Timeline for Do these definitions of integrable quaternionic structure agree?
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| Sep 23, 2017 at 18:47 | comment | added | Robert Bryant | @Mtheorist: Maybe not obvious, but it does follow. You have to use the torsion-free condition, i.e., $\Gamma^\kappa_{\mu\nu}=\Gamma^\kappa_{\nu\mu}$. At, say, the origin of your coordinates $x=0$, you'll then find that the quadratic map $\Phi:\mathbb{R}^{4n}\to\mathbb{R}^{4n}$ defined by $\Phi^\kappa(x) = \Gamma^\kappa_{\mu\nu}(0)x^\mu x^\nu$ must have its differential satisfy $$\Phi'(x)(J_iy)=J_i \bigl(\Phi'(x)(y)\bigr)$$ for all $i=1,2,3$, i.e., $\Phi$ is quaternion differentiable, which forces $\Phi(x)\equiv0$, since, as is well-known, quaternion differentiable maps must be affine linear. | |
| Sep 23, 2017 at 18:07 | comment | added | Mtheorist | Sorry, I am a little confused. In a local patch, $\nabla_{\nu} {(J_i)^{\lambda}}_{\mu}=\partial_{\nu} {(J_i)^{\lambda}}_{\mu}+{\Gamma^{\lambda}}_{\nu \kappa}{(J_i)^{\kappa}}_{\mu}-{\Gamma^{\kappa}}_{\nu \mu}{(J_i)^{\lambda}}_{\kappa}$. The complex structures being parallel and having constant coefficients would imply that ${\Gamma^{\lambda}}_{\nu \kappa}{(J_i)^{\kappa}}_{\mu}={\Gamma^{\kappa}}_{\nu \mu}{(J_i)^{\lambda}}_{\kappa}$. It is not obvious to me how this implies flatness of the connection. | |
| Sep 23, 2017 at 11:00 | vote | accept | Mtheorist | ||
| Sep 23, 2017 at 10:49 | comment | added | Robert Bryant | @Mtheorist: No, that is not true. For example, one can construct the Eguchi-Hanson metric (which is hyperKähler) on $M=TS^2$ by a hyperKähler quotient of $Y=\mathbb{H}^2$ by a standard $S^1$-action. However, there do not exist local coordinates on $M$ in which the three complex structures have constant coefficients: The hyperKähler structure is not flat, and its Levi-Civita connection leaves the complex structures parallel, so it must agree with the Obata connection. If there were coordinates in which the three complex structures had constant coefficients, the Obata connection would be flat. | |
| Sep 23, 2017 at 10:05 | comment | added | Mtheorist | Thank you for your answer. Can't we obtain the three almost complex structures in the first sense on a hyperkaehler manifold with non-flat metric, via the hyperkaehler quotient? In this (projecteuclid.org/download/pdf_1/euclid.jmsj/1227107863) reference, Nakajima and Gocho show that when taking such a quotient of a manifold, $Y$, the three almost complex structures on $Y$ descend to the three almost complex structures on the quotient, $M$. Then wouldn't it be true that if the almost complex structures on $Y$ take the above form with constant coefficients, the same should hold for $M$? | |
| Sep 23, 2017 at 9:57 | history | edited | Robert Bryant | CC BY-SA 3.0 |
fixed a typo, added a reference
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| Sep 23, 2017 at 9:50 | comment | added | Mtheorist | Sorry, I think there is a typo in your last sentence. | |
| Sep 23, 2017 at 9:47 | history | answered | Robert Bryant | CC BY-SA 3.0 |