Knowlpedia

Definition

Hyperkähler isometry

A diffeomorphism of hyperkähler manifolds that preserves the metric and each member of the specified quaternionic triple.

Definition

Let

(M,gM,IM,JM,KM),(N,gN,IN,JN,KN)(M,g_M,I_M,J_M,K_M),\qquad (N,g_N,I_N,J_N,K_N)

be with specified ordered triples. A hyperkähler isometry, in the strict ordered-triple sense, is a diffeomorphism f:M→Nf:M\to N such that

f∗gN=gMf^*g_N=g_M

and

df∘IM=IN∘df,df∘JM=JN∘df,df∘KM=KN∘df.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.

It is therefore both a Riemannian isometry and a . Equivalently, it preserves the metric and each of the three .

It is equivalently a that is a diffeomorphism. Hyperkähler isometries are the isomorphisms in the category with hyperkähler isometric immersions as morphisms.

Strict preservation

A Riemannian isometry of the underlying metrics need not fix a selected ordered hyperkähler triple. An isometry that instead carries the triple to one constant SO(3)SO(3)-rotation is a , a distinct definition. The unqualified term on this page always means strict preservation.

References
  1. Dominic D. Joyce, Compact Manifolds with Special Holonomy, Oxford University Press, 2000. DOI record. Relevant: Chapter 7, parallel quaternionic triples and their Kähler forms.
  2. 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.