Geodesics and curvatures of special sub-Riemannian metrics on Lie groups
Let \(M = G / K\) be a Riemannian symmetric space, where \(M\) is a connected Riemannian manifold, \(G\) is its isometry Lie group (assumed in this paper to be connected and semisimple), and \(K \subset G\) the stabilizer subgroup. Then there is a Riemannian metric \(\{ \cdot, \cdot \}\) on \(G\) which is \(G\)-left-invariant, \(K\)-right-invariant, and which turns the projection \(p : G \to M\) into a Riemannian submersion. If \(\mathfrak{P} \subset TG\) is the horizontal distribution induced by \(p\), then the restriction of \(\{\cdot, \cdot\}\) to \(\mathfrak{P}\) can be viewed as a sub-Riemannian geometry on \(G\). In this paper, the author studies properties of this geometry. In Theorem 4, it is shown that the distribution \(\mathfrak{P}\) is bracket-generating, and that all geodesics of the sub-Riemannian manifold \((G, \mathfrak{P}, \{\cdot,\cdot\}|_{\mathfrak{P}})\) are normal. Moreover, up to reparametrization, every such geodesic \(\gamma\) from the identity is either of the form \(\gamma(t) = \exp(t \xi)\) or \(\gamma(t) = \exp(t(\xi + \eta)) \exp(-t \eta)\), where \(\xi, \eta\) are some elements of the Lie algebra of \(G\). Sub-Riemannian manifolds in general do not have a canonical notion of curvature. However, if one chooses a complement \(\mathfrak{P}_1\) of the horizontal distribution \(\mathfrak{P}\) in the tangent bundle (here such a choice is called a ``rigging''), then it becomes possible to define sectional curvature with respect to this rigging; here the author follows the development of [\textit{A. F. Solov'ev}, Math. Notes 35, 61--68 (1984; Zbl 0553.53017)]. In the present setting, there is a natural choice of rigging \(\mathfrak{P}_1\); namely, the orthogonal complement of \(\mathfrak{P}\) with respect to the Riemannian metric \(\{\cdot,\cdot\}\). Using this rigging, in Theorem 10, a simple formula for the sectional curvatures in \(G\) is obtained. In Section 5, the author studies the special case of \(M = \mathrm{Sp}(1)\) equipped with a bi-invariant Riemannian metric, with \(G = \mathrm{Sp}(1) \times \mathrm{Sp}(1)\) and \(K \subset G\) the diagonal subgroup. Let \(\gamma\) be a geodesic in \((G, \mathfrak{P}, \{\cdot,\cdot\}|_{\mathfrak{P}})\) and \(q = p \circ \gamma\) be its projection onto \(M\). Using the aforementioned characterization of geodesics \(\gamma\), the author computes the geodesic curvature and torsion of the projection \(q\).
scientific article; zbMATH DE number 6892017
scientific article