Geodesic spheres and symmetries in naturally reductive spaces.

D’Atri, J. E.

doi:10.1307/mmj/1029001423

Cited by 22 publications

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“…Here ( , ) denotes the inner product on m induced by the metric g. DAtri and Nickerson [1,2] have proved THEOREM A. Let (M,g) be a naturally reductive homogeneous space.…”

Section: A Generalization Of a Theorem On Naturally Reductive Homogenmentioning

confidence: 99%

A Generalization of a Theorem on Naturally Reductive Homogeneous Spaces

Kowalski¹,

Vanhecke²

1984

Proceedings of the American Mathematical Society

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ABSTRACT. We prove that a homogeneous Riemannian manifold all of whose geodesies are orbits of one-parameter subgroups of isometries has volumepreserving local geodesic symmetries.Let (M, g) be a Riemannian manifold. (M, g) is said to be naturally reductive [6] if there is a connected Lie group G of isometries acting transitively and effectively on M such that M = G/H (H being the isotropy subgroup of G at some point of M) is a reductive homogeneous space with respect to the decomposition g = m + I) of the Lie algebra g of G and moreover

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“…Here ( , ) denotes the inner product on m induced by the metric g. DAtri and Nickerson [1,2] have proved THEOREM A. Let (M,g) be a naturally reductive homogeneous space.…”

Section: A Generalization Of a Theorem On Naturally Reductive Homogenmentioning

confidence: 99%

A Generalization of a Theorem on Naturally Reductive Homogeneous Spaces

Kowalski¹,

Vanhecke²

1984

Proceedings of the American Mathematical Society

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“…The first examples which are not locally symmetric were discovered in [4], [6]. These are the naturally reductive homogeneous spaces.…”

Section: §1 Introductionmentioning

confidence: 99%

Volume-preserving geodesic symmetries on four-dimensional Hermitian Einstein spaces

Cho

Sekigawa

Vanhecke

1997

Nagoya Mathematical Journal

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Abstract. We prove that a four-dimensional Hermitian Einstein space is weakly *-Einsteinian and use this result to show that all geodesic symmetries are volume-preserving (up to sign) if and only if it is local symmetric. §1. IntroductionRiemannian manifolds such that all (local) geodesic symmetries are volume-preserving (up to sign) or equivalently, are divergence-preserving, have been introduced in [5] and are called DΆtri spaces [27]. The first examples which are not locally symmetric were discovered in [4], [6]. These are the naturally reductive homogeneous spaces. Since then, many other classes of examples has been found and studied. The main classes are the following: Riemannian g.o. spaces (i.e., spaces such that every geodesic is an orbit of a one-parameter group of isometries), commutative spaces (i.e., homogeneous spaces whose algebra of all differential operators which are invariant under all isometries is commutative), generalized Heisenberg groups, harmonic spaces (in particular, the Damek-Ricci examples), weakly symmetric spaces, 5C-spaces (i.e., spaces such that the principal curvatures of small geodesic spheres have antipodal symmetry), probabilistic commutative spaces, TC-and Co-spaces (see [1], [2] for more details). Of course, any manifold which is locally isometric to one of these examples is also a DΆtri space. We refer to

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“…~) est l'alg~bre de Lie de G (resp. K) (2) La classification des espaces naturellement r6ductifs n'est connue que jusqu'~t la dimension 5 ([17], [13], [14]). La liste des groupes semi-simples (compacts) naturellement r4ductifs a 6t4 4tablie darts [5].…”

Section: Introductionunclassified