View source for Quaternionic structure

<!--
q0767901.png
$#A+1 = 61 n = 0
$#C+1 = 61 : ~/encyclopedia/old_files/data/Q076/Q.0706790 Quaternionic structure
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

A quaternionic structure on a real vector space  $  V $
is a module structure over the skew-field of quaternions  $  \mathbf H $, 
that is, a subalgebra  $  H $
of the algebra  $   \mathop{\rm End}  V $
of endomorphisms of  $  V $
induced by two anti-commutative complex structures  $  J _ {1} , J _ {2} $
on  $  V $(
cf. [[Complex structure|Complex structure]]). The endomorphisms  $  J _ {1} , J _ {2} $
are called standard generators of the quaternionic structure  $  H $, 
and the basis  $  \{  \mathop{\rm id} , J _ {1} , J _ {2} , J _ {3} = J _ {1} J _ {2} \} $
of  $  H $
defined by them is called the standard basis. A standard basis is defined up to automorphisms of  $  H $. 
The algebra  $  H $
is isomorphic to the algebra of quaternions (cf. [[Quaternion|Quaternion]]). An automorphism  $  A $
of the vector space  $  V $
is called an automorphism of the quaternionic structure if the transformation  $   \mathop{\rm Ad}  A $
of the space of automorphisms induced by it preserves  $  H $, 
that is, if  $  (  \mathop{\rm Ad}  A ) H = A H A  ^ {-1} = H $. 
If, moreover, the identity automorphism is induced on  $  H $, 
then  $  A $
is called a special automorphism of the quaternionic structure. The group of all special automorphisms of the quaternionic structure is isomorphic to the [[General linear group|general linear group]]  $   \mathop{\rm GL} ( m , \mathbf H ) $
over the skew-field  $  \mathbf H $, 
where  $  4 m =  \mathop{\rm dim}  V $. 
The group of all automorphisms of a quaternionic structure is isomorphic to the direct product with amalgamation of the subgroup  $   \mathop{\rm GL} ( m , \mathbf H ) $
and the group of [[unit quaternion]]s  $  H _ {1} \approx  \mathop{\rm Sp} ( 1) $.

A quaternionic structure on a differentiable manifold is a field of quaternionic structures on the tangent spaces, that is, a subbundle  $  \pi :  H \rightarrow M $
of the bundle  $   \mathop{\rm End} ( T ( M)) \rightarrow M $
of endomorphisms of tangent spaces whose fibres  $  {\mathcal H} _ {p} = \pi  ^ {-1} ( p) $
are quaternionic structures on the tangent spaces  $  T _ {p} M $
for all  $  p \in M $. 
A pair of anti-commutative almost-complex structures  $  J _ {1} , J _ {2} $
on the manifold  $  M $
is called a special quaternionic structure. It induces the quaternionic structure  $  H $, 
where

$$ 
H _ {p}  =  \{ {J =
\lambda _ {0}  \mathop{\rm id} +
\lambda _ {1} J _ {1} +
\lambda _ {2} J _ {2} +
\lambda _ {3} J _ {1} J _ {2} } : {
\lambda _ {i} \in \mathbf R } \}
.
$$

A quaternionic structure  $  H $
on a manifold  $  M $
is induced by a special quaternionic structure if and only if the bundle  $  H \rightarrow M $
is trivial. A quaternionic structure on a manifold can be regarded as a  $   \mathop{\rm Sp} ( 1) \cdot  \mathop{\rm GL} ( m , \mathbf H ) $-
structure, and a special quaternionic structure as a  $   \mathop{\rm GL} ( m , \mathbf H ) $-
structure in the sense of the theory of  $  G $-
structures (cf. [[G-structure| $  G $-
structure]]). Hence, in order that a quaternionic structure (or a special quaternionic structure) should exist on a manifold  $  M $, 
it is necessary and sufficient that the structure group of the tangent bundle reduces to the group  $   \mathop{\rm Sp} ( 1) \cdot  \mathop{\rm Sp} ( m) $(
or  $   \mathop{\rm Sp} ( m) $). 
The first prolongation of a special quaternionic structure, regarded as a  $   \mathop{\rm GL} ( m , \mathbf H ) $-
structure, is an  $  e $-
structure (a field of frames), which determines a canonical [[Linear connection|linear connection]] associated with the special quaternionic structure. The vanishing of the [[Curvature|curvature]] and [[Torsion|torsion]] of this connection is a necessary and sufficient condition for the special quaternionic structure to be locally equivalent to the standard flat special quaternionic structure on the vector space  $  \mathbf R  ^ {4m} $.

A quaternionic Riemannian manifold is the analogue of a [[Kähler manifold|Kähler manifold]] for quaternionic structures. It is defined as a [[Riemannian manifold|Riemannian manifold]]  $  M $
of dimension  $  4 m $
whose holonomy group  $  \Gamma $
is contained in the group  $   \mathop{\rm Sp} ( 1) \cdot  \mathop{\rm Sp} ( m) $. 
If  $  \Gamma \subset   \mathop{\rm Sp} ( m) $, 
then the quaternionic Riemannian manifold is called a special or quaternionic Kähler manifold, and it has zero [[Ricci curvature|Ricci curvature]]. A quaternionic Riemannian manifold can be characterized as a Riemannian manifold  $  M $
in which there exists a quaternionic structure  $  H $
that is invariant with respect to Levi-Civita [[Parallel displacement(2)|parallel displacement]]. Similarly, a special quaternionic Riemannian manifold is a Riemannian manifold in which there exists a special quaternionic structure  $  ( J _ {1} , J _ {2} ) $
that is invariant with respect to Levi-Civita parallel displacement:  $  \nabla J _ {1} = \nabla J _ {2} = 0 $, 
where  $  \nabla $
is the operator of [[Covariant differentiation|covariant differentiation]] of the [[Levi-Civita connection|Levi-Civita connection]].

In a quaternionic Riemannian manifold there exists a canonical parallel  $  4 $-
form that defines a number of operators in the ring  $  \Lambda ( M) $
of differential forms on  $  M $
that commute with the Laplace–Beltrami operator (exterior product operator, contraction operators). This enables one to construct an interesting theory of harmonic differential forms on quaternionic Riemannian manifolds [[#References|[2]]] analogous to Hodge theory for Kähler manifolds, and to obtain estimates for the Betti numbers of the manifold  $  M $(
cf. [[Hodge structure|Hodge structure]]; [[Betti number|Betti number]]). Locally Euclidean spaces account for all the homogeneous special quaternionic Riemannian manifolds. As an example of a homogeneous quaternionic Riemannian manifold that is not special one may cite the quaternionic projective space and also other Wolf symmetric spaces which are in one-to-one correspondence with simple compact Lie groups without centre (cf. [[Symmetric space|Symmetric space]]). These account for all compact homogeneous quaternionic Riemannian manifolds. A wide class of non-compact non-symmetric homogeneous quaternionic Riemannian manifolds can be constructed by means of modules over Clifford algebras (see [[#References|[5]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S.-S. Chern, "On a generalization of Kähler geometry" R.H. Fox (ed.) D.C. Spencer (ed.) A.W. Tucker (ed.) , ''Algebraic geometry and topology (Symp. in honor of S. Lefschetz)'' , Princeton Univ. Press (1957) pp. 103–121 {{MR|0087172}} {{ZBL|0078.14103}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> V.Y. Kraines, "Topology of quaternionic manifolds" ''Trans. Amer. Math. Soc.'' , '''122''' (1966) pp. 357–367 {{MR|0192513}} {{ZBL|0148.16101}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> K. Yano, M. Ako, "An affine connection in an almost quaternionic manifold" ''J. Differential Geom.'' , '''8''' : 3 (1973) pp. 341–347 {{MR|355892}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A.J. Sommese, "Quaternionic manifolds" ''Mat. Ann.'' , '''212''' (1975) pp. 191–214 {{MR|0425827}} {{ZBL|0299.53023}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> D.V. Alekseevskii, "Classification of quaternionic spaces with a transitive solvable group of motions" ''Math. USSR Izv.'' , '''9''' : 2 (1975) pp. 297–339 ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''39''' : 2 (1975) pp. 315–362 {{MR|402649}} {{ZBL|0324.53038}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> J.A. Wolf, "Complex homogeneous contact manifolds and quaternionic symmetric spaces" ''J. Math. Mech.'' , '''14''' : 6 (1965) pp. 1033–1047 {{MR|0185554}} {{ZBL|0141.38202}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> D.V. Aleksevskii, "Lie groups and homogeneous spaces" ''J. Soviet Math.'' , '''4''' : 5 (1975) pp. 483–539 ''Itogi Nauk. i Tekhn. Algebra. Topol. Geom.'' , '''11''' (1974) pp. 37–123</TD></TR></table>