Let Sbe a surface defined by a conformal immersion  of a Riemann surface  By the existence of isothermal coordinates on  associated to the immersed surface Sone has the generalized Gauss map from  into the quadric  defined locally in terms of a complex partial derivative of  The main purpose of the article is to relate the normal and Gaussian curvatures of Sto its generalized Gauss map. Using an isometric identification of the quadric  with the product of the two 2-spheres in  and theory of complex functions we get an extension of the celebrated Theorem of Chern and Spanier, Hoffman and Osserman to conformally immersed surface in

normal curvature, Gaussian curvature, surface immersed in