Everything about Quaternionic manifold
Concept in geometry
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. Wikipedia →
🧭 Start here: a 6-step reading path
- Read the overview: Quaternionic manifoldA 5-minute orientation
- Skim a recent paper: The Monge–Ampère equation for $$(n-1)$$-quaternionic PSH functions on a hyperKähler manifoldInfluential work since 2022
- Pick up a classic book: On Clifford-type structuresThe most-reprinted book on the subject
- Read the landmark paper: Quaternionic K�hler manifoldsCited 347 times
- Go sideways: Quaternion-Kähler manifoldA closely related topic
- Go sideways: Hypercomplex manifoldA closely related topic
📖 Definitions
No dictionary entry for this exact phrase.
🗂️ The subject at a glance
🏆 Landmark papers (most cited)
Quaternionic K�hler manifolds
Quaternionic manifolds for type II superstring vacua of Calabi-Yau spaces
Quaternionic structures on a manifold and subordinated structures
Quaternionic manifolds
Totally complex submanifolds of a quaternionic Kaehlerian manifold
ON THE KAUFFMAN BRACKET SKEIN MODULE OF THE QUATERNIONIC MANIFOLD
A twistor construction of Kähler submanifolds of a quaternionic Kähler manifold
The Sharp Lower Bound of the First Eigenvalue of the Sub-Laplacian on a Quaternionic Contact Manifold
🆕 Influential since 2022
The Monge–Ampère equation for $$(n-1)$$-quaternionic PSH functions on a hyperKähler manifold
Solution of the qc Yamabe equation on a 3-Sasakian manifold and the quaternionic Heisenberg group
Riemannian Optimization Framework on the Generalized Quaternionic Stiefel Manifold
The Obata First Eigenvalue Theorem on a Seven-Dimensional Quaternionic Contact Manifold
Quaternionic Stochastic Areas on Quaternionic Full Flag Manifolds and Applications
📚 Books worth reading
- 📕On Clifford-type structuresWiesław Królikowski · 2006
Quaternionic structures in mathematics and physicsMeeting on Quaternionic Structures in Mathematics and Physics (2nd 1999 Rome, Italy) · 2001- 📕Nonlinear elliptic equations and nonassociative algebrasNikolai Nadirashvili · 2014
Ranked by number of editions on Open Library, a rough measure of how lasting a book is.
🔗 Related topics
- Quaternion-Kähler manifold In differential geometry, a quaternion-Kähler manifold (or quaternionic Kähler manifold) is a Riemannian 4n-manifold whose Riemannian holonomy group is…
- Hypercomplex manifold 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…
- Quaternionic projective space In mathematics, quaternionic projective space is an extension of the ideas of real projective space and complex projective space, to the case where coordinates…
- Spinh structure In spin geometry, a spinh structure (or quaternionic spin structure) is a generalization of a spin structure. In mathematics, these are used to describe…
- G-structure on a manifold In differential geometry, a G-structure on an n {\displaystyle n} -manifold M {\displaystyle M} , for a given structure group G {\displaystyle G} , is…
- Principal SU(2)-bundle In mathematics, especially differential geometry, principal SU ( 2 ) {\displaystyle \operatorname {SU} (2)} -bundles (or principal Sp ( 1 ) {\displaystyle…
- Bott periodicity theorem In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by Raoul Bott (1957, 1959)…
- Classical group In mathematics, the classical groups are the matrix groups arising from finite-dimensional vector spaces and from nondegenerate bilinear, sesquilinear…