Hyperkähler isometric immersion

Let

(M,g_M,I_M,J_M,K_M),\qquad (N,g_N,I_N,J_N,K_N)

be hyperkähler manifolds with specified ordered triples. A hyperkähler isometric immersion is a smooth map $f:M\to N$ satisfying

f^*g_N=g_M

and

df\circ I_M=I_N\circ df,\qquad df\circ J_M=J_N\circ df,\qquad df\circ K_M=K_N\circ df.

Thus it is both a triholomorphic map and a Riemannian isometric immersion. No separate immersion hypothesis is needed, because the metric pullback equation makes $df$ injective.

Kähler forms

Let $(\omega_M)_I,(\omega_M)_J,(\omega_M)_K$ and $(\omega_N)_I,(\omega_N)_J,(\omega_N)_K$ be the corresponding Kähler forms. The defining equations imply

f^*(\omega_N)_I=(\omega_M)_I,\qquad f^*(\omega_N)_J=(\omega_M)_J,\qquad f^*(\omega_N)_K=(\omega_M)_K.

Conversely, preservation of the metric and any two members of the ordered triple implies preservation of the third, since $K=IJ$ .

Isomorphisms

Hyperkähler isometric immersions compose. If $f$ is a diffeomorphism, it is a hyperkähler isometry. Allowing an $SO(3)$ -rotation of the target triple gives the distinct notion of a rotating hyperkähler isometry.

References

Dominic D. Joyce, Compact Manifolds with Special Holonomy, Oxford University Press, 2000. DOI record. Relevant: Chapters 6–7, hypercomplex and hyperkähler structures.
Daniel Huybrechts, “Compact Hyperkähler Manifolds: Basic Results,” Inventiones Mathematicae 135 (1999), 63–113. DOI record. Relevant: §1, hyperkähler metrics and induced complex structures.