# 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

## 1 Answer

For a proof of this result (and a more general version), there's a paper by Satoh  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  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.

 Satoh, Hiroyasu, Almost Hermitian structures on tangent bundles, Suh, Young Jin (ed.) et al., Proceedings of the 11th international workshop on differential geometry, Taegu, Korea, November 9–11, 2006. Taegu: Kyungpook National University. 105-118 (2007). ZBL1125.53022.

• 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