- Like the metric

f(θ) = Ω(u, (cos θ)v + (sin θ)w).

f(0) = Ω(u, v) while f(π) = Ω(u, −v)

but Ω(u, v) = − Ω(u, −v)

and so f(0) = −f(π)

but f is real and continuous and therefore there must be φ for which f(φ)=0.

Since v and w are independent x = (cos φ)v + (sin φ)w = 0.

We have then Ω(u, x) = 0 with u not parallel to x.

I think the plainest definition of non-degenerate ω, is that for any non-zero vector u, there is a vector v such that ω(u, v) ≠ 0.