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Tanget Tangent bundle with Sasaki metric is Kähler iff $M$ is locally flat

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tanget Tanget bundle with sasakiSasaki metric is Kahlet iif MKähler iff $M$ is locally flat

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.

tanget bundle with sasaki metric is Kahlet iif M is locally flat

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

this question is from the article Indefinite Kahler Manifolds by Manuel Barros and Alfonso Romero

Tanget bundle with Sasaki metric is Kähler iff $M$ is locally flat

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 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 assertion is from the article Indefinite Kähler manifolds by Manuel Barros and Alfonso Romero.

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tanget bundle with sasaki metric is Kahlet iif M is locally flat

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

this question is from the article Indefinite Kahler Manifolds by Manuel Barros and Alfonso Romero