A note on Canonical Ricci forms on 2-step nilmanifolds

In this note we prove that any left-invariant almost Hermitian structure on a 2-step nilmanifold is Ricci-flat with respect to the Chern connection and that it is Ricci-flat with respect to another canonical connection if and only if it is cosymplectic.

Hence any B ∈ Ω 2 (T M ) can be written as B = B 2,0 + B 1,1 + B 0,2 . Notice that in terms of complex vector fields of type (1, 0) we have In particular the condition B 2,0 = 0 can be written in terms of (1, 0) vector fields as Now we consider connections on M . A connection ∇ on M is called Hermitian if ∇J = 0, ∇g = 0. It is well-known that every almost Hermitian manifold admits Hermitian connections. We denote by C the space of Hermitian connection on M . Gauduchon introduced in [13] the following special class of Hermitian connections: From [13] it follows that any canonical connection ∇ can be written as for some t ∈ R, where d c is the operator acting on r-forms as d c = (−1) r JdJ and N denotes the Nijenhuis For t ∈ R we denote by ∇ t the corresponding canonical connection. In the special case of a quasi-Kähler structure (i.e. ∂ω = 0) the space of canonical connections reduces to a single point, while if J is integrable (i.e. N = 0) equation (2.1) reduces to For the parameters t = 1, 0, −1, the family (2.1) gives the following remarkable cases • t = 1. In this case ∇ 1 is called the Chern connetion. This connection can be defined as the unique Hermitian connection satisfying T 1,1 = 0. • t = 0. In this case ∇ 0 is called the first canonical connection. This connection can be defined as the unique Hermitian connection whose torsion satisfies T 2,0 = 0.
• t = −1. In this case the connection ∇ −1 is important in the complex case where it is known as the Bismut connection. Indeed, if J is integrable, then ∇ −1 can be defined as the unique Hermitian connection having totally skew-symmetric torsion (see [6]).

Canonical Ricci forms
Let (M 2n , g, J) be an almost Hermitian manifold and let C the space of the associated Hermitian connections. For any ∇ ∈ C it is defined the Ricci form ρ(X, . Such a form is always closed and it locally satisfies ρ = dθ, where θ(X) = n r=1 g(∇ X Z r , Z r ) and {Z r } is a local unitary frame. In the case of a canonical connection ∇ t ∈ C we use notation ρ t and θ t . We denote by ♮ the natural isomorphism between vector fields and 1-forms induced by g. Namely, if X a vector field, then we denote by X ♮ the 1-form X ♮ (Y ) = g(X, Y ). We have the following for any vector field X.
Proof. First of all we note that if Z r is a vector field of type (1, 0), then Then we have So in order to prove the statement we have to show that We can write X = n r=1 (X r Z r + X r Z r ) and X ♮ = n r=1 (X r ζ r + X r ζ r ), where {ζ r } is the coframe dual to of {Z r }. Then we get where with B we denote the components of the brackets.
Corollary 3.2. The following formulae hold . It is useful to write down formula (3.1) in real coordinates. In order to do this we write Z r = A remarkable consequence of formula (3.1) is the following Corollary 3.3. All the canonical connection of a cosymplectic structure have the same Ricci form.
Proof. It is enough to show that θ 1 = θ −1 . Since the cosymplectic condition d * ω = 0 implies Now we observe that g(D Zr X 0,1 , Z r ) = − g(X 0,1 , D Zr Z r ) = 0, since the cosymplectic condition forces D Z r Z r to be of type (1, 0) (see e.g. [16]). The last step consists to show that ℑm g(D Zr X 0,1 , Z r ) = 0. Here it is enough to consider the identity n r=1 ℑm {Z r g(X, Z r )} + i ℑm g(D Z r X 0,1 , Z r ) which can be checked performing a direct computation. Then equation (3.2) implies the statement.
Remark 3.4. In the Hermitian case this last result was already known. In fact, it can be deduced from formula (8) of [14]. Another proof of this fact can be found in [17].

Canonical Ricci forms on Lie algebras
Now we restrict our attention to left-invariant almost Hermitian structures on Lie groups (or more generally on left-invariant almost Hermitian structures on quotient of Lie groups by lattices). Since here all the computations are purely algebraic, we may assume to work on a Lie algebra (g, [·, ·]) equipped with an almost Hermitian structure (g, J). An almost Hermitian structure on a Lie algebra is a pair (g, J), where J is an endomorphism of g satisfying J 2 = −Id and g is a J-Hermitian inner product. The bracket of g has not a priori any relation with J. The pair (g, J) induces as usual the fundamental form ω(·, ·) = g(J·, ·).
Proposition 3.1 implies the following Proposition 4.1. Let (g, [·, ·], g, J) be a Lie algebra with an almost Hermitian structure. For any t ∈ R the following formula holds Moreover if (g, [·, ·]) is unimodular (i.e. tr ad X = 0 for any X ∈ g); then and (g, J) is Ricci-flat with respect to any canonical connection if and only if it is cosymplectic.
An interesting consequence of this last proposition is the following Now we can prove Theorem 1 Proof of Theorem 1. Let (g, [·, ·], g, J) is a 2-step nilpotent Lie algebra with an almost Hermitian structure. Then, taking into account that g is unimodular, the 2-step condition implies that [g, g] is contained in the center of g and tr(ad [X,Y ] • J) = 0 for every X, Y ∈ g. Then formula (4.2) reduces to and first claim follows.
In particular in the unimodular case bi-invariant, anti-bi-invariant and anti-abelian almost Hermitian structures are Ricci-flat with respect to any canonical connection, while in the abelian case ρ t is given by the following formula and ρ t = 0 for t = 0 if and only if (g, J) is a cosymplectic structure.
Remark 4.3. We remark the following facts: • The bi-invariant condition [J·, ·] = J[·, ·] is equivalent to require that the simply-connected Lie group associated to (g, J) is a complex Lie group. The fact that a bi-invariant almost Hermitian structure on an unimodular Lie algebra is Ricci-flat with respect any canonical connection has been already proved by Grantcharov in [15]. • The anti-bi-invariant condition [J·, ·] = −J[·, ·] is equivalent to require that any J-compatible inner product on g is quasi-Kähler and flat with respect to the Chern connection ∇ 1 (see [9]). • The abelian condition [J·, J·] = [·, ·] was introduced in [4] and was intensely studied in [3,5,12,8,18]. This condition is equivalent to require that g 1,0 is an abelian Lie algebra. • Finally, the anti-abelian condition [J·, J·] = −[·, ·] was studied in [11].
Remark 4.4. Theorem 1 can be applied to the Heisenberg Lie algebras h n (R) and h n (C). That accords to Theorem 4.1 of [20] and Proposition 4.10 and 4.11 of [10]. Moreover things work differently either in the 3-step nilpotent case or in the 2-step solvable case (see [10]).