On orthogonal decompositions of hermitian Higgs bundles

A hermitian Higgs bundle is a triple $({\mathfrak E},h) = (E,\Phi, h)$, where ${\mathfrak E}=(E,\Phi)$ is a Higgs bundle and $(E,h)$ is a holomorphic hermitian vector bundle. It is well-known that several results on holomorphic vector bundles extend to the Higgs bundles setting, although this is not always the case. In this article we show that some classical propositions, involving orthogonal decompositions of holomorphic hermitian vector bundles and the second fundamental form of its holomorphic subbundles, can be extended to hermitian Higgs bundles. The extended propositions concerning orthogonal decompositions have immediate applications in Higgs bundles, and we mention some of these throughout the article. Moreover, the extended propositions concerning the second fundamental form are generalizations of previously known results on Higgs bundles. In particular, here we include alternative proofs of these extended propositions without using local computations. Finally, as an application of the above results and due to the lack of a certain parallelism condition, we show that a classical theorem concerning the Kobayashi functional for holomorphic vector bundles does not admit a straightforward extension to Higgs bundles.