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Given a hermitian holomorphic vector bundle (E, h) on a complex manifold-with-a-metric (X,g), then consider the following (natural) construction of a metric on the total space of $\mathbb{P}(E)$ : Firstly, the metric $h$ induces a Fubini-study metric on the fibres of $\mathbb{P}(E)$ (just take local orthonormal frames to get a smooth fibre bundle, use the usual Fubini-study metric and then pullback to the holomorphic bundle). Secondly, the Chern connection of $h$ induces a splitting $T\mathbb{P}(E) = T\mathbb{C}\mathbb{P}^{r} \oplus TX$. Now put the direct sum metric. My question is: This seems like a very natural construction. Has this been studied before (in the sense of curvature properties etc)? If so, I'd be most grateful if a reference is pointed out.

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I have seen a description of this metric in Andrei Teleman's paper Families of holomorphic bundles. Commun. Contemp. Math. 10 (2008), no. 4, 523–551.

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