Harmonic sections of homogeneous fibre bundles. (Q1410599)

Let \(\pi : N \rightarrow M\) be a smooth reductive homogeneous fibre bundle equipped with a Kaluza-Klein geometric structure. Let \(\sigma : M \rightarrow N\) be a smooth section. One defines the super-curvature of \(\sigma\) to be the tensor field \(\alpha^v\sigma = \nabla^v \text{ d}^v \sigma\) and the vertical tension field of \(\sigma\) to be \(\tau^v \sigma = \text{ Tr} \nabla^v \text{ d}^v \sigma\), where \(\text{ d}^v\) is the vertical component of the exterior differential, and \(\nabla^v\) is the vertical component of the Levi-Civita connection of \(N\). The section \(\sigma\) is defined to be harmonic if \(\tau^v\sigma=0\) and is super-flat if \(\alpha^v \sigma = 0\). This paper establishes the following sequence of implications for sections of \(N\): \[ \text{super-flat} \Rightarrow \text{ totally\;geodesic} \Rightarrow \text{ harmonic\;map} \Rightarrow \text{ harmonic\;section}. \] For flat bundles, or more generally flat sections, there are the equivalencies: super-flat \(\Leftrightarrow\) totally geodesic, and harmonic map \(\Leftrightarrow\) harmonic section. The author constructs examples to show these can fail to be equivalent for nonflat bundles.