Given a symmetric, bilinear map $B : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^m$, a common notion of nondegeneracy is the following:
$$
\mbox{If } B(x,y) = 0 \mbox{ for all } y, \mbox{then } x=0.
$$
However, there is another notion, let's call nonsingular, defined as follows:
$$
\mbox{If } B(x,y) = 0 \mbox{ then } x=0 \mbox{ or } y=0.
$$
This second notion is important for immersions and embeddings of projective spaces, and so the use of the term *nonsingular* is common in some literature: see e.g. Section 6 of [this paper](http://blms.oxfordjournals.org/content/3/3/257.full.pdf) by James.  I would like to know more about this second notion, but cannot find much because I only see search results concerning the first notion.  I am wondering

1. Is there a different name for this second notion?
2. Are there some known uses for this definition besides immersions/embeddings of projective spaces?