In this paper we show the uniqueness of the constant solution to
a general Yamabe-type equation on a compact pseudohermitian manifold. The
Riemannian version was first proved by Bidaut-Ve´ron and Ve´ron. In another
direction, we make use of the vanishing property of two eigenfunctions of the
sub-Laplacian to conclude the nonvanishing of the first cohomology group of a
compact pseudohermitian manifold. The corresponding version for the Riemannian case was first proved by Gichev.
