Curves in homogeneous spaces and their contact with 1-dimensional orbits (Q639980)

Let \(I_r\subset\mathbb R, r = 1,2\), be two open intervals, and let \(t_r\in I_r\). Two imbeddings \(\beta_r: I_r\to M\) into a differentiable manifold \(M\) are said to be in contact of order \(k\) at the point \(\beta_1(t_1) = \beta_2(t_2)\in M\) if there is a diffeomorphism \(\phi\) of a neighbourhood of \(t_2\) in \(I_2\) on \(I_1\) such that the \(k\)-jets of \(\beta_1\circ\phi\) and \(\beta_2\) at \(t_2\) coincide. Let now \(G\) be a real Lie group, \(H\subset G\) its closed subgroup, and let \(\mathfrak h \subset\mathfrak g\) be their Lie algebras. The authors consider the natural transitive action of \(G\) on the manifold \(M = G/H\). Any \(v\in\mathfrak g\) and \(x\in M\) determine the so-called exponential curve \(\beta_v(s) = (\exp(sv))x, s\in\mathbb R\), in \(M\). Let \(I\subset\mathbb R\) be a neighborhood of 0 in \(\mathbb R\), and let \(\alpha: I\to M\) be a smooth curve in \(M\). The authors construct a family of subspaces \(S_k^{\alpha}(t)\subset\mathfrak g,\, k\in\mathbb N, t\in I\), such that \(\alpha\) and \(\beta_v(s) = (\exp(sv))\alpha(t_0)\) are in contact of order \(k\) at \(\alpha(t_0)\) if and only if \(v\in S_k^{\alpha}(t_0)\). This family is invariant under the adjoint representation of \(G\). A curve \(\alpha: I\to M\) is called \(k\)-comparable at \(\alpha(t)\) if there exists an exponential curve passing through \(\alpha(t)\) which is in contact of order \(k\) with \(\alpha\) at \(\alpha(t)\). The following stabilization theorem is proved: if a curve \(\alpha\) is \((k+2)\)-comparable at \(\alpha(t_0)\) and if \(S_k^{\alpha}(t_0) = S_{k+1}^{\alpha}(t_0)\), then \(S_{k+1}^{\alpha}(t_0) = S_{k+2}^{\alpha}(t_0)\).