I'm having a hard time proving the following:
If $M$ is an n-dimensional indefinite Riemannian manifold whose metric g has index s, then the metric of Sasaki $g^{D}$ is an indefinite metric on $TM$ whose index is 2s. Let $J′$ be the natural almost complex structure on TM, then $TM(J′,g^{D})$ is an indefinite almost Kahler manifold (that means the fundamental 2- form is closed). Moreover $(TM,g^{D} )$ is Kahler if and only if M is locally flat
If $M$ is an $n$-dimensional indefinite Riemannian manifold whose metric $g$ has index $s$, then the metric of Sasaki $g^{D}$ is an indefinite metric on $TM$ whose index is $2s$. Let $J'$ be the natural almost complex structure on TM, then $TM(J',g^{D})$ is an indefinite almost Kähler manifold (that means the fundamental 2-form is closed). Moreover $(TM,g^{D} )$ is Kähler if and only if $M$ is locally flat.
this questionThis assertion is from the article Indefinite Kahler ManifoldsIndefinite Kähler manifolds by Manuel Barros and Alfonso Romero.