  We present a discussion about the local isometric rigidity problem in
codimension 2 with a concrete example. We show the necessity of extending the
notions of genuine and honest rigidity in order to have the transitivity
property. In order to do so, we show the necessity of studying the isometric
immersions in semi-Euclidean spaces. We show that this extension comes with a
natural type of singularity in the inner product.
