Two-dimensional manifolds with metrics of revolution

This is a study of the topological and metric structure of two-dimensional manifolds with a metric that is locally a metric of revolution. In the case of compact manifolds this problem can be thoroughly investigated, and in particular it is explained why there are no closed analytic surfaces of revolution in $\mathbb R^3$ other than a sphere and a torus (moreover, in the smoothness class $C^\infty$ such surfaces, understood in a certain generalized sense, exist in any topological class).