---
title: Gauduchon Connections in Hermitian Geometry
url: https://www.emergentmind.com/topics/gauduchon-connections
type: topic
---

# Gauduchon Connections in Hermitian Geometry

A Gauduchon connection is a distinguished one-parameter family of Hermitian connections $\nabla^s$ on a complex manifold with Hermitian metric, interpolating between the Chern and the Bismut (Strominger) connections. These connections play a central role in complex geometry, especially in the study of special Hermitian metrics and their curvature properties on non-Kähler manifolds. The structure and classification of manifolds admitting flat or Kähler-like Gauduchon connections has been an area of active research, with deep links to questions in differential geometry, representation theory, and complex analysis.

## 1. Definition and Fundamental Structure

Let $(M^n, g, J)$ be a Hermitian manifold of complex dimension $n$; $J$ is the complex structure and $g$ the Hermitian metric. The Levi–Civita connection $\nabla$ of $g$ need not preserve $J$, so the Hermitian connections—connections compatible with both $g$ and $J$—form an affine space. Two prominent canonical Hermitian connections are:

- The **Chern connection** $\nabla^c$, uniquely determined by
  - $\nabla^c g = 0$,
  - $\nabla^c J = 0$,
  - Torsion $T^c$ of type $(2,0)$.

- The **Bismut (Strominger) connection** $\nabla^b$, defined by
  - $\nabla^b g = 0$,
  - $\nabla^b J = 0$,
  - Torsion $T^b$ totally skew-symmetric; $T^b(X,Y,Z) = d\omega(JX, JY, JZ)$ for fundamental form $\omega(X,Y) = g(JX, Y)$.

The **Gauduchon connections** form a real one-parameter family interpolating between $\nabla^c$ and $\nabla^b$:
\[
\nabla^s = (1 - \tfrac{s}{2}) \nabla^c + \tfrac{s}{2} \nabla^b, \qquad s \in \mathbb{R}
\]
Special cases:
- $s = 0$: Chern connection,
- $s = 2$: Bismut connection,
- $s = 1$: first canonical (Lichnerowicz) Hermitian connection.

Each $\nabla^s$ is metric and complex compatible, with torsion
\[
T^s = (1 - \tfrac{s}{2}) T^c + \tfrac{s}{2} T^b
\]
and possesses explicit local expressions in a unitary frame [1709.02530][2204.08170][2211.05973].

## 2. Curvature, Torsion, and Kähler-like Properties

For any Hermitian connection $\nabla^s$, the curvature tensor is defined in the standard way:
\[
R^s(X, Y) Z = \nabla^s_X \nabla^s_Y Z - \nabla^s_Y \nabla^s_X Z - \nabla^s_{[X, Y]} Z
\]
However, in general, $\nabla^s$ is not torsion-free, and $R^s$ does not satisfy the Riemannian first Bianchi identity except in special geometric circumstances.

A connection $\nabla^s$ is said to be **Kähler-like** if its curvature satisfies both the first Bianchi identity and the complex symmetries characteristic of Kähler or Chern curvature:
\[
R^s(x,y,z,w) + R^s(y,z,x,w) + R^s(z,x,y,w) = 0
\quad\text{and}\quad
R^s(Jx,Jy,z,w) = R^s(x,y,z,w)
\]
If $\nabla^s$ is Kähler-like or even flat ($R^s \equiv 0$), strong restrictions on the underlying metric and manifold structure ensue [1809.02632][2108.08181][2204.08170].

## 3. Rigidity and Classification Results

### Compact Manifolds

A central development is the **rigidity** of compact Hermitian manifolds with flat Gauduchon connections. Precisely, apart from the two extremal cases $s=0$ (Chern) and $s=2$ (Bismut), any compact Hermitian manifold with flat $\nabla^s$ is necessarily Kähler:
- **Boothby-Wang**: Compact Chern-flat Hermitian manifolds are finite quotients of complex Lie groups with left-invariant Hermitian metrics [1709.02530].
- **Wang–Yang–Zheng (Bismut-flat case)**: Compact Bismut-flat Hermitian manifolds have universal covers of the form $G \times \mathbb{R}^k$, $G$ a simply-connected compact semisimple Lie group with bi-invariant metric and left-invariant complex structure [1709.02530].
  
These claims are formalized:
\[
\text{If } s \notin \{0,2\} \text{ and }\nabla^s\text{ is flat on compact }(M^n,g),\text{ then }g\text{ is Kähler}
\]
This is established via integral identities involving the torsion $T$ and its trace $n$, leading to a universal Kähler-rigidity outside the interval $s \in [4-2\sqrt{3},\,4+2\sqrt{3}]$ [1709.02530][2204.08170].

### Surfaces and Dimension-Dependence

For compact Hermitian surfaces $(n=2)$, the only non-Kähler, non-flat examples with flat $\nabla^s$ arise for $s=2$ (Bismut), realized on isosceles Hopf surfaces. For all other $s$, the metric is flat Kähler (e.g., complex tori, hyperelliptic surfaces) [1709.02530][2201.13083].

When the Gauduchon connection is Kähler-like (not necessarily flat), similar rigidity results hold. For $s \notin \{0,2\}$, any compact Hermitian manifold with Kähler-like $\nabla^s$ is Kähler [2204.08170][2108.08181]; this confirms conjectures by Angella, Otal, Ugarte, and Villacampa, as well as Yang and Zheng.

## 4. Holomorphic Sectional Curvature and Cohomological Aspects

The holomorphic sectional curvature (HSC) for $\nabla^s$ is defined as
\[
H^s(v) = \frac{R^s(v, \bar v, v, \bar v)}{|v|^4}
\]
where $v$ is a nonzero (1,0)-vector. A monotonicity principle holds:
\[
H^s(v) \leq H^c(v) \quad \forall v; \quad H^s(v) = H^c(v) \text{ iff } s=1 \text{ or } g \text{ is Kähler}
\]
If a compact Hermitian surface has pointwise constant $H^s$, then either the metric is Kähler, or the manifold is an isosceles Hopf surface with admissible metric, realized at $s=-1$ or $s=3$ (the Bismut or minimal connection) [2201.13083][2211.05973].

Cohomologically, the first Ricci form of $\nabla^s$ is a deformation of the Chern-Ricci form:
\[
\operatorname{Ric}^{s,(1)} = \operatorname{Ric}^{c,(1)} + (s-1) \big(\partial \bar\partial^* \omega + \bar\partial \partial^* \omega\big)
\]
yielding $t$-Ricci-flat Hermitian metrics on non-Kähler Calabi–Yau type manifolds [2211.05973].

## 5. Realizations on Lie Groups and Homogeneous Spaces

For a Lie group $G$ with left-invariant (almost) Hermitian structure, left-invariant Hermitian connections correspond to elements in the space
\[
\wedge^{1,1} \mathfrak{g}^* \otimes \mathfrak{g}
\]
The Gauduchon family of connections retains this identification, and explicit formulas for torsion and curvature are available in terms of Lie algebra structure constants. On $G=H \times A$ (with $H$ compact, $A$ abelian, and $J$ totally real on $H$), the Bismut connection may coincide with the trivial connection, SKT condition can be realized, and further flat Gauduchon connections can exist at special parameter values [2308.00126][1805.04719].

In the left-invariant context, classification results mirror the compact case. For real dimension 4, or in the existence of a $\nabla^s$-parallel invariant frame, left-invariant Hermitian metrics with flat $\nabla^s$ force $g$ to be Kähler for $s \neq 0,2$ [1805.04719].

## 6. Methodologies and Invariant Identities

Key technical tools involve structure equations under local $\nabla^s$-parallel unitary frames and pointwise or integral identities relating torsion and its trace to curvature. The integral identity
\[
(8s - s^2 - 4)|n|^2 = s^2 |T|^2
\]
plays a central role, especially upon integration over compact manifolds. In dimension two, Bochner-type arguments on logarithmic functions of torsion components complete rigidity proofs [1709.02530].

Beyond flatness, the concept of Kähler-likeness (algebraic curvature identities) allows application of maximum principles and the trace constraints to extend rigidity beyond the cases accessible to direct curvature vanishing [2204.08170][2108.08181].

## 7. Dualities, Open Problems, and Research Directions

A duality phenomenon among Gauduchon connections is encoded by the transformation $r \mapsto r/(2r - 1)$; self-dual points are $r=0$ (Lichnerowicz) and $r=1$ (Chern). It is shown that two Gauduchon connections can have equal holomorphic sectional curvature if and only if $s = t$, $s + t = 2$, or the metric is Kähler [2211.05973].

Open research problems include:
- Complete classification of noncompact Hermitian manifolds with flat $\nabla^s$ outside the compact setting,
- Clarification of geometric and physical origins of the duality among the Gauduchon parameters,
- Further investigation of special Hermitian connections in relation to pluriclosed flow, SKT metrics, and moduli of canonical connections [1709.02530][2108.08181].

The growing taxonomy and structure theory of Gauduchon connections elucidate the internal organization of Hermitian geometry, bridging the complexities of Kähler and non-Kähler manifolds, and deepening the interface with Lie theory, cohomology, and geometric analysis.

Source: https://www.emergentmind.com/topics/gauduchon-connections