  In this paper, we give two classes of positive semi-definite metrics on
2-manifolds. The one is called a class of Kossowski metrics and the other is
called a class of Whitney metrics: The pull-back metrics of wave fronts which
admit only cuspidal edges and swallowtails in $\boldsymbol{R}^3$ are Kossowski
metrics, and the pull-back metrics of surfaces consisting only of cross cap
singularities are Whitney metrics. Since the singular sets of Kossowski metrics
are the union of regular curves on the domains of definitions, and Whitney
metrics admit only isolated singularities, these two classes of metrics are
disjoint. In this paper, we give several characterizations of intrinsic
invariants of cuspidal edges and cross caps in these classes of metrics.
Moreover, we prove Gauss-Bonnet type formulas for Kossowski metrics and for
Whitney metrics on compact 2-manifolds.
