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 Dirichlet energy functional. Furthermore, we also determine when these Hermitian structures are locally conformally Kähler, balanced, strong Kähler with torsion, Gauduchon or k-Gauduchon ($k\ge 2$). Finally, we study the Bismut connection associated to $(J_{a,b}, g_{a,b})$ and we provide formulas for the Bismut-Ricci tensor ${\text {Ric}}^B$ and the Bismut-Ricci form $\rho ^B$. We show that these tensors vanish if and only if each Sasakian factor is $\eta$-Einstein with appropriate constants and we also exhibit some examples fulfilling these conditions, thus providing new examples of Calabi-Yau with torsion manifolds.

Appendix

As mentioned before, we provide here a proof of Proposition 4.1, for the sake of completeness.

Proof of Proposition 4.1

Let $\{e_1,\ldots ,e_{2n}\}$ be an orthonormal local frame satisfying $Je_{2i-1}=e_{2i}$ for $1\le i\le n$. Using this frame, we compute by definition both sides of the equality we want to prove. For $X\in \mathfrak {X}(M)$, using that $J(\nabla _U J)=-(\nabla _U J)J$ for all $U\in \mathfrak {X}(M)$, we have

$$\begin{aligned}{}[J,\nabla ^*\nabla J](X)&=\sum _{i=1}^{2n} \underbrace{J(\nabla _{e_i} (\nabla _{e_i} J))(X)}_{\textcircled {1}}-\underbrace{(\nabla _{e_i} (\nabla _{e_i} J))(JX)}_{\textcircled {2}}-2J(\nabla _{\nabla _{e_i} e_i} J)(X), \end{aligned}$$

Recall that the integrability of J is equivalent to $\nabla _{JU} J=J(\nabla _U J)$ for any $U\in \mathfrak {X}(M)$. Using this fact when writing $(\nabla _{e_i} J)=-(\nabla _{J^2 e_i} J)$, we obtain

$$\begin{aligned} \textcircled {1}&=-J(\nabla _{e_i} J(\nabla _{Je_i} J))(X)\\&=-J\nabla _{e_i} (J(\nabla _{Je_i} J)X)-(\nabla _{Je_i} J)(\nabla _{e_i} X)\\&=-J\nabla _{e_i} J\nabla _{Je_i} JX-J\nabla _{e_i} \nabla _{Je_i} X-\nabla _{Je_i} J\nabla _{e_i} X+J\nabla _{Je_i} \nabla _{e_i} X\\&=-J\nabla _{e_i} J\nabla _{Je_i} JX-\nabla _{Je_i} J\nabla _{e_i} X-J R(e_i,Je_i)X-J\nabla _{[e_i,Je_i]} X, \end{aligned}$$

$$\begin{aligned} \textcircled {2}&=-(\nabla _{e_i} J(\nabla _{Je_i} J))(JX)\\&=-\nabla _{e_i} (J(\nabla _{Je_i} J)(JX))+J(\nabla _{Je_i} J)(\nabla _{e_i} JX)\\&=\nabla _{e_i} J \nabla _{Je_i} X-\nabla _{e_i} \nabla _{Je_i} JX+J\nabla _{Je_i} J\nabla _{e_i} JX+\nabla _{Je_i} \nabla _{e_i} JX\\&=\nabla _{e_i} J \nabla _{Je_i} X+J\nabla _{Je_i} J\nabla _{e_i} JX-R(e_i,Je_i)JX-\nabla _{[e_i,Je_i]} JX. \end{aligned}$$

Hence,

$$\begin{aligned}{}[J,\nabla ^* \nabla J](X)&=-2[J,P](X)+\sum _{i=1}^{2n} (\nabla _{[e_i,Je_i]} J)X-2\sum _{i=1}^{2n} J(\nabla _{\nabla _{e_i} e_i} J)(X)\\&\quad -\sum _{i=1}^{2n} (J\nabla _{e_i} J\nabla _{Je_i} JX+\nabla _{Je_i} J\nabla _{e_i} X+\nabla _{e_i} J \nabla _{Je_i} X+J\nabla _{Je_i} J\nabla _{e_i} JX).\end{aligned}$$

Note that in the chosen J-adapted frame $\{e_i\}$ the last sum equals zero, since replacing $e_i$ by $Je_i$ gives the same terms with opposite sign. Thus,

$$\begin{aligned}{}[J,\nabla ^* \nabla J](X) =-2[J,P](X)+\sum _{i=1}^{2n} (\nabla _{[e_i,Je_i]} J)X-2\sum _{i=1}^{2n} J(\nabla _{\nabla _{e_i} e_i} J)(X). \end{aligned}$$

(7.1)

Now, using (4.2) with our J-adapted frame, we get

$$\begin{aligned} 2(\nabla _{\delta J} J)(X)&=2\sum _{i=1}^{2n} (\nabla _{(\nabla _{e_i} J)e_i} J)(X)\\&=2\sum _{i=1}^{2n} (\nabla _{\nabla _{e_i} Je_i} J)(X)-2\sum _{i=1}^{2n} (\nabla _{J \nabla _{e_i} e_i} J)(X)\\&=\sum _{i=1}^{2n} (\nabla _{\nabla _{e_i} Je_i} J+\nabla _{\nabla _{Je_i} e_i} J)(X)+\sum _{i=1}^{2n} (\nabla _{[e_i, Je_i]} J)(X)\\&\quad -2\sum _{i=1}^{2n} J(\nabla _{ \nabla _{e_i} e_i} J)(X). \end{aligned}$$

Replacing $e_i$ by $Je_i$ in the first sum we obtain the same terms with opposite sign, and thus this sum equals zero. Therefore,

$$\begin{aligned} 2(\nabla _{\delta J} J)(X) = \sum _{i=1}^{2n} (\nabla _{[e_i, Je_i]} J)(X)-2\sum _{i=1}^{2n} J(\nabla _{ \nabla _{e_i} e_i} J)(X). \end{aligned}$$

(7.2)

Comparing (7.1) with (7.2) we obtain $[J,\nabla ^* \nabla J]=2\nabla _{\delta J}J-2[J,P]$, as we wanted to prove. $\square$