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 X, Y 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.

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。