The Hessian of a harmonic reduction

In this work, following ideas of Eells-Lemaire, we define the notion of Hessian of a section σ ∈ Г(M, P/H). We obtain an explicit formula for the Hessian and in the case that σ is a harmonic reduction, we show the Hessian is symmetric. The notion of stability for harmonic reductions is also introduced.