Differential geometry of projective bundles Can someone give me a reference about projective bundles from a differential-geometric point of view? I am not very familiar with algebraic geometry. I would like, for example, some theory about when a projective bundle comes from a vector bundle, and when not (I guess it has to do with cohomology, just like projective representations). 
Better if the material is a little didactic!
Thanks.
 A: The smooth and topological answers should agree so I will just concentrate on the topological case. Let me restrict attention to the complex case although the real case looks similar. 
The classifying space of $\mathbb{P}^{n-1}$ bundles is $BPGL_n(\mathbb{C})$. There is a natural functor from rank $n$ complex vector bundles to $\mathbb{P}^{n-1}$-bundles and it corresponds by the Yoneda lemma to the natural map $BGL_n(\mathbb{C}) \to BPGL_n(\mathbb{C})$ on classifying spaces. There is a short exact sequence of topological groups
$$1 \to \mathbb{C}^{\times} \to GL_n(\mathbb{C}) \to PGL_n(\mathbb{C}) \to 1$$
which in turn induces a longish exact sequence the end of which reads
$$\dots \to H^1(X, GL_n(\mathbb{C})) \to H^1(X, PGL_n(\mathbb{C})) \to H^2(X, \mathbb{C}^{\times}).$$
So the obstruction to a $\mathbb{P}^{n-1}$-bundle coming from a rank $n$ vector bundle is a class in $H^2(X, \mathbb{C}^{\times})$, which one can think of as a topological analogue of the Brauer group. In fact Serre showed that a natural notion of "topological Brauer group" gives precisely the torsion subgroup of $H^2(X, \mathbb{C}^{\times})$. For more details see these notes by Jeremy Booher. 
(It's tempting to just think of $H^2(X, \mathbb{C}^{\times})$ as $H^3(X, \mathbb{Z})$ but then the analogy to the Brauer group and projective representations is less clear. Also I think equivariantly they actually differ.) 
