Why is the subspace defined by ⟨ω,x⟩=constant linear in Hawking and Ellis?

[Manicwhale]

I'm reading Hawking and Ellis, and on p.16 they say that

"The subspace of $$T_p$$ defined by $$\langle \omega,x \rangle$$=(constant) for a given one-form $$\omega$$, is linear."

But in what sense is this true? For if the constant is non-zero, the 0 of $$T_p$$ is not in the subspace, nor does it satisfy the usual linearity condition.

By analogy with euclidean space, the points $$r \cdot a = d$$ form a plane, but not technically a subspace of the original space (since it is "shifted"). Am I missing something?

 

[George Jones]

No, I don't think that you are missing anything, especially considering the sentence that follows the sentence which you quoted.

 

[Manicwhale]

Thanks!

Turns out that's the least of my difficulties in this book, but it's quite interesting.