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On distinguished curves in parabolic geometries. (English) Zbl 1070.53021
Consider a Lie group \(G\) with Lie algebra \({\mathfrak{g}}\), a Lie subgroup \(H\subset G\), and the Maurer-Cartan form \(\omega_0:T(G)\to{\mathfrak{g}}\). A Cartan geometry with flat model \(G/H\) is a triple \(({\mathcal{G}},M,\omega)\), where \({\mathcal{G}}\to M\) is a principal \(H\)-fibre bundle and \(\omega:T({\mathcal{G}})\to{\mathfrak{g}}\) a 1-form giving for each \(u\in{\mathcal{G}}\) a linear isomorphism between \(T_u({\mathcal{G}})\) and \({\mathfrak{g}}\) and sharing with the Maurer-Cartan form the following properties: (i) \(\omega(\zeta_Z)=Z\) for \(Z\in{\mathfrak{h}}\), where \(\zeta_Z\) is the fundamental vector field corresponding to \(Z\) and (ii) \(\left(r_h\right)^\ast\omega=\text{Ad}(h^{-1})\circ\omega\) for each \(h\in H\), \(r_h\) denoting the corresponding right translation of \({\mathcal{G}}\). Therefore, Cartan geometries whose flat models are for instance the Euclidean, affine, projective or Möbius space are Riemannian geometry, geometry of affine connections, or projective and conformal structures respectively. Distinguished curves as mentioned in the title are intended to serve as a generalization of the notion of geodesic in Cartan geometries.
The authors confine themselves to parabolic geometries, i. e., to Cartan geometries with flat model \(G/P\), where \(G\) is semisimple and \(P\subset G\) a parabolic subgroup. The parabolic subgroup \(P\subset G\) determines a grading \[ {\mathfrak{g}}={\mathfrak{g}}_{-k}\oplus\dots\oplus{\mathfrak{g}}_k \tag{grad} \] of \({\mathfrak{g}}\) such that \({\mathfrak{p}}={\mathfrak{g}}_0\oplus\cdots\oplus{\mathfrak{g}}_k\), \({\mathfrak{p}}_+={\mathfrak{g}}_1\oplus\cdots\oplus{\mathfrak{g}}_k\) is the nilradical of \({\mathfrak{g}}\). The space \({\mathfrak{n}}={\mathfrak{g}}_{-k}\oplus\cdots\oplus{\mathfrak{g}}_{-1}\) is isomorphic as a \(P\)-module to \({\mathfrak{g}}/{\mathfrak{p}}\) and \({\mathfrak{g}}_0\) is a Lie subalgebra of \({\mathfrak{g}}\) corresponding to a Lie subgroup \(G_0\subset G\). Any manifold \(S\) acted on by \(P\) gives rise to an associated fibre bundle \({\mathcal{S}}={\mathcal{G}}\times_PS\). The special cases \(S={\mathfrak{g}}/{\mathfrak{p}}\), \(S=G\), \(S=G/P\) give the bundles \(TM={\mathcal{G}}\times_P{\mathfrak{g}}/{\mathfrak{p}}\) (tangent bundle of \(M\)), \({\widetilde{{\mathcal{G}}}}={\mathcal{G}}\times_PG\), and \({\mathcal{S}}={\mathcal{G}}\times_PG/P={\widetilde{{\mathcal{G}}}}\times_GG/P\). \({\widetilde{{\mathcal{G}}}}\) is a principal bundle and the 1-form \(\omega_0:T(G)\to{\mathfrak{g}}\) extends to a principal connection form \(\widetilde{\omega}:T{\widetilde{{\mathcal{G}}}}\to{\mathfrak{g}}\). Since \({\mathcal{S}}\) is an associated bundle of \({\widetilde{{\mathcal{G}}}}\), parallel transport is well defined in \({\mathcal{S}}\). Moreover, \({\mathcal{S}}\to M\) has a canonical section \({\mathbf{O}}\) defined with the help of the fixed point \(o=e\,P\in G/P\) of the \(P\)-action and the tangent bundle \(TM\) can be identified with the pull back \({\mathbf{O}}_\ast(V{\mathcal{S}})\) of the vertical bundle of \({\mathcal{S}}\). Let \(c(t)\) be a smooth curve \(s(t)\) in \(M\) defined on an interval \(I\subset{\mathbb{R}}\). The section \({\mathbf{O}}\) and the parallel transport of \({\mathcal{S}}\) can be used to construct for any \(t_0\in I\) a vertical development \(\text{dev}(c,t_0)(s)\), i. e., a curve in the fibre \({\mathcal{S}}_{c(t_0)}\) obtained by marching from \({\mathbf{O}}(c(t_0))\) to \({\mathbf{O}}(c(t_0+s))\) and then back to the fibre \({\mathcal{S}}_{c(t_0)}\) along the horizontal lift of \(c(t)\).
On the other hand, vertical curves in \({\mathcal{S}}_{c(t_0)}\) are in 1-1 correspondence with curves in \(G/P\) modulo \(P\) and any choice of a \(P\)-invariant class \({\mathcal{C}}\) of curves \(\widetilde{c}(t)\) in \(G/P\) with \(\widetilde{c}(0)=e\) determines a class of curves in \({\mathcal{S}}_{c(t_0)}\) passing through \({\mathbf{O}}(c(t_0+s))\). Curves in \(M\) whose vertical developments belong to this class are called distinguished curves of type \({\mathcal{C}}\). Natural choices of such \(P\)-invariant sets of curves are orbits \(t\to\text{exp}(t\,X)\,P\) of 1-parameter subgroups, where \(X\in{\mathfrak{g}}\) belongs to a distinguished subset \(A\subset{\mathfrak{g}}\). This class is denoted by \({\mathcal{C}}_A\). Much of the paper is concerned with the problem, to find the order of initial \(k\)-jets that determine the geodesics in a unique way. The authors restrict their considerations to types \({\mathcal{C}}_A\), where \(A\subset{\mathfrak{n}}\) is a \(G_0\)-invariant subset and the corresponding distinguished curves are called generalized geodesics of type \({\mathcal{C}}_A\) or simply generalized geodesics if \(A={\mathfrak{n}}\). In section 2, for any \(A\subset{\mathfrak{n}}\), the space \(T^r_{{\mathcal{C}}_A}M\subset J^r_0({\mathbb{R}},M)\) of \(r\)-jets of geodesics of type \({\mathcal{C}}_A\), called the space of \(r\)-velocities of geodesics type \({\mathcal{C}}_A\), is considered. It is shown that any two geodesics having the same \((k+2)\)-jet at some point \(t_0\in{\mathbb{R}}\) must coincide, where \(k\) is the order of the grading of the Lie algebra \({\mathfrak{p}}\) of \(P\). A description of the bundle \(T^2_{{\mathcal{C}}_A}\) is given as follows: The standard fibre is the \(P\)-orbit of the \(G_0\)-invariant subspace \(A\times\{0\}\subset{\mathfrak{g}}_{-1}\times{\mathfrak{g}}_{-1}\).
Section 3 is concerned with the problem to identify distinguished curves modulo change of parametrization. In section 4, the estimates of the order of jets are improved. Theorem 4.2 asserts that each generalized geodesic of type \({\mathcal{C}}_{{\mathfrak{g}}_{-k}}\) is determined by its 2-jet at a single point and that parametrizations are unique up to projective changes. More generally, for geodesics of type \({\mathcal{C}}_{{\mathfrak{g}}_{-j}}, 1\leq{}j\leq{}k,\) the \(r\)-jet will determine the curve provided that \(r\,j\geq{}k+1\). Finally, two examples are given.

MSC:
53C22 Geodesics in global differential geometry
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C30 Differential geometry of homogeneous manifolds
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