It is well known that a smooth manifold $M$ is orientable if the first Stiefel-Whitney class of the tangent bundle vanishes. In particular, this implies that if $M$ is equipped with a pseudo-Riemannian metric of signature $(t,s)$, then the frame bundle $O(M)$ admits a reduction to a principal $SO(t,s)$ bundle under the embedding $SO(t,s)\hookrightarrow O(t,s)$.
Now, let $SO^+(t,s)$ denote the Lie subgroup of $SO(t,s)$ consisting of the linear isometries of $\mathbb{R}^{t,s}$ which preserve the orientation of any maximally negative definite subspace of $\mathbb{R}^{t,s}$. The pseudo Riemannian manifold is then orientable and time orientable if the frame bundle $O(M)$ admits a reduction to a principal $SO^+(t,s)$ bundle under the embedding $SO^+(t,s)\hookrightarrow O(t,s)$. Is there a characteristic class related to this result?
I am asking because any orientable and time orientable pseudo Riemmanian manifolds admits a $\text{Spin}^+(t,s)$ structure if and only if the second Stiefel-Whitney class of $TM$ vanishes. If instead $M$ is just orientable, then I imagine that $M$ would admit a $\text{Spin}(t,s)$ structure if and only if the second Stiefel-Whitney class vanishes. However, I am unsure of when general orientable pseudo Riemannian manifolds which are also spin admit a $\text{Spin}^+(t,s)$ structure, i.e. when can we conclude that a pseudo Riemannian manifold that is spin, is also $\text{spin}^+$?
In the Lorentzian case, then we get a time orientation for free, as a necessary and sufficient condition for the existence of a Lorentz metric is a nowhere vanishing vector field, which would easily supply us with a time orientation.
I would guess that if there exists a vector bundle isomorphism:
$$TM\cong E^t\oplus E^s$$
where the vector bundle $E^t$ has vanishing first Stiefel-Whitney class, then $TM$ is time orientable, but I am honestly unsure. Any source, or collection of sources that systematically treats existence of spin structures, and time orientations in the pseudo Riemannian case would also be greatly appreciated.
Best Answer
For the first question I believe the answer is yes. Almost certainly it's in the literature but I do not know this literature very well.
The idea is as you suggested. Maximal-rank timelike (or spacelike) vector subspaces form a convex space. So you paste together local decompositions using a partition of unity to decompose the tangent bundle of your Lorentz manifold into an orthogonal direct-sum of a maximal timelike sub-bundle and a maximal space-like subbundle.
You then have characteristic classes for the sub-bundles, and $w_1$ of your timelike sub-bundle is the obstruction you are looking for.