In differential geometry, a hypercomplex manifold is a manifold with the tangent bundle equipped with an action by the algebra of quaternions in such a way that the quaternions define integrable almost complex structures.
If the almost complex structures are instead not assumed to be integrable, the manifold is called quaternionic, or almost hypercomplex. [1]
Every hyperkähler manifold is also hypercomplex. The converse is not true. The Hopf surface
(with acting as a multiplication by a quaternion , ) is hypercomplex, but not Kähler, hence not hyperkähler either. To see that the Hopf surface is not Kähler, notice that it is diffeomorphic to a product hence its odd cohomology group is odd-dimensional. By Hodge decomposition, odd cohomology of a compact Kähler manifold are always even-dimensional. In fact Hidekiyo Wakakuwa proved [2] that on a compact hyperkähler manifold . Misha Verbitsky has shown that any compact hypercomplex manifold admitting a Kähler structure is also hyperkähler. [3]
In 1988, left-invariant hypercomplex structures on some compact Lie groups were constructed by the physicists Philippe Spindel, Alexander Sevrin, Walter Troost, and Antoine Van Proeyen. In 1992, Dominic Joyce rediscovered this construction, and gave a complete classification of left-invariant hypercomplex structures on compact Lie groups. Here is the complete list.
where denotes an -dimensional compact torus.
It is remarkable that any compact Lie group becomes hypercomplex after it is multiplied by a sufficiently big torus.
Hypercomplex manifolds as such were studied by Charles Boyer in 1988. He also proved that in real dimension 4, the only compact hypercomplex manifolds are the complex torus , the Hopf surface and the K3 surface.
Much earlier (in 1955) Morio Obata studied affine connection associated with almost hypercomplex structures (under the former terminology of Charles Ehresmann [4] of almost quaternionic structures). His construction leads to what Edmond Bonan called the Obata connection [5] [6] which is torsion free, if and only if, "two" of the almost complex structures are integrable and in this case the manifold is hypercomplex.
There is a 2-dimensional sphere of quaternions satisfying . Each of these quaternions gives a complex structure on a hypercomplex manifold M. This defines an almost complex structure on the manifold , which is fibered over with fibers identified with . This complex structure is integrable, as follows from Obata's theorem (this was first explicitly proved by Dmitry Kaledin [7] ). This complex manifold is called the twistor space of . If M is , then its twistor space is isomorphic to .
In mathematics, a 3-sphere, glome or hypersphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensions is an ordinary sphere, the boundary of a ball in four dimensions is a 3-sphere. A 3-sphere is an example of a 3-manifold and an n-sphere.
In mathematics, hypercomplex number is a traditional term for an element of a finite-dimensional unital algebra over the field of real numbers. The study of hypercomplex numbers in the late 19th century forms the basis of modern group representation theory.
In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and complex algebraic varieties, functions of several complex variables, and holomorphic constructions such as holomorphic vector bundles and coherent sheaves. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric aspects of complex analysis.
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnoldus Schouten and David van Dantzig in 1930, and then introduced by Erich Kähler in 1933. The terminology has been fixed by André Weil. Kähler geometry refers to the study of Kähler manifolds, their geometry and topology, as well as the study of structures and constructions that can be performed on Kähler manifolds, such as the existence of special connections like Hermitian Yang–Mills connections, or special metrics such as Kähler–Einstein metrics.
In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in , such that the transition maps are holomorphic.
In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian differentiable manifold whose Ricci tensor is proportional to the metric. They are named after Albert Einstein because this condition is equivalent to saying that the metric is a solution of the vacuum Einstein field equations, although both the dimension and the signature of the metric can be arbitrary, thus not being restricted to Lorentzian manifolds. Einstein manifolds in four Euclidean dimensions are studied as gravitational instantons.
In differential geometry, a hyperkähler manifold is a Riemannian manifold endowed with three integrable almost complex structures that are Kähler with respect to the Riemannian metric and satisfy the quaternionic relations . In particular, it is a hypercomplex manifold. All hyperkähler manifolds are Ricci-flat and are thus Calabi–Yau manifolds.
In differential geometry, a quaternion-Kähler manifold (or quaternionic Kähler manifold) is a Riemannian 4n-manifold whose Riemannian holonomy group is a subgroup of Sp(n)·Sp(1) for some . Here Sp(n) is the sub-group of consisting of those orthogonal transformations that arise by left-multiplication by some quaternionic matrix, while the group of unit-length quaternions instead acts on quaternionic -space by right scalar multiplication. The Lie group generated by combining these actions is then abstractly isomorphic to .
In mathematics, quaternionic projective space is an extension of the ideas of real projective space and complex projective space, to the case where coordinates lie in the ring of quaternions Quaternionic projective space of dimension n is usually denoted by
In complex geometry, a Hopf manifold is obtained as a quotient of the complex vector space (with zero deleted) by a free action of the group of integers, with the generator of acting by holomorphic contractions. Here, a holomorphic contraction is a map such that a sufficiently big iteration maps any given compact subset of onto an arbitrarily small neighbourhood of 0.
In differential geometry, a quaternion-Kähler symmetric space or Wolf space is a quaternion-Kähler manifold which, as a Riemannian manifold, is a Riemannian symmetric space. Any quaternion-Kähler symmetric space with positive Ricci curvature is compact and simply connected, and is a Riemannian product of quaternion-Kähler symmetric spaces associated to compact simple Lie groups.
In mathematics, a nearly Kähler manifold is an almost Hermitian manifold , with almost complex structure , such that the (2,1)-tensor is skew-symmetric. So,
In mathematics, and in particular gauge theory and complex geometry, a Hermitian Yang–Mills connection is a Chern connection associated to an inner product on a holomorphic vector bundle over a Kähler manifold that satisfies an analogue of Einstein's equations: namely, the contraction of the curvature 2-form of the connection with the Kähler form is required to be a constant times the identity transformation. Hermitian Yang–Mills connections are special examples of Yang–Mills connections, and are often called instantons.
Edmond Bonan is a French mathematician, known particularly for his work on special holonomy.
In algebraic geometry and differential geometry, the nonabelian Hodge correspondence or Corlette–Simpson correspondence is a correspondence between Higgs bundles and representations of the fundamental group of a smooth, projective complex algebraic variety, or a compact Kähler manifold.
In differential geometry, a quaternionic manifold is a quaternionic analog of a complex manifold. The definition is more complicated and technical than the one for complex manifolds due in part to the noncommutativity of the quaternions and in part to the lack of a suitable calculus of holomorphic functions for quaternions. The most succinct definition uses the language of G-structures on a manifold. Specifically, a quaternionic n-manifold can be defined as a smooth manifold of real dimension 4n equipped with a torsion-free -structure. More naïve, but straightforward, definitions lead to a dearth of examples, and exclude spaces like quaternionic projective space which should clearly be considered as quaternionic manifolds.
In mathematics, and especially differential geometry and mathematical physics, gauge theory is the general study of connections on vector bundles, principal bundles, and fibre bundles. Gauge theory in mathematics should not be confused with the closely related concept of a gauge theory in physics, which is a field theory which admits gauge symmetry. In mathematics theory means a mathematical theory, encapsulating the general study of a collection of concepts or phenomena, whereas in the physical sense a gauge theory is a mathematical model of some natural phenomenon.
In mathematics, hyperbolic complex space is a Hermitian manifold which is the equivalent of the real hyperbolic space in the context of complex manifolds. The complex hyperbolic space is a Kähler manifold, and it is characterised by being the only simply connected Kähler manifold whose holomorphic sectional curvature is constant equal to -1. Its underlying Riemannian manifold has non-constant negative curvature, pinched between -1 and -1/4 : in particular, it is a CAT(-1/4) space.
In complex geometry, the Kähler identities are a collection of identities between operators on a Kähler manifold relating the Dolbeault operators and their adjoints, contraction and wedge operators of the Kähler form, and the Laplacians of the Kähler metric. The Kähler identities combine with results of Hodge theory to produce a number of relations on de Rham and Dolbeault cohomology of compact Kähler manifolds, such as the Lefschetz hyperplane theorem, the hard Lefschetz theorem, the Hodge-Riemann bilinear relations, and the Hodge index theorem. They are also, again combined with Hodge theory, important in proving fundamental analytical results on Kähler manifolds, such as the -lemma, the Nakano inequalities, and the Kodaira vanishing theorem.
