Harmonicity and gauge transformations in dimension 3 (Q1407633)

Let (\(M,\eta \)) be a contact manifold. Its contact bundle \(H(M)\) leads to a natural CR-structure and an almost contact structure (\(\Phi \), \(\xi \), \(\eta \)) on \(M\). The author considers the Webster metric \(g\) on \(M\). After a special gauge transformation for the contact form \(\eta \), defined by \(\widetilde{\eta }=\varepsilon e^{f}\eta \) where \(f\in C^{\infty }(M)\) and \(\varepsilon =\pm 1\), a new almost contact structure (\(\widetilde{\Phi },\widetilde{\xi },\widetilde{\eta }\)) on \(M\) is obtained. Moreover, a new Riemannian metric \(\widetilde{g}\) is obtained from \(g\) such that the compatibility conditions with respect the new structure (\(\widetilde{\Phi },\widetilde{\xi },\widetilde{\eta }\)) are fulfilled and the restrictions of \(g\) and \(\widetilde{g}\) on \(H(M)\) are connected by a conformal transformation. Let \(M\) be a smooth contact \(3\) dimensional manifold endowed with the Webster metric \(g\) and let \(\eta\) be the nondegenerate contact 1-form. Then \(H(M)=\ker \eta \) and the restriction \(J\) of \(\Phi \) on \(H(M)\) is a complex structure on \(H(M)\). Moreover, (\(X,Y,\xi \)), where \(X\in H(M),Y=\Phi (X)\) and \(\xi \) is the Reeb vector field, is a local \(g\)-orthonormal basis for \(\chi (M)\). The following result is proved: Let \(M\) be a contact 3-dimensional manifold on which a special gauge transformation is considered. Then, the identity map \(1_{M}:(M,g)\rightarrow (M,\widetilde{g})\) is harmonic if and only if either \(f\) is a constant function, or there exists a local unit vector field \(X\) belonging to the contact distribution \(H(M)\), satisfying \([X,\xi ]=v\Phi X\) and \([\Phi X,\xi ]=0\) with \(v\) constant greater (strictly) than 2 and \(f\) being given by \(df=\pm \sqrt{v-2}X^{b}\) (here \(f\) is the \(C^{\infty }\) function defining the special gauge transformation used and \(X^{b}\) denotes the 1-form obtained from \(X\) by the musical isomorphism).