Harmonic maps on contact metric manifolds
The authors initiate a study of harmonic maps between almost contact metric manifolds. Let \(M(\varphi, \xi, \eta, g)\) and \(M'(\varphi', \xi', \eta', g')\) be two almost contact metric manifolds [cf. \textit{D. E. Blair}, Contact manifolds in Riemannian geometry, Lect. Notes Math. 509, Springer (1976; Zbl 0319.53026)] and let \(f : M \to M'\) be a smooth map. \(f\) is said to be \(+\varphi\)-holomorphic if \(f_* \circ \varphi = \varphi \circ f_*\), and \(-\varphi\)-holomorphic if \(f_* \circ \varphi = - \varphi \circ f_*\). By an analogy to the Kählerian case, \(f\) (here, it is sufficient for \(M'\) to be Riemannian) is called \(\varphi\)-pluriharmonic if its second fundamental form \(\alpha\) satisfies the condition (*) \(\alpha (X,Y) + \alpha(\varphi X, \varphi Y) = 0\) for any vector fields \(X\), \(Y\) on \(M\), and \(f\) is called \(D\)-pluriharmonic if (*) holds for any \(X,Y \in D = \text{ker }\eta\). It is proved that if \(M\) and \(M'\) are contact metric, then any \(\pm \varphi\)-holomorphic map \(f\) is harmonic. Any \(\varphi\)-pluriharmonic map \(f\) is harmonic. When \(M\), \(M'\) are Sasakian manifolds, then a \(\pm \varphi\)-pluriharmonic map \(f\) is \(D\)-pluriharmonic; and \(f\) is \(\varphi\)-pluriharmonic if and only if it is isometric. Harmonic maps on globally \(\varphi\)-symmetric Sasakian manifolds of constant \(\varphi\)-sectional curvature are also studied; furthermore, certain examples on principal circle bundles are discussed.
scientific article; zbMATH DE number 871700
scientific article