Harmonic maps between two second order holomorphic jets bundle

Violeta Zaluţchi

In the geometry of the holomorphic jets bundle of order two $J^{(2,0)}M$, we have studied a special linear connection, named the Chern-Lagrange connection and, with respect to this, the geodesic curves are characterized in \cite{Za1,Za2,Za3}. In the present paper, we consider a holomorphic function $f:M\rightarrow N$ between two complex manifolds, which carries the curves from $J^{(2,0)}M$ into curves on $J^{(2,0)}N$, and we find when this mapping is harmonic. We prove that if a curve on $J^{(2,0)}M$ is a geodesic and $f$ defines a harmonic map between $J^{(2,0)}M$ and $J^{(2,0)}N$, then the $f$-image of this curve on $J^{(2,0)}N$ is a geodesic too.