ON CERTAIN CONDITIONS TO BE HARMONIC MAPS BETWEEN QUATERNIONIC KAEHLER MANIFOLDS

Abstract

Let (M, g), (N, h) be quaternionic Kaehler manifolds of dimensions 4m, 4n with almost Hermitian structures {I, J, K}, {F, G, H}, and local 1-forms {p￣, q￣, r￣}, {p, q, r}, respectively. Suppose that

φ_*I=Fφ_*, φ_*J=Gφ_*,

p￣=φ^*p, q￣=φ^*q, r￣=φ^*r.

If there exists an m-dimensional submanifold S of M such that the restriction of φ to S is harmonic, then the map φ : M → N is harmonic.

Let (M, g), (N, h) be quaternionic Kaehler manifolds of dimensions 4m, 4n with almost Hermitian structures {I, J, K}, {F, G, H}, and local 1-forms {p￣, q￣, r￣}, {p, q, r}, respectively. Suppose that

φ_*I=Fφ_*, φ_*J=Gφ_*,

p￣=φ^*p, q￣=φ^*q, r￣=φ^*r.

If there exists an m-dimensional submanifold S of M such that the restriction of φ to S is harmonic, then the map φ : M → N is harmonic.