Harmonic complex structures and special Hermitian metrics on products of Sasakian manifolds

Abstract

It is well-known that the product of two Sasakian manifolds carries a 2-parameter family of Hermitian structures (J a,b , g a,b ). We show in this article that the complex structure J a,b is harmonic with respect to g a,b , i.e. it is a critical point of the energy functional. Furthermore, we also determine when these Hermitian structures are locally conformally Kähler, balanced, strong Kähler with torsion, Gauduchon or k-Gauduchon (k ≥ 2). Finally, we provide an expression for the Bismut connection associated to (J a,b , g a,b ) in terms of the characteristic connections on each Sasakian factor.

Harmonic almost complex structures on almost abelian Lie groups and solvmanifolds

“…More recently, He and Li introduced in [22] the harmonic heat flow for almost complex structures compatible with a fixed Riemannian metric, which is a tensor-valued version of the harmonic map heat equation first studied by Eells-Sampson [12]. Generalizing the case of Calabi-Eckmann manifolds we showed in [5] that in a 2-parameter family of Hermitian structures (J a,b , g a,b ) on a product of Sasakian manifolds, the integrable almost complex structure J a,b is harmonic with respect to g a,b .…”

“…Theorem 4.2. Let (J, • , • ) be an almost Hermitian structure on a 2n-dimensional almost abelian Lie algebra g = R ⋉ L R 2n−1 , with L given as in (5) and n ≥ 3. Then we have the following relations between the Gray-Hervella classes:…”

“…Left-invariant SKT structures on almost abelian Lie groups have been studied in [6], where the following result is proved 3 : Theorem 5.13. [6, Theorem 4.6] Let (J, • , • ) be a Hermitian structure on a 2n-dimensional almost abelian Lie algebra g = Re 0 ⋉ L R 2n−1 , where L is given by (5).…”

Complex structures on the product of two Sasakian manifolds

“…This construction was further generalized in [11] to the product of two Sasakian manifolds. It was recently considered also in [1].…”

“…All these constructions use the tensorial definition of Sasakian manifolds and are heavily computational. With these techniques, the authors of [1] can prove that the considered two-parameter family of Hermitian structures is neither Kähler nor locally conformally Kähler.…”