# On curvatures of homogeneous sub-Riemannian manifolds

## TL;DR

This paper explores curvature tensors in homogeneous sub-Riemannian manifolds, proposing methods to compute Solov'ev curvatures using invariant distributions and foliations related to Lie group symmetries.

## Contribution

It introduces a rigging approach for calculating curvatures in homogeneous sub-Riemannian manifolds using invariant distributions and Lie group symmetries.

## Key findings

- Method for calculating Solov'ev sectional and Ricci curvatures.
- Application of foliation on cotangent bundle related to Lie groups.
- Examples including contact manifolds and Carnot groups.

## Abstract

The author discusses in some detail the old definitions of the curvature tensors for rigged metrized distributions on manifolds given by Schouten, Wagner, and Solov'ev. To calculate the Solov'ev sectional and Ricci curvatures for homogeneous sub-Riemannian manifolds, the author suggests to use in some cases special riggings of invariant completely non- holonomic distributions on manifolds. As a justification, we find a foliation on the cotangent bundle over a Lie group G whose leaves are tangent to invariant Hamiltonian vector fields for the Pontryagin-Hamilton func- tion. This function was applied in the Pontryagin maximum principle for the time-optimal problem. The foliation is entirely described by the co-adjoint representation of the Lie group G. Also we use the canonical symplectic form and its values for the above mentioned invari- ant Hamiltonian vector fields. In particular, the above rigging method is applicable to contact sub-Riemannian manifolds, sub-Riemannian Carnot groups, and homogeneous sub-Riemannian manifolds possessing a subme- try onto a Riemannian manifold. At the end, some examples are presented.