Isometric realization of cross caps as formal power series and its applications

Two cross caps in Euclidean $3$-space are said to be formally isometric if their Taylor expansions of the first fundamental forms coincide by taking a suitable local coordinate system. For a given $C^\infty$ cross cap $f$, we give a method to find all cross caps which are formally isometric to $f$. As an application, we give a countable family of intrinsic invariants of cross caps which recognizes formal isometry classes completely.