Geodesic spheres and symmetries in naturally reductive spaces.

SynopsisThis is a contribution to the general problem of determining the extent to which the geometry of a riemannian manifold is determined by properties of its geodesic spheres. In particular we show that total umbilicity of geodesic spheres determines riemannian manifolds of constant sectional curvature; quasi-umbilicity of geodesic spheres determines Kähler and nearly-Kähler manifolds of constant holomorphic sectional curvature; and the condition that geodesic spheres have only two different principal curvatures, one having multiplicity 3, determines manifolds locally isometric to the quaternionic projective spaces. The use of Jacobi vector fields leads to a unified treatment of these different cases.

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The Classification of 7- and 8-dimensional Naturally Reductive Spaces

Canadian Journal of Mathematics ◽

10.4153/s0008414x19000300 ◽

2019 ◽

Vol 72 (5) ◽

pp. 1246-1274 ◽

Cited By ~ 2

Author(s):

Reinier Storm

Keyword(s):

New Method ◽

Structure Theory ◽

New Construction ◽

Naturally Reductive

AbstractA new method for classifying naturally reductive spaces is presented. This method relies on a new construction and the structure theory of naturally reductive spaces recently developed by the author. This method is applied to obtain the classification of all naturally reductive spaces in dimension 7 and 8.

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Geodesic orbit and naturally reductive nilmanifolds associated with graphs

Mathematische Nachrichten ◽

10.1002/mana.201800456 ◽

2020 ◽

Vol 293 (4) ◽

pp. 754-760

Author(s):

Y. Nikolayevsky

Keyword(s):

Naturally Reductive

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Naturally Reductive Homogeneous Riemannian Manifolds

Canadian Journal of Mathematics ◽

10.4153/cjm-1985-028-2 ◽

1985 ◽

Vol 37 (3) ◽

pp. 467-487 ◽

Cited By ~ 29

Author(s):

Carolyn S. Gordon

Keyword(s):

Homogeneous Space ◽

Lie Group ◽

Riemannian Manifolds ◽

Homogeneous Spaces ◽

Invariant Metrics ◽

List Type ◽

Left Invariant Metrics ◽

Homogeneous Riemannian Manifolds ◽

Transitive Subgroup ◽

Naturally Reductive

The simple algebraic and geometric properties of naturally reductive metrics make them useful as examples in the study of homogeneous Riemannian manifolds. (See for example [2], [3], [15]). The existence and abundance of naturally reductive left-invariant metrics on a Lie group G or homogeneous space G/L reflect the structure of G itself. Such metrics abound on compact groups, exist but are more restricted on noncompact semisimple groups, and are relatively rare on solvable groups. The goals of this paper are(i) to study all naturally reductive homogeneous spaces of G when G is either semisimple of noncompact type or nilpotent and(ii) to give necessary conditions on a Riemannian homogeneous space of an arbitrary Lie group G in order that the metric be naturally reductive with respect to some transitive subgroup of G.

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