What is the analog of a vector bundle connection for a projective bundle?

Question

For a vector bundle, a connection is defined as a linear operator $\nabla:\Gamma(E)\rightarrow \Gamma(E\otimes T^*M)$, and the horizontal subspace is locally spanned by solutions of the equation $\nabla s_i=0$. Is there a similar way of viewing connections on projective bundles? If a projective bundle descends from a vector bundle, what is the relation between the connections? Is there a good source to learn this information from (there are many good sources for vector bundles and principal bundles, but I have not seen any good sources on projective bundles).

Answer 1

You can define a connection on any surjective submersion $f:Y\rightarrow X$ as a splitting of the exact sequence of vector bundles over $Y$

$0\rightarrow VY\rightarrow TY\xrightarrow{Tf} f^*TX\rightarrow 0$,

where the vertical subspace $VY$ is defined to be $VY:=\ker(Tf)$. However, the definition is really only useful when $Y$ has the structure of a fibre bundle.

When $Y$ is a vector bundle it is possible to identify $VY\cong f^*E$ and in this case the usual idea of a connection on a vector bundle follows. In particular, the definition of a connection you give as an associated covariant derivative. Likewise when $Y$ is a principal $G$-bundle it is possible to identify $VY\cong f^*ad(Y)$, and the usual idea of a connection on a principal bundle follows.

A connection on a projective bundle $f:Y\rightarrow X$ is therefore a splitting of the above exact sequence. I'm not sure if there is a nice identification of the vertical subspace in this case. Neither am I sure whether defining it as a covariant derivative is particularly fruitful in this case.

If $Y$ is the projective bundle associated to a vector bundle $E\rightarrow M$ then it is easy to see that any vector bundle connection $\nabla$ on $E$ induces a connection on $E_0=E-M$ and this will descend to one on $Y$ if and only if it is equivariant with respect to the action of scalar multiplication on $E$.

It may be useful to view a rank $n$ vector bundle $E$ as the associated bundle $P\times_{Gl_n(\mathbb{K})} \mathbb{K}^n$ to some principal $Gl_n(\mathbb{K})$-bundle $P$, and any vector bundle connection on $E$ as one induced from a principal connection on $P$. In this case we have the principal $PGl_n(\mathbb{K})$-bundle $Q=P/\Delta$, where $\Delta$ denotes scalar multiples of the identity in $Gl_n(\mathbb{K})$, and the projective bundle associated to $E$ is $PE=Q\times_{PGl_n(\mathbb{K})} \mathbb{K}P^n\cong P\times_{Gl_n(\mathbb{K})} \mathbb{K}P^n$. It follows that any principal connection on $Q$ will induce a connection on $PE$.

I would suggest Michor's book "Topics in Differential Geometry" for some good reading that goes well beyond my very limited knowledge.