Talk:Hyperkähler manifold

message: A complex structure is, I think, an atlas of holomorphic maps to C^n. What I don't understand is how you can multiply two complex structures or form linear combinations. --MarSch 09:37, 7 September 2005 (UTC)
replies: text: The atlas needs holomorphic transition maps as well. Locally, a complex structure is equivalent to choosing a linear isomorphism J on each fiber of the tangent bundle such that J^2=-1, where 1 is the identity map. So on each fiber the algebraic operations you're wondering about are those of matrix algebra. Orthografer 05:44, 29 March 2006 (UTC)

text: I agree that the quaternionic action is what I naturally consider when I hear the word hyperkahler, but I've never gone through the equivalence in my head. For instance, I think it would be wonderful if you could explain in the page why it's wrong to say the holonomy group (which I have understood to be global, though I'm no expert) should be precisely Sp(n). I gave a rationale for excluding the 4-torus and I think your reason for including it would be a welcome addition, instead of simply changing to definition to taste. I've been harassing my hyperkahler friends to tell me things, but they always make me pin down a definition of hyperkahler first. Maybe this debate is worthy of inclusion in the article by someone who knows about it. Orthografer 04:27, 10 April 2006 (UTC)

text: Tiphareth: what do you think of my changes? Orthografer 20:39, 20 April 2006 (UTC)
thread_title: Initial discussions

message: Is there a reason that the spelling "hyperkähler" should not be dismissed as incorrect in favor of "hyper-Kähler"?  — Preceding unsigned comment added by Seub (talk • contribs) 23:16, 12 March 2015 (UTC)
thread_title:  Orthography 

message: I'm not a fan of the definition given of a hyperkähler manifold in this article. By the way, it is only given in the preamble, is that normal?
The reason that I'm not a fan is the same reason that I wouldn't like the definition of a Kähler manifold as "a Riemannian manifold with holonomy contained in U(n)" (which is is not the definition given in the corresponding Wikipedia article). 
Let me explain in case it's not clear: when you have a Kähler manifold, you know what the complex structure is. Saying that you have a Riemannian metric with holonomy in U(n) just tells you that there exists a complex structure compatible with the metric, turning your manifold into a Kähler manifold. However, several different complex structures can exist. And they can be "really different" in the sense that there might not exists self-automorphisms of your manifold that take a complex structure to the other. In practice, you really want to know what complex structure you are dealing with.
Similarly, when you have a hyperkähler manifold, you want to know hat I, J and K are (at least the triple, but a lot of the time each one of them, in my modest experience).
So, in a nutshell, I would say that a Riemannian manifold with holonomy in Sp(k) is not a hyperKähler manifold per se, it is a Riemannian manifold which admits a compatible quaternionic (or hypercomplex) structure turning it into a hyperkähler manifold, however several different such quaternionic structures may exist.
I am not a seasoned contributor to Wikipedia, should I go ahead and edit the article, or would that be bad-mannered?--Seub (talk) 23:57, 13 March 2015 (UTC)
thread_title:  Definition 

message: Substituted at 02:13, 5 May 2016 (UTC)
thread_title: Assessment comment