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.
`(\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