Integrability conditions on a sub-Riemannian structure on \$\$\mathbb {S}^3\$\$ S 3

Integrability conditions on a sub-Riemannian structure on \$\$\mathbb {S}^3\$\$ S 3 This paper deals with integrability conditions for a sub-Riemannian system of equations for a step 2 distribution on the sphere \$\$\mathbb {S}^3\$\$ S 3 . We prove that a certain sub-Riemannian system \$\$Xf =a\$\$ X f = a , \$\$Yf =b\$\$ Y f = b on \$\$\mathbb {S}^3\$\$ S 3 has a solution if and only if the following integrability conditions hold: \$\$X^2 b + 4b = (XY + [X, Y]) a \$\$ X 2 b + 4 b = ( X Y + [ X , Y ] ) a , \$\$Y^2 a + 4a = (YX-[X, Y]) b\$\$ Y 2 a + 4 a = ( Y X - [ X , Y ] ) b . We also provide an explicit construction of the solution f in terms of the vector fields X, Y and functions a and b. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

Integrability conditions on a sub-Riemannian structure on \$\$\mathbb {S}^3\$\$ S 3

, Volume 7 (1) – Feb 13, 2016
10 pages      /lp/springer-journals/integrability-conditions-on-a-sub-riemannian-structure-on-mathbb-s-3-s-enrh720ExL
Publisher
Springer Journals
Subject
Mathematics; Analysis; Mathematical Methods in Physics
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-016-0126-8
Publisher site
See Article on Publisher Site

Abstract

This paper deals with integrability conditions for a sub-Riemannian system of equations for a step 2 distribution on the sphere \$\$\mathbb {S}^3\$\$ S 3 . We prove that a certain sub-Riemannian system \$\$Xf =a\$\$ X f = a , \$\$Yf =b\$\$ Y f = b on \$\$\mathbb {S}^3\$\$ S 3 has a solution if and only if the following integrability conditions hold: \$\$X^2 b + 4b = (XY + [X, Y]) a \$\$ X 2 b + 4 b = ( X Y + [ X , Y ] ) a , \$\$Y^2 a + 4a = (YX-[X, Y]) b\$\$ Y 2 a + 4 a = ( Y X - [ X , Y ] ) b . We also provide an explicit construction of the solution f in terms of the vector fields X, Y and functions a and b.

Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Feb 13, 2016