Rotating hyperkähler isometry

Definition

Let

(M,g_M,(J_M)_1,(J_M)_2,(J_M)_3),\qquad (N,g_N,(J_N)_1,(J_N)_2,(J_N)_3)

be hyperkähler manifolds with specified ordered triples. A rotating hyperkähler isometry is a Riemannian isometry $f:M\to N$ for which there is a constant matrix $A=(A_{ba})\in SO(3)$ satisfying

df\circ (J_M)_a =\sum_{b=1}^3 A_{ba}(J_N)_b\circ df \qquad(a=1,2,3).

For disconnected manifolds, the rotation may instead be specified separately and constantly on each connected component.

Kähler-form formulation

If $(\omega_M)_a$ and $(\omega_N)_b$ are the three Kähler forms, the definition is equivalently

f^*(\omega_N)_b =\sum_{a=1}^3 A_{ba}(\omega_M)_a.

The matrix is orientation-preserving because it must respect quaternion multiplication on the sphere of induced complex structures.

Rotating hyperkähler isometries compose, with their rotation matrices composing. The special case $A=I_3$ is a hyperkähler isometry in the strict ordered-triple sense.

Distinction from metric isometries

A Riemannian isometry of the underlying metrics need not preserve the parallel three-plane spanned by $I,J,K$ , so it need not be rotating hyperkähler. Conversely, a rotating isometry with $A\ne I_3$ is not triholomorphic for the specified ordered triples.

References

Dominic D. Joyce, Compact Manifolds with Special Holonomy, Oxford University Press, 2000. DOI record. Relevant: Chapter 7, parallel quaternionic triples and their Kähler forms.
Daniel Huybrechts, “Compact Hyperkähler Manifolds: Basic Results,” Inventiones Mathematicae 135 (1999), 63–113. DOI record. Relevant: §1, hyperkähler structures and the sphere of induced complex structures.