
Izv. Saratov Univ. Math. Mech. Inform., 2018, Volume 18, Issue 3, Pages 263–273
(Mi isu761)




Scientific Part
Mathematics
Classification of prolonged bimetric structures on distributions of nonzero curvature of subRiemannian manifolds
S. V. Galaev^{} ^{} Saratov State University, 83, Astrakhanskaya Str.,
Saratov, 410012, Russia
Abstract:
The notion of the interior geometry of a subRiemannian 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 subRiemannian 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 subRiemannian manifold $M$ are the following: the Schouten curvature tensor; the 1form $\eta$ 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 subRiemannian manifolds are defined. Using the interior connection on the subRiemannian manifold $M$, an almost contact structure with a bimetric 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 subRiemannian manifolds. The second classification is grounded on the properties of the fundamental $F$ of type $(0, 3)$ associated with the bimetrical structure. According to the second classification, there exist $2^{11}$ classes of bimetrical structures, among that $11$ are basis classes $F_i$, $i=1,\ldots,11$. The paper considers the case of a subRiemannian manifold with nonzero Schouten curvature tensor and with zero Lie derivative $L_{\vec\xi}g$. It is proved that the prolonged almost contact bimetrical structures corresponding to subRiemannian 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 nonzero invariant а $\omega=d\eta$ belong to the class $F_1\oplus F_2\oplus F_3\oplus F_7\oplus \ldots\oplus F_{10}$.
Key words:
subRiemannian manifold of contact type; interior geometry of subRiemannian manifold; prolonged almost contact structure with bimetric; distribution of nonzero curvature.
DOI:
https://doi.org/10.18500/181697912018183263273
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514.76
Citation:
S. V. Galaev, “Classification of prolonged bimetric structures on distributions of nonzero curvature of subRiemannian manifolds”, Izv. Saratov Univ. Math. Mech. Inform., 18:3 (2018), 263–273
Citation in format AMSBIB
\Bibitem{Gal18}
\by S.~V.~Galaev
\paper Classification of prolonged bimetric structures on distributions of nonzero curvature of subRiemannian manifolds
\jour Izv. Saratov Univ. Math. Mech. Inform.
\yr 2018
\vol 18
\issue 3
\pages 263273
\mathnet{http://mi.mathnet.ru/isu761}
\crossref{https://doi.org/10.18500/181697912018183263273}
\elib{https://elibrary.ru/item.asp?id=35728991}
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