We find that there is a serious mistake in the proof of Lemma 3 of paper [1]. The author wants to prove, copying the original method in [2], the global Borel theorem. But a Sample example shows that this is impossible.
We find that: the Theorems 1 and 2, called 'the generalization of Morse Lemma' in [1], are just a copy of a well known result in Singularity Theory, and the conclusion of Theorem 2 is not true. Since the Morse Lemma is an important result in Singularity Theory, we will discuss to the author of [1] for the problem of generalizing Morse Lemma.
