Classification of prolonged bi-metric structures on distributions of non-zero curvature of sub-Riemannian manifolds

S. V. Galaev

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 18 (2018) no. 3, pp. 263-273

Résumé

The notion of the interior geometry of a sub-Riemannian manifold $M$ is introduced, that is the aggregate of those manifold properties that depend only on the framing $D^ \perp$ of the distribution $D$ of the sub-Riemannian manifold as well as on the parallel transport of the vectors tangent to the distribution $D$ along the curves tangent to this distribution. The main invariants of the interior geometry of a sub-Riemannian manifold $M$ are the following: the Schouten curvature tensor; the 1-form $\eta$ defining the distribution $D$; the Lie derivative $L_{\vec\xi}g$ of the metric tensor $g$ along a vector field $\vec\xi$; the tensor field P that with respect to adaptive coordinates has the components $P_{ad}^c=\partial_n\Gamma_{ad}^c$. Depending on the properties of these invariants, 12 classes of sub-Riemannian manifolds are defined. Using the interior connection on the sub-Riemannian manifold $M$, an almost contact structure with a bi-metric is defined on the distribution $D$, which is called the prolonged structure in the paper. The comparison of two classifications of the prolonged structures is given. Accordance with the first classification, there are 12 classes of the prolonged structures corresponding to the 12 classes of the initial sub-Riemannian manifolds. The second classification is grounded on the properties of the fundamental $F$ of type $(0, 3)$ associated with the bi-metrical structure. According to the second classification, there exist $2^{11}$ classes of bi-metrical structures, among that $11$ are basis classes $F_i$, $i=1,\ldots,11$. The paper considers the case of a sub-Riemannian manifold with non-zero Schouten curvature tensor and with zero Lie derivative $L_{\vec\xi}g$. It is proved that the prolonged almost contact bi-metrical structures corresponding to sub-Riemannian structures with the invariant $\omega=d\eta$ equal to zero, belong to the class $F_1\oplus F_2\oplus F_3$, and the ones with non-zero invariant а $\omega=d\eta$ belong to the class $F_1\oplus F_2\oplus F_3\oplus F_7\oplus \ldots\oplus F_{10}$.