Metric connection on $\mathbb{R}^4$ that is locally Kähler but not globally Kähler

Asked Jan 16 '18 at 19:05

Active Jan 18 '18 at 17:51

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in a comment to this question When can a Connection Induce a Riemannian Metric for which it is the Levi-Civita Connection?

Robert Bryant mentions that it is possible to construct a metric connection on $\mathbb{R}^4$ that is locally Kähler, but not Kähler. I haven't been able to figure out how to construct such connection, can anyone give me a hint?

Thanks.

edited Jan 18 '18 at 17:51

Ali Taghavi

asked Jan 16 '18 at 19:05

student

Hint: inside $SO(4)$ there are two copies of $U(2)$ whose product is $SO(4)$ see e.g Van Elfrinkhof's formula in https://en.wikipedia.org/wiki/Rotations_in_4-dimensional_Euclidean_space. – Holonomia Jan 20 '18 at 11:44

Metric connection on $\mathbb{R}^4$ that is locally Kähler but not globally Kähler

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