Maps from 2-planes to projective spaces using quaternions and octonions.
problem Constructing maps between geometric spaces.
method Using quaternions and octonions, maps are constructed from Gr2(Rn) to RPk. result Maps induce isomorphisms at the fundamental group level and are submersions for certain values of n and k. This paper is devoted to the specific class of pseudoconformal mappings of quaternion and octonion variables. Normal families of functions are defined and investigated. Four criteria of a family being normal are proven. Then groups of pseudoconformal diffeomorphisms of quaternion and octonion manifolds are investigated…
Abstract: Investigates octonion product deformations and related geometries.
problem Exploring geometries and deformations from the 7-sphere S7. method Analyzing the spontaneous compactification M4imesS7 and solutions of Lagrangian equations. result Obtains a family of geometries including those with torsion and G2-structures. We prove that the multiplication maps sn×sn→sn (n=1,3,7) for unit complex, quaternion and octonion numbers are, up to isometries of domain and range, the unique Lipschitz constant minimizers in their homotopy classes. Other geometrically natural maps, such as pro…
Minimal submanifolds in octonionic hyperbolic spaces have large volume.
problem Characterizing minimal submanifolds in locally symmetric spaces.
method Analyzing higher expansion properties and volume constraints.
result Codimension two minimal submanifolds have at least linear volume in the ambient space.
The study connects submanifolds of S7 with octonionic maps and eigenmaps.
problem Characterizing minimal isoparametric submanifolds of S7.
method Using octonionic multiplication and vector bundles, the study establishes conditions for harmonic octonionic maps.
result Minimal isoparametric submanifolds of S7 have specific eigenvalues and eigenvectors related to their octonionic maps.
New octonionic Kähler metrics solve an octonionic Calabi-Yau theorem.
problem Finding metrics on 16D manifolds.
method Introduced octonionic Kähler metrics and solved an octonionic Monge-Ampère equation.
result Solved an octonionic Calabi-Yau theorem.
We use reduced homogeneous coordinates to study Riemannian geometry of the octonionic (or Cayley) projective plane. Our method extends to the para-octonionic (or split octonionic) projective plane, the octonionic projective plane of indefinite signature, and the hyperbolic dual of the octonionic projective plane; we di…
Research examines octonionic slice regular functions and their automorphisms and invariants.
problem Analyzing slice regular functions in the octonionic algebra.
method Investigates automorphisms and invariants of octonionic slice regular functions.
result Characterizes the automorphisms and invariants of octonionic slice regular functions.
The abstract defines G2-structures and connects them to octonion algebras.
problem Classifying G2-structures and understanding their geometric properties. method Established an isomorphism between G2-structures and octonion algebras over C∞(M). result The classification of G2-structures agrees with a parametrisation of octonion algebras with isometric norm. We use a G2-structure on a 7-dimensional Riemannian manifold with a fixed metric to define an octonion bundle with a fiberwise non-associative product. We then define a metric-compatible octonion covariant derivative on this bundle that is compatible with the octonion product. The torsion of the G2-structure is then sh…
Study on octonionic Nahm's equations and their moduli space properties.
problem Properties of octonionic Nahm's equations and their moduli space.
method Analyzing basic properties, constructing solutions, introducing symmetry, proving theorems.
result Moduli space of smooth solutions to octonionic Nahm's equations over [0,1] is a star-shaped smooth manifold.
Study subelliptic heat kernel on octonionic anti-de Sitter space.
problem Heat kernel of octonionic anti-de Sitter space.
method Lift Laplacian of octonionic hyperbolic space and use sub-Laplacian.
result Two integral representations for subelliptic heat kernel.
Constructs projective plane over octonions, proving no higher real division algebras.
problem Existence of higher-dimensional real division algebras.
method Using Adams' solution of the Hopf invariant 1 problem, constructs projective plane over octonions.
result No higher-dimensional real division algebras exist.
The paper explores spinors and polyforms using quaternions and octonions.
problem Understanding spinors and polyforms in Clifford algebras.
method Generalizes Pauli matrices to quaternions and octonions, and relates these to spinor models.
result Explicitly describes Weyl spinors of Spin(4,4) related to quaternions and octonions.
New mechanics on non-associative octonions discovered.
problem Discrete mechanics on non-associative groups.
method Generalized Lagrangian and Hamiltonian mechanics to non-associative objects.
result Discrete mechanics on unitary octonions achieved.
Study of SL(2) over octonions using twistor geometry.
problem No invertible 2x2 matrices over octonions, use Spin(9,1) x SL(2,R) orbit.
method Twistor geometry in eight dimensions.
result Interpretation of open orbit in 32D representation space.
James's octonionic Stiefel spaces questions answered partially.
problem Two fundamental questions about octonionic Stiefel spaces.
method Partial answers to James's questions about octonionic Stiefel spaces.
result Partial answers to James's questions about octonionic Stiefel spaces.
In four-dimensional gauge theory there exists a well-known correspondence between instantons and holomorphic curves, and a similar correspondence exists between certain octonionic instantons and triholomorphic curves. We prove that this latter correspondence stems from the dynamics of various dimensional reductions of …
We prove that Riemannian Spin(7) holonomy manifolds carry octonionic-Kähler structure.
Reviews interactions between Spin(9) and octonionic geometries.
problem Understanding the role of Spin(9) in octonionic geometry.
method Analyzes canonical 8-forms, vector fields, Hopf fibrations, and manifolds.
result Discovers new insights into the geometry of octonionic Hopf fibrations.
We identify R^7 as the pure imaginary part of octonions. Then the multiplication in octonions gives a natural almost complex structure for the unit 6-sphere. It is known that a cone over a surface M in S^6 is an associative submanifold of R^7 if and only if M is almost complex in S^6. In this paper, we show that the Ga…
Harmonic unit normal sections studied for Grassmannians induced by cross products.
problem Energy of maps assigning unit vectors to subspaces of Grassmannians.
method Analyzing cross products to induce harmonic sections into sphere bundles.
result All unit normal sections of Grassmannians associated with cross products are harmonic.
Study subelliptic heat kernel on lifted sphere from octonionic projective space.
problem Analyzing sub-Laplacian on lifted sphere from octonionic projective space.
method Explicit formulas for heat kernel and Green function derived.
result Explicit formulas for heat kernel and Green function.
The dimensions of the spaces of k-homogeneous Spin(9)-invariant valuations on the octonionic plane are computed using results from the theory of differential forms on contact manifolds as well as octonionic geometry and representation theory. Moreover, a valuation on Riemannian manifolds of particular inte…
New formula found for a unique invariant 8-form on Riemannian manifolds with Spin(9) structure.
problem Finding a new explicit algebraic formula for a unique invariant 8-form.
method Generalizing the standard Kähler 2-form expression, constructing the invariant 8-form from octonion-valued coordinate 1-forms.
result A new explicit algebraic formula for the Spin(9)-invariant 8-form. We give an inductive construction for irreducible Clifford systems on Euclidean vector spaces. We then discuss how this notion can be adapted to Riemannian manifolds, and outline some developments in octonionic geometry.
Study of a G2-equivariant octonionic operator and its right spectrum.
problem Understanding the spectrum of a G2-equivariant octonionic operator. method Computed the ordinary real spectrum and analyzed the octonionic right-eigenvalue problem using G2-decomposition and residual symmetry analysis. result Explicit spectral loci (quartic curve and circle) in each complex slice of the octonionic space.
We develop and study quaternionic and octonionic analogies of Cartan angular and Toledo invariants that are well known in the complex hyperbolic space. Using such invariants we study quasifuchsian deformations (including bendings) of quaternionic and octonionic hyperbolic manifolds.
634 vertex-transitive and over 10^103 non-vertex-transitive 27-vertex triangulations of octonionic projective plane.
problem Constructing and classifying triangulations of the octonionic projective plane.
method Combinatorial construction and analysis of symmetry groups.
result Found 634 vertex-transitive and over 10^103 non-vertex-transitive 27-vertex triangulations.
We use the octonion algebra to construct singular solutions of Hessian fully nonlinear uniformly elliptic equations in 21 or more dimensions. The regularity of these solutions is the least possible one. The same is proven for Isaacs equtions.
The paper defines and characterizes 2-Ruled hypersurfaces in Minkowski 4-space using octonions.
problem Characterizing 2-Ruled hypersurfaces in Minkowski 4-space.
method Definition and analysis of 2-Ruled hypersurfaces using octonions.
result Characterizations of Gaussian and mean curvatures of 2-Ruled hypersurfaces.
The study finds Lie algebra formulae and classifies polar actions on a hyperbolic plane.
problem Finding Lie algebra formulae and classifying actions on hyperbolic planes.
method Using octonions and triality, explicit Lie brackets were found for Lie algebras of isometry groups.
result Explicit formulae for Lie brackets of f4 and f4∗ Lie algebras. Study G2-structures in N=1 AdS4 solutions of M-theory.
problem Characterize G2-structures in N=1 AdS4 solutions of M-theory.
method Reformulate Killing spinor equations using octonion bundle structure; study G2-structures and their torsion.
result Define single complexified G2-structure or two real G2-structures on M.
Researchers describe even Clifford structures on specific Grassmannians.
problem Understanding even Clifford structures on Grassmannians.
method Explicit description of structures on real, complex, and quaternionic Grassmannians.
result Explicit description of non-flat parallel even Clifford structures of ranks 8, 6, and 5.
Special orthogonal representations from octonions have geometric properties linked to binary cubics.
problem Understanding geometric properties of special orthogonal representations from octonions.
method Using octonions and their derivations, spinors, and covariants to show geometric properties.
result Covariants and Mathews identities of these representations are related to the Fano plane and (Z2)3. Introduces Plücker coordinates for a complex projective octonion plane, solving an overdetermined system of relations.
problem Understanding the complex projective octonion plane and its quotient space EIII.
method Introduces Plücker coordinates and uses Clifford algebra to solve the overdetermined system of relations.
result Shows that EIII can be decomposed into F4-orbits and provides detailed analysis near the subvariety X∞.
Understanding the exceptional Lie groups as the symmetry groups of simpler objects is a long-standing program in mathematics. Here, we explore one famous realization of the smallest exceptional Lie group, G2. Its Lie algebra acts locally as the symmetries of a ball rolling on a larger ball, but only when the ratio of r…
Formula derived for Lie algebra E8's bracket.
problem Calculating the bracket of the exceptional Lie algebra E8.
method Based on triality and oct-octonions, following Barton-Sudbery description.
result Explicit formula for E8's bracket.
Stable planes are locally isomorphic to classical projective planes.
problem Characterizing stable planes that are locally isomorphic to classical projective planes.
method Analyzing properties of stable planes and comparing them to classical projective planes over specific fields.
result Simply connected stable planes with connected lines are isomorphic to open subplanes of classical projective planes.
Study of Schrödinger flows on S6 using octonions.
problem Schrödinger flows on S6 and related geometric properties. method Using G2-structure on O, study of G2-binormal motion of curves in R7. result Equivalence of G2-binormal motion to Schrödinger flows and nonlinear Schrödinger-type system. The study restricts stable minimal immersions in product spaces to specific configurations.
problem Prohibiting stable minimal immersions in certain product spaces.
method Analyzing stable minimal immersions in products of complex, quaternionic, and octonionic projective spaces.
result The only stable compact minimal immersions in the product of a quaternionic projective space with any other Riemannian manifold are the products of quaternionic projective subspaces with compact stable minimal immersions of the second manifold.
We deal with Riemannian properties of the octonionic Hopf fibration S^{15}-->S^8, in terms of the structure given by its symmetry group Spin(9). In particular, we show that any vertical vector field has at least one zero, thus reproving the non-existence of S^1 subfibrations. We then discuss Spin(9)-structures from a c…
The purpose of this paper is to provide an octonionic description of the Lie group SL(2,O). The main result states that it can be obtained as a free group generated by invertible and determinant preserving transformations from h2(O) onto itself. An interesting characterization is giv…
New triangulations of octonionic projective plane found with restricted symmetry groups.
problem Finding symmetry groups of 27-vertex triangulations of manifolds like the octonionic projective plane.
method Using Smith and Bredon's results on transformation groups to restrict possible symmetry groups.
result List of 26 subgroups of S27 containing all possible symmetry groups of 27-vertex triangulations of manifolds like the octonionic project plane.
New Lie groupoid and algebroid constructed for octonionic Hopf foliation.
problem No known Lie group action generates the singular octonionic Hopf foliation.
method Constructs a G2-equivariant Lie groupoid and Lie algebroid.
result Minimal Lie algebroid and groupoid generate the singular octonionic Hopf foliation.
Witten's approach to Khovanov homology of knots is based on the five-dimensional system of partial differential equations, which we call Haydys-Witten equations. We argue for a one-to-one correspondence between its solutions and solutions of the seven-dimensional system of equations. The latter can be formulated on any…
The octonionic flag manifold Fl(O) is the space of all pairs in OP2×OP2 (where OP2 denotes the octonionic projective plane) which satisfy a certain "incidence" relation. It comes equipped with the projections π1,π2:Fl(O)→OP2, which are $\mat…