# Complex structures on $TM$ and $T^*M$

There we go. I'm asking this question to know the different complex structures can be defined on $$TM$$ and $$T^*M$$ (I don't mind because my manifold will be Kähler).

I know there are related questions, for example those ones in MO (the former cites the latter) and also others in M.SE, although I can't find them. However, none of them clarify my doubts.

Suppose $$M$$ is a manifold. It is well known that $$T^*M$$ is a symplectic manifold in a natural way. It seems that doesn't happen if we are interested in complex structures, i.e., $$T^*M$$ isn't a (an almost) complex manifold in a natural way.

Now, suppose that $$M$$ is a Riemannian manifold. Róbert Szöke, in his paper Complex structures on tangent bundles of Riemannian manifolds (you should be able to find the paper here) says that there are a natural complex structure on $$TM$$. Namely, that the metric gives rise to a direct sum decomposition of the bundle $$T(TM)$$ into the vertical and the horizontal subbsendles, so for any $$p\in TM$$ we have the isomorphism $$T_p(TM)\cong T_{\pi(p)}M\oplus T_{\pi(p)}M$$.

Question 1. Im' not sure the metric give you that split. I guess that the split is actually given by the Levi-Civita connection associated with the Riemannian metric of $$M$$. Am I right?

Furthermore, imagine that $$M$$ is also complex (in my case it will be Kähler). If $$J$$ is the complex structure on $$M$$, I have checked that $$TJ$$ (the differential map of $$J$$) is an almost complex structure on $$TM$$. I don't know if it will be integrable, it should be though (call it poetic justice).

Question 2. I can't find the question here in M.SE, but there, there was a comment saying that if $$M$$ is complex, then $$T^*M$$ is too. Is $$TJ$$ the complex structure that person referred to? Or can we define (in a more or less natural way) other complex structures using $$J$$? In that case, does it matter if I consider $$TM$$ or $$T*M$$, suposseing $$M$$ is still Riemannian?

Let me add some context for my question. I want to reproduce Hitchin's result about the c-Map but defining all the structures globally. Because he points that it is possible, but he doesn't prove the result and works only locally.

Here is the paper I say and here you can find another paper studying the c-map globally in the lagange of principal bundles, which I want to avoid.

• Notice that raising/lowering indices with a metric $g$ determines an isomorphism $TM \cong T^*M$, so it's enough to ask these questions, e.g., about $TM$. – Travis Willse Feb 25 '19 at 20:46
• @Travis Perfect. Thanks – Dog_69 Feb 25 '19 at 21:03

(1) You are correct that the almost complex structure $$J : TTM \to TTM$$ is derived from the Levi-Civita connection of the given metric. This connection can be viewed as a particular distribution $$H \subset TTM$$ complementary to the vertical distribution $$V := \ker T\pi$$, where $$\pi : TM \to M$$ is the canonical projection. In particular, we can define the almost complex structure on $$TTM = H \oplus V$$ by declaring $$J(X, Y) := (-Y, X) .$$ This also shows that actually need less information than a metric to define this almost complex structure: All we need is a distribution $$H$$ complementary to $$V$$, and we can identify any such $$H$$ with a linear connection $$\nabla$$ on $$M$$ and vice versa.
As Dombrowski points out in $$\S$$ 5 of the article mentioned below, this complex structure is integrable iff $$\nabla$$ it is flat and torsion-free. So, for Levi-Civita connections (and as Szöke mentions) the complex structure is integrable iff $$g$$ is (locally) flat, and in particular it is not integrable for nonflat Kähler metrics.
(2) For any metric, the index-raising and lowering operators define a canonical isomorphism $$TM \cong T^* M$$, so it's enough to ask the questions, e.g., about $$TM$$.
• Fantastic answer. It will help me a lot, I'm sure. I'll read Dombrowski article tomorrow (in Spain is time to sleep). Just one question regarding 2). $TJ$ is one of the complex structures I can define on $TM$ using the original. Do you know if it is "the standard"? I mean, if $M$ is a complex manifold, how can I define complex structures on $TM$? Do you recommend me some reference for this issue? And thanks again. – Dog_69 Feb 25 '19 at 23:05
• Thank you, and I'm glad you found it useful. I don't know offhand what are all the functorial constructions that assign to any complex manifold $(M, J)$ a complex structure on $TM$. Of course, if you have one such construction $(M, J) \rightsquigarrow \Bbb J$, "conjugating" gives another such construction, i.e., $(M, J) \rightsquigarrow -\Bbb J$. It's an interesting problem, but I don't know any reference that treats this problem---that certainly doesn't mean there isn't a literature about it, though. And you're welcome. – Travis Willse Feb 26 '19 at 0:26