Let $(M,g)$ be a $2n$-dimensional almost Hermitian manifold with a almost complex structure $\cal J$ (not necessary integrable). i.e., $${\cal J}^2=-I,\quad\qquad g({\cal J} X,{\cal J} Y)=g(X,Y).$$ Suppose that $\{X_i,{\cal J}X_i\}$ be any local orthonormal ${\cal J}$-frame and the following relations hold for $i\neq j$ $$K(X_i,X_j)-K(X_j,{\cal J}X_j)=\frac{1}{2n-2}\{\rho(X_i,X_i)-\rho({\cal J}X_j,{\cal J}X_j)\}\tag{1}$$ $$K(X_i,X_j)-K(X_i,{\cal J}X_j)=\frac{1}{2n-2}\{\rho(X_j,X_j)-\rho({\cal J}X_j,{\cal J}X_j)\}\tag{2}$$ where $\rho$ and $K$ are Ricci curvature tensor and sectional curvature respectively. Then

Can be deduce that $(M,\cal J,g)$ is of constant curvature?

This question comes from the study of conformally flat almost Hermitian manifolds.

Your suggestions will be appreciated.

Let $(M,g)$ be a $2n$-dimensional almost Hermitian manifold ($n\geq 2$) with a almost complex structure $\cal J$ (not necessary integrable). i.e., $${\cal J}^2=-I,\quad\qquad g({\cal J} X,{\cal J} Y)=g(X,Y).$$ Suppose that $\{X_i,{\cal J}X_i\}$ be any local orthonormal ${\cal J}$-frame and the following relations hold for $i\neq j$ $$K(X_i,X_j)-K(X_j,{\cal J}X_j)=\frac{1}{2n-2}\{\rho(X_i,X_i)-\rho({\cal J}X_j,{\cal J}X_j)\}\tag{1}$$ $$K(X_i,X_j)-K(X_i,{\cal J}X_j)=\frac{1}{2n-2}\{\rho(X_j,X_j)-\rho({\cal J}X_j,{\cal J}X_j)\}\tag{2}$$ where $\rho$ and $K$ are Ricci curvature tensor and sectional curvature respectively. Then

Can be deduce that $(M,\cal J,g)$ is of constant curvature?

This question comes from the study of conformally flat almost Hermitian manifolds.

Your suggestions will be appreciated.