Chern flat, Chern–Ricci flat twisted product in almost Hermitian geometry

A.1 The Chern connection

An almost complex structure on M is an endomorphism J of TM, J ∈ Γ ⁢ ( End ⁢ ( T ⁢ M ) ) , satisfying J 2 = - Id T ⁢ M , where TM is the real tangent vector bundle of M. The pair ( M , J ) is called an almost complex manifold. Let ( M , J ) be an almost complex manifold. A Riemannian metric G on M is called J-invariant if J is compatible with G. In this case, the pair ( J , G ) is called an almost Hermitian structure. The complexified tangent vector bundle is given by T ℂ ⁢ M = T ⁢ M ⊗ ℝ ℂ for the real tangent vector bundle TM. By extending J ℂ -linearly and G ℂ -bilinearly to T ℂ ⁢ M , they are also defined on T ℂ ⁢ M and we observe that the complexified tangent vector bundle T ℂ ⁢ M can be decomposed as T ℂ ⁢ M = T 1 , 0 ⁢ M ⊕ T 0 , 1 ⁢ M , where T 1 , 0 ⁢ M , T 0 , 1 ⁢ M are the eigenspaces of J corresponding to eigenvalues - 1 and - - 1 , respectively:

Let Λ 1 ⁢ M denote the dual of the real tangent vector bundle TM. We have that

where

It can be seen that ( T 1 , 0 ⁢ M ) * = Λ 1 , 0 ⁢ M , ( T 0 , 1 ⁢ M ) * = Λ 0 , 1 ⁢ M . Now let us define

Then we have Λ r ⁢ M ⊗ ℝ ℂ = ⊕ p + q = r Λ p , q ⁢ M .

Notice that on an almost complex manifold M, we can split the exterior differential operator

into four components

with

For any p-form ψ, there holds that

(A.1)

for any vector fields X 1 , … , X p + 1 ∈ Γ ⁢ ( T ℂ ⁢ M ) (cf. [11]).

Let ( M , J , G ) be an almost Hermitian manifold. There exists a unique affine connection ∇ preserving J and G on M whose torsion has vanishing ( 1 , 1 ) -part (cf. [3]), which is called the Chern connection. Now let ∇ be the Chern connection on M.

Let { e σ } be a local ( 1 , 0 ) -frame with respect to an almost Hermitian metric G and let { θ σ } be a local associated coframe with respect to { e σ } , i.e., θ γ ⁢ ( e σ ) = δ σ γ for σ , γ = 1 , … , n . We denote the structure coefficients of Lie bracket by

Notice that J is integrable if and only if the B σ ⁢ γ α ¯ ’s vanish.

A.2 The curvature on almost complex manifolds

Since the Chern connection ∇ preserves J, we have

where

Note that the mixed derivatives ∇ σ ⁡ e γ ¯ do not depend on the metric G, which means that Γ σ ⁢ γ ¯ α ¯ C = B σ ⁢ γ ¯ α ¯ do not depend on G (cf. [7]). On the other hand, the Levi-Civita connection D is determined by

Let { Ξ σ γ } be the connection form, which is defined by Ξ σ γ = Γ δ ⁢ σ γ C ⁢ θ δ + Γ δ ¯ ⁢ σ γ C ⁢ θ δ ¯ . The torsion T C of the Chern connection ∇ is given by T σ C = d ⁢ θ σ - θ β ∧ γ β σ , T σ ¯ C = d ⁢ θ σ ¯ - θ β ¯ ∧ γ β ¯ σ ¯ , which has no ( 1 , 1 ) -part and the only non-vanishing components are as follows:

and

These tell us that T C = ( T α C ) splits into T C = T ′ C + T ′′ C , where T ′ C ∈ Γ ⁢ ( Λ 2 , 0 ⁢ M ⊗ T 1 , 0 ⁢ M ) , T ′′ C ∈ Γ ⁢ ( Λ 0 , 2 ⁢ M ⊗ T 1 , 0 ⁢ M ) . Since the torsion T C of the Chern connection ∇ has no ( 1 , 1 ) -part

we obtain that

By taking conjugate, we have that

From (A.1), a direct computation yields for the associated smooth real ( 1 , 1 ) -form Φ with respect to G,

which implies that the quasi-Kählerity condition ∂ ¯ ⁢ Φ = 0 is equivalent to T ′ C ≡ 0 .

The curvature Ω C of the Chern connection ∇ splits in Ω C = H C + R C + H ¯ C and the curvature form can be expressed by Ω σ γ C = d ⁢ Ξ σ γ + Ξ δ γ ∧ Ξ σ δ .

In terms of e σ , we have

We define that

Let P C and S C denote the first and second Chern–Ricci curvature respectively locally given by

We define the Chern scalar curvature s G C of the almost Hermitian metric G with respect to the Chern connection ∇ :