Notes in the isometric deformations problem in codimension 2

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.