# Tangent 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.

• By $D$, you mean the Levi-Civita connection, right? Otherwise, for an arbitrary affine connection, the requirement for the Sasaki metric to be Kahler is for $D$ to be flat and to be Codazzi-coupled to the metric $g$. – Gabe K May 16 '20 at 22:06
• actully $D$ represents covariant differentiation: $Dv^{i}=dv^{i}+Γ_{jk}^{i}v^{j}dx^{k}$ – MOULI Kawther May 16 '20 at 22:13
• Right. Choosing a connection is equivalent to choosing the connection coefficients $\Gamma_{jk}^i$. The point is that the theorem is only true when the connection used is the Levi-Civita connection. – Gabe K May 16 '20 at 22:37
• assuming the connection used is the Levi-Civita connection connection, can you please help me with the proof? – MOULI Kawther May 16 '20 at 23:02

For a proof of this result (and a more general version), there's a paper by Satoh [1] which has a lot of detail. The main idea is that for $$TM$$ to be Hermitian, the Nijenhuis tensor of $$J^\prime$$ needs to vanish. However, when you calculate the Nijenhuis tensor, you find that it vaishes if and only if the torsion and curvature of $$D$$ vanish (i.e. D is flat). See the equation on the bottom of page 8 of [1] for the exact formula.
For a Riemannian manifold, $$g$$ is flat iff the Levi-Civita connection is a flat connection, so I presume this is what the authors are using. For a more general connection $$D$$ (i.e. not the Levi-Civita connection), $$(TM, J^\prime,g^D)$$ is Kahler iff $$(M,g,D)$$ is a so-called Hessian manifold, which means that $$D$$ is flat and satisfies $$(D_X g)(Y,Z)=(D_Y g)(X,Z)$$ for all vector fields $$X,Y$$ and $$Z$$. This relationship between the metric and connection is also known as Codazzi-coupling. The proof of this is also in that reference.
• thanks a lot, can we use the same argument if $TM$ is associated with $J^h$ instead where h is the horisental lifting? – MOULI Kawther May 17 '20 at 3:18