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Two-step homogeneous geodesics in pseudo-Riemannian manifolds
Annals of Global Analysis and Geometry ( IF 0.6 ) Pub Date : 2021-01-18 , DOI: 10.1007/s10455-020-09751-4
Andreas Arvanitoyeorgos , Giovanni Calvaruso , Nikolaos Panagiotis Souris

Given a homogeneous pseudo-Riemannian space \((G/H,\langle \ , \ \rangle),\) a geodesic \(\gamma :I\rightarrow G/H\) is said to be two-step homogeneous if it admits a parametrization \(t=\phi (s)\) (s affine parameter) and vectors XY in the Lie algebra \({\mathfrak{g}}\), such that \(\gamma (t)=\exp (tX)\exp (tY)\cdot o\), for all \(t\in \phi (I)\). 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 \(\langle \ ,\ \rangle\) on the unimodular Lie group \(SL(2,{{\mathbb{R}}})\) such that \(\big (SL(2,{{\mathbb{R}}}),\langle \ ,\ \rangle \big )\) is a two-step g.o. space.



中文翻译:

伪黎曼流形中的两步齐次测地线

给定齐次伪黎曼空间\((G / H,\ langle \,\ \ rangle),\)测地线\(\ gamma:I \ rightarrow G / H \)被认为是两步齐次的接受参数化\(t = \ phi(s)\)s仿射参数)和李代数\ {{\ mathfrak {g}} \}中的向量X,  Y,使得\(\ gamma(t)= \ exp(tX)\ exp(tY)\ cdot o \),对于所有\(t \ in \ phi(I)\)。这样,两步均质测地线是均质测地线的自然概括(即,测地线是等参数的一参数组的轨道)。我们获得了两步齐次测地线的特征,包括还原性齐次空间和一般情况,并且进行了两步走空间的研究,即均质伪拟黎曼流形的所有测地线都是两步齐次的。我们还完全确定单模李群\ {SL(2,{{\ mathbb {R}}}} \\)上的左不变度量\ {\ langle \,\ \ rangle \)使得\(\ big( SL(2,{{\ mathbb {R}}}),\ langle \,\ \ rangle \ big)\)是两步走的空间。

更新日期:2021-01-18
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