Characterizations of Riemannian maps between Kaehler manifolds by certain curves (Q6607857)

Let \((M,g_M)\) and \((N,g_N)\) denote two Riemannian manifolds and \(F:(M,g_M) \to (N,g_N)\) a smooth map (of constant rank). The tangent bundle \(TM\) splits into the direct sum of the horizontal bundle (the union of \((\ker F_{*p})^\perp, p \in M\)) and the vertical bundle (the union of \(\ker F_{*p}, p \in M\)). The map \(F\) is said to be Riemannian if \(g_N(F_* X, F_* Y )=g_M(X,Y)\) for all horizontal vector fields \(X,Y\). As such, Riemannian maps generalize both isometric immersions and Riemannian submersions. In the paper under review, the authors consider Riemannian maps from two Kähler manifolds \((M,J)\) and \(\tilde{M},\tilde{J})\) and investigate their geometric properties as determined by the behaviour of special curves (namely, geodesics, circles and Frenet curves) under such maps. For example, they prove that if the image by a Riemannian map of a Kähler circle on \(M\) is a Kähler circle on \(M\), then the map is isotropic.