Quaternionic structure

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

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[2]	V.Y. Kraines, "Topology of quaternionic manifolds" Trans. Amer. Math. Soc. , 122 (1966) pp. 357–367 MR0192513 Zbl 0148.16101
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[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
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