常曲率流形中2-调和等距浸入与k-调和映照的关系

RELATION BETWEEN 2-HARMONIC ISOMETRIC IMMERSION AND k-HARMONIC MAPPING IN A RIMANN MANIFOLD WITH CONSTANT CURVATURE

设M和N是两个Riemann流形,如果映照f:M→N是能量泛函E_K(f)=1/2∫_M‖(d+d~*)~Kf‖~2*1的临界点,则称f为k-调和映照。本文讨论了2-调和等距浸入与K-映照之间的关系,获得了如下定理:设f:M→N是Riemann流形间的2-调和等距浸入,且M紧致,N具有常截面曲率,则f是k-调和映照(k≠2)当且仅当M是极小的。

Let M and N be two Riemannian manifolds, a mapping f: M→N is calleda k-harmonic mapping if it is a critical point of the energy intergral E_h (f)where E_h (f) =1/2 ∫_M‖(d+d~*)~kf‖~2*1 We discuss the relation between 2-harmonic isometric immersion andk- harmonic mapping, and obtain the following Theorem Let f: M→N be a 2--harmonic isometric immersion between twoRiemannian manifolds M and N, if M is compact and N has constant sectionalcurvture, then f is a k-harmonic mapping (k≠2 ) iff M is minimal.