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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" />.

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

<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 <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 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]].

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]]]).

====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>