Poincaré Lemma on Some SubRiemannian Manifolds

Hsi-Chun Wu

Department of Mathematics

National Central University

flyinmoon0217@yahoo.com.tw

    Let `\mathbf{X}=\{X_1,X_2,\ldots, X_m\}` be `m` linearly independent vector fields defined on an `n`-dimensional manifold `\mathcal M_n` with `m\le n`, and assume that `\mathbf{X}` satisfies the bracket generating property: the vector fields `\mathbf{X}` and finitely many steps of their Lie brackets span `T\mathcal{M}_n`. Therefore, `\mathcal{M}_n` can be recognized as a subRiemannian manifold by Chow's theorem [6] and the Carnot-Carathéodory distance. Let `V=(a_1,a_2,\ldots, a_m)` be a vector-valued function defined on `{\mathcal M}_n` where `a_j`, `j=1,\ldots, m` are smooth functions. The function `V` is said to be conservative if there exists a function `f`, called the potential function, that satisfies the following system
` X_1 f = a_1,\quad X_2 f = a_2,\quad \cdots \quad X_m f = a_m.`
It is known that in [2], in virtue of the curl operator [7], a characterization of conservative vector fields, called the integrability condition, on the Heisenberg group `\mathcal{H}^1` is provided as \begin{cases} X^2_1b = (X_1X_2 + [X_1, X_2])a, \\ X^2_2a = (X_2X_1 + [X_2, X_1])b, \end{cases}. where `X_1 = \partial_{x} - 2y\partial_z, X_2 = \partial_{y} + 2x\partial_z`, `a, b` are smooth functions, and `[\cdot,\cdot]` is the Lie bracket. In [3], a potential function on `\mathcal{H}^` is given by
`f(x,y,z)=\int_0^1\langle U(\gamma(t)),\dot{\gamma}(t) \rangle dt,`
where `\langle\cdot,\cdot\rangle` is a subRiemannian metric, `U=aX_1+bX_2`, and `\gamma` is a geodesic connecting `(x,y,z)` and the origin.
In this talk, I will discuss integrability conditions and potential functions on two important examples in subRiemannian manifolds: the Heisenberg groups `\mathcal{H}^n` and the quaternion Heisenberg group `qH^1` [4,5]. The integrability conditions can be found by using the curl tensor [1]
`(\text{curl }U)(X,Y) = Yg(U,X) - Xg(U,Y) + g(U,[X,Y]),`
where `U,X,Y` are vector fields and `g` is a Riemannian metric. The potential functions related to conservative vector fields are able to be solved explicitly in integral forms.

Keyword: Bracket generating property, Heisenberg group, Curl, Integrability condition, Poincaré lemma

References

[1] O. Calin and D. C. Chang, Geometric mechanics on Riemannian manifolds: applications to partial differential equations, Birkhäuser Boston, 2005. doi: https://doi.org/10.1007/b138771
[2] O. Calin, D. C. Chang, and M. Eastwood, Integrability conditions for Heisenberg and Grushin-type distributions, Anal. Math. Phys., 4 (2014), 99-114. doi: 10.1007/s13324-014-0073-1
[3] O. Calin, D. C. Chang, and J. Hu, Poincaré's lemma on the Heisenberg group, Adv. in Appl. Math., 60 (2014), 90-102. http://dx.doi.org/10.1016/j.aam.2014.08.003
[4] D. C. Chang, Y. S. Lin, H. C. Wu, and N. Yang, Poincaré lemma on Heisenberg groups, Applied Analysis and Optimization, 1 (2017), 283-300.
[5] D. C. Chang, N. Yang, and H. C. Wu, Poincaré lemma on quaternion-like Heisenberg groups, Canad. Math. Bull., 61 (2018), 495-508. http://dx.doi.org/10.4153/CMB-2017-027-4
[6] W. L. Chow, Uber Systeme van Linearen partiellen Differentialgleichungen erster Ordnung, {Math. Ann., 117 (1939), 98-105}.
[7] B. Franchi, N. Tchou, and M. C. Tesi, Div-curl type theorem, H-convergence and Stokes formula in the Heisenberg group, Comm. Contemp. Math., 8 (2006), 67-99.