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Cuspidal edges and swallowtails are typical non-degenerate singular points onwave fronts in the Euclidean $3$-space. Their first fundamental forms belong toa class of positive semi-definite metrics called "Kossowski metrics". A pointwhere a Kossowski metric is not positive definite is called a singular point ora semi-definite point of the metric. Kossowski proved that real analyticKossowski metric germs at their non-parabolic singular points(the definition of"non-parabolic singular point" is stated in the introduction here) can berealized as wave front germs (Kossowski's realization theorem).  On the other hand, in a previous work with K. Saji, the third and the fourthauthors introduced the notion of "coherent tangent bundle". Moreover, theauthors, with M. Hasegawa and K. Saji, proved that a Kossowski metriccanonically induces an associated coherent tangent bundle.  In this paper, we shall explain Kossowski's realization theorem from theviewpoint of coherent tangent bundles. Moreover, as refinements of it, we givea criterion that a given Kossowski metric can be realized as the induced metricof a germ of cuspidal edge (resp. swallowtail or cuspidal cross cap). Severalapplications of these criteria are given. Also, some remaining problems onisometric deformations of singularities of analytic maps are given at the endof this paper.

Isometric deformations of wave fronts at non-degenerate singular points