Classiﬁcation of Prolonged Bi-metric Structures on Distributions of Non-zero Curvature of Sub-Riemannian Manifolds
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⊥ 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 η deﬁning the distribution D; the Lie derivative Lξ~g ~ the tensor ﬁeld P that with respect to adaptive coordinates has the of the metric tensor g along a vector ﬁeld ξ; c = ∂ Γc. Depending on the properties of these invariants, 12 classes of sub-Riemannian components Pad n ad 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 211 classes of bi-metrical structures, among that 11 are basis classes Fi, i = 1,..., 11. The paper considers the case of a sub-Riemannian manifold with non-zero Schouten curvature tensor and with zero Lie derivative Lξ~g. It is proved that the prolonged almost contact bi-metrical structures corresponding to sub-Riemannian structures with the invariant ω = dη equal to zero, belong to the class F1 ⊕ F2 ⊕ F3, and the ones with non-zero invariant а ω = dη belong to the class F1 ⊕ F2 ⊕ F3 ⊕ F7 ⊕... ⊕ F10.