisfy (I) provide counterexamples to Theorem 5.4 in that paper; they will discuss this in detail in a forthcoming article [12]. Other geometric properties

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Jan 1, 1983

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