Abstract
This paper deals with integrability conditions for a sub-Riemannian system of equations for a step 2 distribution on the sphere \(\mathbb {S}^3\). We prove that a certain sub-Riemannian system \(Xf =a\), \(Yf =b\) on \(\mathbb {S}^3\) has a solution if and only if the following integrability conditions hold: \(X^2 b + 4b = (XY + [X, Y]) a \), \(Y^2 a + 4a = (YX-[X, Y]) b\). We also provide an explicit construction of the solution f in terms of the vector fields X, Y and functions a and b.
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This research project is partially supported by Hong Kong RGC competitive earmarked research Grants \(\#\)601813, \(\#\)601410. The D.-C. Chang is also partially supported by an NSF Grant DMS-1408839.
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Calin, O., Chang, DC. & Hu, J. Integrability conditions on a sub-Riemannian structure on \(\mathbb {S}^3\) . Anal.Math.Phys. 7, 9–18 (2017). https://doi.org/10.1007/s13324-016-0126-8
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DOI: https://doi.org/10.1007/s13324-016-0126-8