Poincaré’s Lemma on the Heisenberg Group
نویسندگان
چکیده
It is well known that the system ∂xf = a, ∂yf = b on R 2 has a solution if and only if the closure condition ∂xb = ∂ya holds. In this case the solution f is the work done by the force U = (a, b) from the origin to the point (x, y). This paper deals with a similar problem, where the vector fields ∂x, ∂y are replaced by the Heisenberg vector fields X1, X2. In this case the subRiemannian system X1f = a, X2f = b has a solution f if and only if the following integrability conditions hold X2 1 b = (X1X2 + [X1,X2])a, X 2 2a = (X2X1 + [X2,X1])b. The question addressed by this paper is whether we can provide a Poincaré-type Lemma for the Heisenberg distribution. The positive answer is given by Theorem 2, which provides a result similar to the Poincaré’s Lemma in the integral form. The solution f in this case is the work done by the force vector field aX1 + bX2 along any horizontal curve from the origin to the current point.
منابع مشابه
B-FOCAL CURVES OF BIHARMONIC B-GENERAL HELICES IN Heis
In this paper, we study B-focal curves of biharmonic B -general helices according to Bishop frame in the Heisenberg group Heis Finally, we characterize the B-focal curves of biharmonic B- general helices in terms of Bishop frame in the Heisenberg group Heis
متن کاملTranslation invariant surfaces in the 3-dimensional Heisenberg group
In this paper, we study translation invariant surfaces in the 3-dimensional Heisenberg group $rm Nil_3$. In particular, we completely classify translation invariant surfaces in $rm Nil_3$ whose position vector $x$ satisfies the equation $Delta x = Ax$, where $Delta$ is the Laplacian operator of the surface and $A$ is a $3 times 3$-real matrix.
متن کاملPoincaré ’ S Lemma on Some Non - Euclidean Structures
In this paper we prove the Poincaré lemma on some n-dimensional corank 1 sub-Riemannian structures, formulating the (n−1)n(n +3n−2) 8 necessarily and sufficiently ’curl-vanishing’ compatibility conditions. In particular, this result solves partially an open problem formulated by Calin and Chang. Our proof is based on a Poincaré lemma stated on Riemannian manifolds and a suitable Cesàro-Volterra...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2000