I'm reading Hawking and Ellis, and on p.16 they say that "The subspace of [tex]T_p[/tex] defined by [tex]\langle \omega,x \rangle[/tex]=(constant) for a given one-form [tex]\omega[/tex], is linear." But in what sense is this true? For if the constant is non-zero, the 0 of [tex]T_p[/tex] is not in the subspace, nor does it satisfy the usual linearity condition. By analogy with euclidean space, the points [tex]r \cdot a = d[/tex] form a plane, but not technically a subspace of the original space (since it is "shifted"). Am I missing something?