Some remarks on the nonorientable surfaces (Q2774629)

A dianalytic structure is defined by an atlas \(\mathcal A\) such that every transition function is a conformal mapping or a mapping whose complex conjugate is conformal. The main result is the following reinforcement of a classical result due to Klein.NEWLINENEWLINENEWLINETheorem: If \(X\) is a nonorientable surface endowed with a dianalytic structure and if \(f:W\to X\) is a covering of \(X\) with \(W\) orientable, then \(W\) admits a structure of Riemann surface with respect to which the projection \(f\) is dianalytic. For every symmetry \(k\) of \(W\) (if it exists any), there is a dianalytic atlas of the space \(W/\langle k\rangle\) such that the canonical projection \(W\to W/\langle k\rangle\) is a dianalytic function. Moreover, \(W/\langle k\rangle\) is a nonorientable cover of \(X\). If the dianalytic structure of \(X\) has been induced by the isothermal metric \(ds=\lambda|dz|\), then the analytic structure of \(W\) is induced by the isotermal metric \(d\sigma=\lambda_1|dw|\), where \(\lambda_1(w)=\lambda(z(w))\big|\frac{\partial z}{\partial w}+\frac{\partial\bar z}{\partial\bar w}\). NEWLINENEWLINENEWLINEThe effect of these results on the vector bundle of covariant tensors of second order on \(X\) is studied. Some examples are given too.