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Two-step homogeneous geodesics in pseudo-Riemannian manifolds
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Two-step homogeneous geodesics in pseudo-Riemannian manifolds
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Journal Article
Two-step homogeneous geodesics in pseudo-Riemannian manifolds
2021
Overview
Given a homogeneous pseudo-Riemannian space (G/H,⟨,⟩), a geodesic γ:I→G/H is said to be two-step homogeneous if it admits a parametrization t=ϕ(s) (s affine parameter) and vectors X, Y in the Lie algebra g, such that γ(t)=exp(tX)exp(tY)·o, for all t∈ϕ(I). As such, two-step homogeneous geodesics are a natural generalization of homogeneous geodesics (i.e., geodesics which are orbits of a one-parameter group of isometries). We obtain characterizations of two-step homogeneous geodesics, both for reductive homogeneous spaces and in the general case, and undertake the study of two-step g.o. spaces, that is, homogeneous pseudo-Riemannian manifolds all of whose geodesics are two-step homogeneous. We also completely determine the left-invariant metrics ⟨,⟩ on the unimodular Lie group SL(2,R) such that (SL(2,R),⟨,⟩) is a two-step g.o. space.
Publisher
Springer Nature B.V
Subject
