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A quaternionic structure on a real vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q0767901.png" /> is a module structure over the skew-field of quaternions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q0767902.png" />, that is, a subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q0767903.png" /> of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q0767904.png" /> of endomorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q0767905.png" /> induced by two anti-commutative complex structures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q0767906.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q0767907.png" /> (cf. [[Complex structure|Complex structure]]). The endomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q0767908.png" /> are called standard generators of the quaternionic structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q0767909.png" />, and the basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679011.png" /> defined by them is called the standard basis. A standard basis is defined up to automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679012.png" />. The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679013.png" /> is isomorphic to the algebra of quaternions (cf. [[Quaternion|Quaternion]]). An automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679014.png" /> of the vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679015.png" /> is called an automorphism of the quaternionic structure if the transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679016.png" /> of the space of automorphisms induced by it preserves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679017.png" />, that is, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679018.png" />. If, moreover, the identity automorphism is induced on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679019.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679020.png" /> 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]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679021.png" /> over the skew-field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679022.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679023.png" />. The group of all automorphisms of a quaternionic structure is isomorphic to the direct product with amalgamation of the subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679024.png" /> and the group of unit quaternions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679025.png" />.
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A quaternionic structure on a differentiable manifold is a field of quaternionic structures on the tangent spaces, that is, a subbundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679026.png" /> of the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679027.png" /> of endomorphisms of tangent spaces whose fibres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679028.png" /> are quaternionic structures on the tangent spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679029.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679030.png" />. A pair of anti-commutative almost-complex structures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679031.png" /> on the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679032.png" /> is called a special quaternionic structure. It induces the quaternionic structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679033.png" />, where
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679034.png" /></td> </tr></table>
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679035.png" /> on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679036.png" /> is induced by a special quaternionic structure if and only if the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679037.png" /> is trivial. A quaternionic structure on a manifold can be regarded as a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679038.png" />-structure, and a special quaternionic structure as a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679039.png" />-structure in the sense of the theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679040.png" />-structures (cf. [[G-structure(2)|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679041.png" />-structure]]). Hence, in order that a quaternionic structure (or a special quaternionic structure) should exist on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679042.png" />, it is necessary and sufficient that the structure group of the tangent bundle reduces to the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679043.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679044.png" />). The first prolongation of a special quaternionic structure, regarded as a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679045.png" />-structure, is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679046.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679047.png" />.
+
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
  
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]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679048.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679049.png" /> whose holonomy group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679050.png" /> is contained in the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679051.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679052.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679053.png" /> in which there exists a quaternionic structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679054.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679055.png" /> that is invariant with respect to Levi-Civita parallel displacement: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679056.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679057.png" /> is the operator of [[Covariant differentiation|covariant differentiation]] of the [[Levi-Civita connection|Levi-Civita connection]].
+
$$
 +
H _ {p}  = \{ {J =
 +
\lambda _ {0}  \mathop{\rm id} +
 +
\lambda _ {1} J _ {1} +
 +
\lambda _ {2} J _ {2} +
 +
\lambda _ {3} J _ {1} J _ {2} } : {
 +
\lambda _ {i} \in \mathbf R } \}
 +
.
 +
$$
  
In a quaternionic Riemannian manifold there exists a canonical parallel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679058.png" />-form that defines a number of operators in the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679059.png" /> of differential forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679060.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076790/q07679061.png" /> (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]]]).
+
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====
 
====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>
 
<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>

Latest revision as of 15:10, 19 January 2021


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). 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). 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 $ \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 quaternions $ 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). 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 associated with the special quaternionic structure. The vanishing of the curvature and 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 for quaternionic structures. It is defined as a 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. 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. 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 of the 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 [2] analogous to Hodge theory for Kähler manifolds, and to obtain estimates for the Betti numbers of the manifold $ M $( cf. Hodge structure; 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). 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 [5]).

References

[1] 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 MR0087172 Zbl 0078.14103
[2] V.Y. Kraines, "Topology of quaternionic manifolds" Trans. Amer. Math. Soc. , 122 (1966) pp. 357–367 MR0192513 Zbl 0148.16101
[3] K. Yano, M. Ako, "An affine connection in an almost quaternionic manifold" J. Differential Geom. , 8 : 3 (1973) pp. 341–347 MR355892
[4] A.J. Sommese, "Quaternionic manifolds" Mat. Ann. , 212 (1975) pp. 191–214 MR0425827 Zbl 0299.53023
[5] 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 MR402649 Zbl 0324.53038
[6] J.A. Wolf, "Complex homogeneous contact manifolds and quaternionic symmetric spaces" J. Math. Mech. , 14 : 6 (1965) pp. 1033–1047 MR0185554 Zbl 0141.38202
[7] 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
How to Cite This Entry:
Quaternionic structure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quaternionic_structure&oldid=23947
This article was adapted from an original article by D.V. Alekseevskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article