Let $(M,J,g)$ be a Riemannian manifold where $j$ is tensor field of type $(1,1)$ satisfying $J^2=1$. Then we have the following $$T(JX,JY)+T(X,Y)=-\frac{1}{2}N_{J}(X,Y),$$for all $X,Y \in\chi{M}$, where $T$ denotes the torsion tensor.
Could someone please help me prove this result?
I am trying to prove by using the definition of Nijenhuis tensor as following
$ N(JX,JY)=J^2[X,Y]+[JX,JY]-J[JX,Y]-J[X,JY]\\
= [X,Y]+[JX,JY]-J[JX,Y]-J[X,JY] $
$ T(X,Y)= \nabla_XY-\nabla_YX-[X,Y] .....(1)$\
and $T(JX,JY)=\nabla_{JX}JY-\nabla_{JY}JX-[JX,JY]....(2) $\ Ading (1) an (2) I coulnot get required result.So please help me how to proceed.
Where did you found that relation? What you can prove (strongly relateded to almost complex structures on manifolds) is the following:
$\textbf{Claim}$: Let $(M,g,\nabla)$ be a Riemannian manifold endowded with a connection, and $J\in T^1_1(M)$ s.t. $J \circ J= -id_{T^1_1(M)}$ and $J$ compatible with the connection (i.e. $\nabla_X JY=J \nabla_X Y$), then
\begin{equation} N_J(X,Y)= T_{\nabla}(X,Y) + JT_{\nabla}(JX,Y) +JT_{\nabla}(X,JY) - T_{\nabla}(JX,JY) \end{equation} Proof. Expanding the RHS \begin{align*} RHS =&\nabla_X Y - \nabla_YX -[X,Y]+ J \nabla_{JX} Y - J\nabla_Y JX -J[JX,Y]+ \\&+J\nabla_{X} JY - J\nabla_{JY} X -J[X,JY]- \nabla_{JX} JY + \nabla_{JY} JX +[JX,JY]\\ =&\nabla_X Y - \nabla_YX -[X,Y]+ J \nabla_{JX} Y +\nabla_Y X -J[JX,Y]+ \\&-\nabla_{X} Y - J\nabla_{JY} X -J[X,JY]- J\nabla_{JX} Y + J\nabla_{JY} X +[JX,JY]\\ =&\ -[X,Y] -J[JX,Y] -J[X,JY] +[JX,JY]\\ =& J^2[X,Y] -J[JX,Y] -J[X,JY] +[JX,JY]=N_J(X,Y) \end{align*}
Not sure you can simplify the expression by some extra requirements to what you wrote down.