describing 4-vectors in space-time

[jfy4]

I have a question...

I would like to generically describe a 4-vector locally in space-time. Would i go about that by simply taking a 4-vector and multiplying it by a metric? like

u^{\alpha}=u_{\beta}g^{\alpha\beta}

with u^{\alpha} the new 4-vector in the space specified by the metric?

[haushofer]

I'm not sure what you mean; what you've just written down is the notion of vector spaces and dual vector spaces being isomorphic, with the isomorphism given by the metric.

"Local vector fields" are vector fields which can change from point to point on your manifold. On curved manifolds these vectors live in the tangent space at that point.