Structures Which are Harmonic with Respect to Walker Metrics

Abstract

Let $${(W, q, \mathcal{D})}$$ be a Walker manifold. We find all Walker metrics which are harmonic [in the sense of Chen and Nagano in (J Math Soc Jpn 36:295–313, 1984)] w.r.t. q. On the total space of the tangent bundle of W, we obtain necessary and sufficient conditions concerning the harmonicity of certain metrics w.r.t. the Sasaki (resp. horizontal, vertical) lift of q. The harmonicity in the sense of Garcia-Rio et al. in (Ill J Math 41(1):23–30, 1997) of the three endomorphism fields of an almost hyper-para-Hermitian structure is characterized. As an application, we deal with mixed 3-structures, a notion for which we quote Ianus in (Mediterr J Math 3(3–4):581–592, 2006).