\((1,1)\)-geodesic maps into Grassmann manifolds
Let \(M\) be a Kähler manifold, \(D\) be the Levi-Civita connection extended complex-linearly to the complexification of the tangent bundle, \(N\) be a Riemannian manifold, and \(\varphi:M\to N\) be a smooth map. The Hessian \(Dd\varphi\) may be decomposed to \((2,0)\), \((1,1)\), \((0,2)\) parts. The map \(\varphi\) is called pluriharmonic if its \((1,1)\) part is zero (the definition does not depend on the choice of the metric). A pluriharmonic isometric immersion is called \((1,1)\)-geodesic. If \(N\) is also Kähler then any holomorphic or antiholomorphic immersion is \((1,1)\)-geodesic. By [\textit{S. Udagawa}, Proc. Lond. Math. Soc., III. Ser. 57, No. 3, 577-598 (1988; Zbl 0667.53046)], if the complex dimension of \(M\) exceeds some constant depending only on \(N\) then any pluriharmonic immersion is either holomorphic or antiholomorphic. The authors construct examples of \((1,1)\)-geodesic immersions to complex Grassmann manifolds which are either holomorphic or antiholomorphic, in particular in the limit dimension for \(Gr(2,2+p)\). They are of the type \((\xi+\eta)^\perp\), where \(\xi\) is a holomorphic immersion and \(\eta\) is an antiholomorphic map to suitable Grassmannians. For real Grassmannians, it is proved that the normal Gauss map of a minimally immersed Kähler manifold or an extrinsic Hermitian symmetric space in an Euclidean space is \((1,1)\)-geodesic.
scientific article; zbMATH DE number 856569
scientific article