Lie groups with flat Gauduchon connections

Abstract

We pursue the research line proposed in Yang and Zheng (Acta. Math. Sinica (English Series), 34(8):1259–1268, 2018) about the classification of Hermitian manifolds whose s-Gauduchon connection $\nabla ^s =(1-\frac{s}{2})\nabla ^c + \frac{s}{2}\nabla ^b$ is flat, where $s\in {\mathbb {R}}$ and $\nabla ^c$ and $\nabla ^b$ are the Chern and the Bismut connections, respectively. We focus on Lie groups equipped with a left invariant Hermitian structure. Such spaces provide an important class of Hermitian manifolds in various contexts and are often a valuable vehicle for testing new phenomena in complex and Hermitian geometry. More precisely, we consider a connected 2n-dimensional Lie group G equipped with a left-invariant complex structure J and a left-invariant compatible metric g and we assume that its connection $\nabla ^s$ is flat. Our main result states that if either n=2 or there exits a $\nabla ^s$-parallel left invariant frame on G, then g must be Kähler. This result demonstrates rigidity properties of some complete Hermitian manifolds with $\nabla ^s$-flat Hermitian metrics.

Appendix A: Lie groups with a flat left-invariant Kähler structure

In this section we recall some results on flat left-invariant Lie groups due to Milnor [14] and Barberis-Dotti-Fino [3]. These results lead to Corollary 1.6 stated in Sect. 1.

Theorem A.1

[14, Theorem 1.5 on p. 298] A connected Lie group G with a left invariant Riemannian metric is flat if and only if its Lie algebra $\mathfrak {g}$ admits an orthogonal decomposition

$$\begin{aligned} \mathfrak {g}=\mathfrak {h} \oplus \mathfrak {i}, \end{aligned}$$

(35)

where $\mathfrak {h}$ is an abelian Lie subalgebra and $\mathfrak {i}$ is an abelian ideal. Moreover, ${\text {ad}}(X)$ is skew-adjoint for any $X \in \mathfrak {h}$.

In the above theorem, the abelian ideal $\mathfrak {i}$ is $\mathfrak {i}=\{Y \in \mathfrak {g}\ |\ \nabla _{Y}=0 \}$, where $\nabla$ is the Levi-Civita connection. It is further observed (Proposition 2.1 in [3]) that $\mathfrak {i}=\mathfrak {c} \oplus [\mathfrak {g}, \mathfrak {g}]$ where $\mathfrak {c}$ is the center of $\mathfrak {g}$ and $[\mathfrak {g}, \mathfrak {g}]$ is of even dimension.

Lie groups with a flat left-invariant Kähler structure were classified in Proposition 2.1 and Corollary 2.2 in [3]. We summarize and slightly rephrase their results as follows.

Theorem A.2

[3, Proposition 2.1, Corollary 2.2, and Proposition 3.1] Let (G, J, g) be connected Lie group with a left invariant complex structure and a flat left invariant Kähler metric. Then the Lie algebra $\mathfrak {g}$ of G admits an orthogonal decomposition

$$\begin{aligned} \mathfrak {g}=\mathfrak {h} \oplus \mathfrak {c} \oplus [\mathfrak {g}, \mathfrak {g}], \end{aligned}$$

(36)

which satisfies the following properties:

• (i) $\mathfrak {h}$ is a Lie subalgebra, $\mathfrak {c}$ is the center of $\mathfrak {g}$, and $[\mathfrak {g}, \mathfrak {g}]$ is J-invariant.

• (ii) The adjoint action ${\text {ad}}$ injects $\mathfrak {h}$ to skew-symmetric transformations on $[\mathfrak {g}, \mathfrak {g}]$ which commutes with J. Furthermore, we have the ‘simultaneous diagonalization’ of the restriction of ${\text {ad}}_{X}=\nabla _X$ on $\mathfrak {h}$ in the following sense:

  There exists an orthonormal basis with $[\mathfrak {g}, \mathfrak {g}]={\text {span}} \{e_1, Je_1, \ldots , e_p, Je_p\}$ and an injective linear map $q=(q_1, \ldots , q_p): \mathfrak {h} \rightarrow \mathbb {R}^p$ such that for any $X \in \mathfrak {h}$ and $1 \le i \le p$, ${\text {ad}}_{X}$ has the block diagonal form:

  $$\begin{aligned} ad_X \left[ \begin{array}{cc} e_i \\ Je_i \end{array} \right] = \left[ \begin{array}{cc} 0 &{} q_i(X) \\ -q_i (X) &{} 0 \end{array} \right] \left[ \begin{array}{cc} e_i \\ Je_i \end{array} \right] . \end{aligned}$$

• (iii) There are further orthogonal decompositions $\mathfrak {h}=(\mathfrak {h} \cap J\mathfrak {h}) \oplus \mathfrak {h}_1$ and $\mathfrak {c}=(\mathfrak {c} \cap J\mathfrak {c}) \oplus \mathfrak {c}_1$, and J restricts to an isomorphism between $\mathfrak {h}_1$ and $\mathfrak {c}_1$.

It is straightforward to write down all such Lie algebras in real dimension 4, which is exactly covered in Corollary 1.6.

It might be interesting to compare the kernel space $\{Y \in \mathfrak {g}\ |\ \nabla _{Y}=0 \}$ in Theorem A.2 with the Lie algebra homomorphism $p: {\mathfrak {g}} \rightarrow {\mathfrak {u}}(n)$ defined in (29) in Sect. 4. Let $\nabla =\nabla ^{s}$, then our Theorem 1.3 can be reinterpreted as: for a $\nabla ^{s}$-flat Hermitian Lie group (G, J, g) with a left invariant structure, if the kernel space $\{Y \in \mathfrak {g}\ |\ \nabla _{Y}=0 \}=\mathfrak {g}$, then (G, J, g) is Kähler and isomorphic to a complex vector group. We will study the structure of a $\nabla ^{s}$-flat left invariant Hermitian Lie group in the spirit of Theorem A.2 in a future work.