Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sub-Riemannian geometry of parallelizable spheres   The first aim of the present paper is to compare various sub-Riemannian
structures over the three dimensional sphere $S^3$ originating from different
constructions. Namely, we describe the sub-Riemannian geometry of $S^3$ arising
through its right Lie group action over itself, the one inherited from the
natural complex structure of the open unit ball in $\comp^2$ and the geometry
that appears when considering the Hopf map as a principal bundle. The main
result of this comparison is that in fact those three structures coincide.
  In the second place, we present two bracket generating distributions for the
seven dimensional sphere $S^7$ of step 2 with ranks 6 and 4. These yield to
sub-Riemannian structures for $S^7$ that are not present in the literature
until now. One of the distributions can be obtained by considering the CR
geometry of $S^7$ inherited from the natural complex structure of the open unit
ball in $\comp^4$. The other one originates from the quaternionic analogous of
the Hopf map.

1
The categories of this paper are: cs
37
The categories of this paper are: supr-con
14
The categories of this paper are: math-ph
5
The categories of this paper are: gr-qc
17
The categories of this paper are: q-fin
11
The categories of this paper are: hep-lat
19
The categories of this paper are: eess
31
The categories of this paper are: ao-sci
23
The categories of this paper are: solv-int
29
The categories of this paper are: funct-an
26
The categories of this paper are: bayes-an
27
The categories of this paper are: comp-gas
12
The categories of this paper are: nucl-th
4
The categories of this paper are: cond-mat
16
The categories of this paper are: stat
8
The categories of this paper are: hep-ex