Lie groups with flat Gauduchon connections

Abstract

We pursuit the research line proposed in [24] about the classification of Hermitian manifolds whose s-Gauduchon connection ∇ s = (1 − s 2 )∇ c + s 2 ∇ b is flat, where s ∈ Ê and ∇ c and ∇ 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 leftinvariant compatible metric g and we assume that its connection ∇ s is flat. Our main result states that if either n=2 or there exits a ∇ s -parallel left invariant frame on G, then g must be Kähler. This result demonstrates rigidity properties of some complete Hermitian manifolds with ∇ s -flat Hermitian metrics.In the following given a real number s, we call ∇ s as the s-Gauduchon connection. Note that ∇ 0 = ∇ c and ∇ 2 = ∇ b . Moreover, ∇ 1 corresponds to the projection of the Levi-Civita connection onto the space of (1, 0)-vector fields. This connection is called the first canonical Hermitian connection in [8] and [13], the associated connection in [9] and the complexified Levi-Civita connection in [12]. In [24], it is proposed the study of compact Hermitian manifolds with flat s-Gauduchon connections, as a special case of a problem posed by Yau (Problem 87 in [26]). The case s = 0 (i.e. when the Chern connection is flat) was studied in the 1950s by W. Boothby [5] and H. C. Wang [20]. Boothby proved that if (M n , g) is a compact Chern-flat 2010 Mathematics Subject Classification. 53C55 (Primary).

On Strominger space forms

“…We will follow the notations of [33], [40], and [27]. Let g be the Lie algebra of G. Leftinvariant metric or complex structure on G corresponds to inner product or complex structure on on g. The latter means linear transformation J on g satisfying J 2 = −I and the integrability condition [x, y] − [Jx, Jy] + J[Jx, y] + J[x, Jy] = 0, ∀ x, y ∈ g.…”

“…, e n } be a unitary basis of g ′ . We will follow the notations in [33] (see also [36] and [40]), and denote by Denote by ϕ the coframe dual to e, then we have the structure equation…”

Complex nilmanifolds with constant holomorphic sectional curvature

“…For instance, there are examples of non-abelian group G where (G, J, g) is Kähler and flat, thus holomorphically isometric to the complex Euclidean space C n . See [5, Propsition 3.1] or [12,Appendix] for example. See also [9] for the characterization of Lie groups with flat left invariant metrics.…”

Complex nilmanifolds and Kähler-like connections

“…Let us start with a Lie-Hermitian manifold (G, J, g). We will follow the notations of [12]. Let e be a unitary frame of left invariant vector fields of type (1, 0) on G, with ϕ the dual coframe.…”