| dc.description.abstract | The classical result of Eisenhart states that if a Riemannian metric g admits a Riemanian met-ric that is not constantly proportional to g and has the same (parameterized) geodesics as g in a neighborhood of a given point, then g is a direct product of two Riemannian metrics in this neigh-bourhood. We extend this result to sub-Riemannian metrics on a class of step 2 distributions.
The thesis is devoted to study of the properties of sub-Riemannian metrics that ensure that it admits the product structure. It consist of two parts devoted to two different set of properties.
In the first part, inspired by the classical Eisenhart and De Rham decomposition theorem in Riemannian geometry, we make an attempt to show that the property that a sub-Riemannian metric admits a nonconstantly proportional metric with the same geodesics (this property is called affine non-rigidity) implies that it admits the product structure.
In the second part, we replace affine non-rigidity by a weaker set of the following two properties (both of which follows from affine non-rigidity):
1. the Tanaka symbol of the underlying distribution is decomposable in the natural sense (as a fundamental graded Lie algebra) ;
2. the Jacobi equation along generic extremals are decoupled.
Our work in the first part solves the main conjecture and shows that a step 2 distribution with its Tanaka symbol decomposable into 2 nonzero ad-surjective indecomposable fundamental graded Lie algebras, along with the affine non-rigidity of a sub-Riemannian metric on it, must admit a product structure.
In the second part, first, we prove that the decomposability of Jacobi curve is necessary for affine non-rigidity on (D, M, g), and second, we found a combination of necessary conditions for the affine non-rigidity which is actually sufficient for (D, M, g) to admit a product structure in the case study of (4, 6) distribution. To show this sufficiency, we use Zelenko-Li’s theory of a normal moving frames of the corresponding Jacobi curves, The analysis of the structure equations for such frames allows us to conclude the existence of a product structure for distributions under consideration. | |
