It is fun to give an answer when there is no question :-)) LOL...

Anyway, pick your vector bundle $V$ on a manifold $X$ and consider its top exterior power $\Lambda^m V$. Now you have Plucker map from $X$ to the projective space $P(\Gamma (X,\Lambda^m V)^\ast)$ of the dual space of the global sections. It is given by mapping of $p\in X$ to the restriction map of the global sections $\Gamma (X,\Lambda^m V)\rightarrow V_p$. Notice that $V_p$ is a one-dimensional vector space without a natural basis - different choices of basis give different functionals, so the map naturally goes to projective space.

You'd better have your global sections finite-dimensional, which you get by adding holomorphicity + compactness. If you want embedding, you have to play with ampleness, etc.

It is fun to give an answer when there is no question :-)) LOL...

Anyway, pick your vector bundle $V$ on a manifold $X$ and consider its top exterior power $\Lambda^m V$. Now you have Plucker map from $X$ to the projective space $P(\Gamma (X,\Lambda^m V)^\ast)$ of the dual space of the global sections. It is given by mapping of $p\in X$ to the restriction map of the global sections $\Gamma (X,\Lambda^m V)\rightarrow \Lambda^m V_p$. Notice that $\Lambda^m V_p$ is a one-dimensional vector space without a natural basis - different choices of basis give different functionals, so the map naturally goes to projective space.

You'd better have your global sections finite-dimensional, which you get by adding holomorphicity + compactness. If you want embedding, you have to play with ampleness, etc.