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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
Quaternionic_manifold
Type of Riemannian manifold
the quaternionic relations I 2 = J 2 = K 2 = I J K = − 1 {\displaystyle I^{2}=J^{2}=K^{2}=IJK=-1} . In particular, it is a hypercomplex manifold. All
Hyperkähler_manifold
Manifold
manifold or equivalently one whose first Chern class vanishes. Complex dimension Complex analytic variety Quaternionic manifold Real-complex manifold
Complex_manifold
Function theory with quaternion variable
In mathematics, quaternionic analysis is the study of functions with quaternions as the domain and/or range. Such functions can be called functions of
Quaternionic_analysis
differential geometry, a quaternion-Kähler manifold (or quaternionic Kähler manifold) is a Riemannian 4n-manifold whose Riemannian holonomy group is a subgroup
Quaternion-Kähler_manifold
Manifold equipped with a quaternionic structure
not assumed to be integrable, the manifold is called quaternionic, or almost hypercomplex. Every hyperkähler manifold is also hypercomplex. The converse
Hypercomplex_manifold
Four-dimensional number system
Quaternionic manifold – Concept in geometry Quaternionic matrix – Concept in linear algebra Quaternionic polytope – Concept in geometry Quaternionic projective
Quaternion
Concept in mathematics
\mathbb {H} .} Quaternionic projective space of dimension n is usually denoted by H P n {\displaystyle \mathbb {HP} ^{n}} and is a closed manifold of (real)
Quaternionic_projective_space
Manifold of all orthonormal k-frames in n-dimensional Euclidean space
Stiefel manifold V k ( C n ) {\displaystyle V_{k}(\mathbb {C} ^{n})} of orthonormal k-frames in C n {\displaystyle \mathbb {C} ^{n}} and the quaternionic Stiefel
Stiefel_manifold
Smooth manifold
vanishing pure spinor then M is a generalized Calabi–Yau manifold. Almost quaternionic manifold – Concept in geometryPages displaying short descriptions
Almost_complex_manifold
RPn Complex projective space, CPn Quaternionic projective space, HPn Flag manifold Grassmann manifold Stiefel manifold Lie groups provide several interesting
List_of_manifolds
Type of topological space
compact manifolds. Real projective space RPn is a n-dimensional manifold. Complex projective space CPn is a 2n-dimensional manifold. Quaternionic projective
Topological_manifold
Even-dimensional Hopf manifolds admit hypercomplex structure. The Hopf surface is the only compact hypercomplex manifold of quaternionic dimension 1 which
Hopf_manifold
Smooth manifold with an inner product on each tangent space
hyperbolic space, quaternionic hyperbolic space, and Cayley hyperbolic space, which are instead analogues of hyperbolic space. Grassmannian manifolds also carry
Riemannian_manifold
{\displaystyle \mathbb {CP} ^{2}} ( n = 4 {\displaystyle n=4} ), of the quaternionic projective plane H P 2 {\displaystyle \mathbb {HP} ^{2}} ( n = 8 {\displaystyle
Eells–Kuiper_manifold
Type of Riemannian manifold with constant Jacobi operator spectrum
mathematics, particularly in differential geometry, an Osserman manifold is a Riemannian manifold in which the characteristic polynomial of the Jacobi operator
Osserman_manifold
Concept in differential geometry
incompatibility (help) Kraines, Vivian Yoh (1965), "Topology of quaternionic manifolds", Bull. Amer. Math. Soc., 71, 3, 1 (3): 526–7, doi:10.1090/s0002-9904-1965-11316-7
Holonomy
Differential geometry concept
the simple complex Lie groups. Quaternionic discrete series representation Besse, Arthur L. (2008), Einstein Manifolds, Classics in Mathematics, Berlin:
Quaternion-Kähler symmetric space
Quaternion-Kähler_symmetric_space
Study of complex manifolds and several complex variables
the quaternionic relations I 2 = J 2 = K 2 = I J K = − Id {\displaystyle I^{2}=J^{2}=K^{2}=IJK=-\operatorname {Id} } . Thus, hyper-Kähler manifolds are
Complex_geometry
Special tangential 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
Spinh_structure
Mathematical group
\operatorname {Sp} (n)} is given by the quaternionic skew-Hermitian matrices, the set of n × n {\displaystyle n\times n} quaternionic matrices that satisfy A + A
Symplectic_group
Structure group sub-bundle on a tangent frame bundle
In differential geometry, a G-structure on an n {\displaystyle n} -manifold M {\displaystyle M} , for a given structure group G {\displaystyle G} , is
G-structure_on_a_manifold
Describes a periodicity in the homotopy groups of classical groups
theories, (real) KO-theory and (quaternionic) KSp-theory, associated to the real orthogonal group and the quaternionic symplectic group, respectively.
Bott_periodicity_theorem
mathematics, a hypertoric variety or toric hyperkähler variety is a quaternionic analog of a toric variety constructed by applying the hyper-Kähler quotient
Hypertoric_variety
multiple of the metric. Hyperbolic space Quaternionic hyperbolic space Besse, Arthur (1987), Einstein manifolds, Springer, p. 180. Cano, Navarrete & Seade
Complex_hyperbolic_space
American mathematician
Zbl 0553.32008. Galicki, K.; Lawson, H. Blaine Jr. (1988). "Quaternionic reduction and quaternionic orbifolds". Mathematische Annalen. 282 (1): 1–21. doi:10
H._Blaine_Lawson
Non-tensorial representation of the spin group
conditions. When S {\displaystyle S} is of quaternionic type, the representation carries an invariant quaternionic structure but no invariant real structure
Spinor
Mathematical object
of cardinal directions. This means that a 3-sphere is an example of a 3-manifold. In coordinates, a 3-sphere with center (C0, C1, C2, C3) and radius r is
3-sphere
French mathematician
equations for almost quaternionic Hermitian manifolds", Complex Structures and Vector Fields: 114–135. Dominic Joyce, Compact manifolds with special holonomy
Edmond_Bonan
Characteristic class for real vector bundles
giving the signature see Hirzebruch signature theorem. There is also a quaternionic Pontryagin class, for vector bundles with quaternion structure. Chern–Simons
Pontryagin_class
Riemannian manifold equipped with a differential p-form
Paris. 260: 5445–5448. Kraines, Vivian Yoh (1965). "Topology of quaternionic manifolds". Bull. Amer. Math. Soc. 71, 3, 1 (3): 526–527. doi:10
Calibrated_geometry
Space of vacuum states
N=2 Supergravity that in this case, the Higgs branch must be a quaternionic Kähler manifold. In extended supergravities with N>2 the moduli space must always
Moduli_(physics)
Z2,0, repeated. KSp0(X) is the ring of stable equivalence classes of quaternionic vector bundles over X. Bott periodicity implies that the K-groups have
List_of_cohomology_theories
Form of differential geometry
mathematics, systolic geometry is the study of systolic invariants of manifolds and polyhedra, as initially conceived by Charles Loewner and developed
Systolic_geometry
Russian-French mathematician
isometry group of the quaternionic hyperbolic space are arithmetic.[GS92] In 1978, Gromov introduced the notion of almost flat manifolds.[G78] The famous quarter-pinched
Mikhael Gromov (mathematician)
Mikhael_Gromov_(mathematician)
Ring homomorphism from the cobordism ring of manifolds to another ring
homomorphism from the ring of smooth compact manifolds up to the equivalence of bounding a smooth manifold with boundary (i.e., up to suitable cobordism)
Genus of a multiplicative sequence
Genus_of_a_multiplicative_sequence
Theory of supergravity in four dimensions
the relevant scalar manifold must be a quaternionic Kähler manifold. But since these manifolds are not themselves Kähler manifolds, they cannot occur as
4D_N_=_1_supergravity
Relates the geometric vector bundles to algebraic projective modules
variant, concerning smooth vector bundles on a smooth manifold (real, complex, or quaternionic). His topological variant is about continuous (real or
Serre–Swan_theorem
Mathematical result in differential geometry
(1963), states that for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the space of solutions)
Atiyah–Singer_index_theorem
Fiber bundle of the 3-sphere over the 2-sphere, with 1-spheres as fibers
\mathbb {CP} ^{n}} with circles as fibers, and there are also real, quaternionic, and octonionic versions of these fibrations. In particular, the Hopf
Hopf_fibration
Mathematics term
≥ 2. For n ≥ 2, the noncompact Lie group Sp(n, 1) of isometries of a quaternionic hermitian form of signature (n,1) is a simple Lie group of real rank
Kazhdan's_property_(T)
Fiber bundle whose fibers are group torsors
S^{4n+3}} is a principal S p ( 1 ) {\displaystyle Sp(1)} -bundle over quaternionic projective space H P n {\displaystyle \mathbb {H} \mathbb {P} ^{n}}
Principal_bundle
Special type of principal bundle
four-dimensional sphere S 4 {\displaystyle S^{4}} , which include the quaternionic Hopf fibration, can be used to describe hypothetical magnetic monopoles
Principal_SU(2)-bundle
itself. Analogous Veronese embeddings are constructed for complex and quaternionic projective spaces, as well as for the Cayley plane. Lectures on Discrete
Veronese_map
Mathematical concept
Projective Hilbert space Quaternionic projective space Real projective space Complex affine space K3 surface Besse, Arthur L. (1978), Manifolds all of whose geodesics
Complex_projective_space
(pseudo-)Riemannian manifold whose geodesics are reversible
mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of isometries contains an inversion
Symmetric_space
Supergravity in eleven dimensions
squashed 7-sphere, which can be acquired by embedding the 7-sphere in a quaternionic projective space, with this giving a gauge group of SO ( 5 ) × SU ( 2
Eleven-dimensional supergravity
Eleven-dimensional_supergravity
Hypercomplex number system
been shown that the pairs of zero divisors in the unit sedonions form a manifold isomorphic to the Lie group G2 in the space S 2 {\displaystyle \mathbb
Sedenion
Fundamental construction of differential calculus
derivative corresponds to the integral, whence the term differintegral. In quaternionic analysis, derivatives can be defined in a similar way to real and complex
Generalizations of the derivative
Generalizations_of_the_derivative
of harmonic spinors on a spin manifold. In 3 and 4 dimensions Clifford analysis is sometimes referred to as quaternionic analysis. When n = 4, the Dirac
Clifford_analysis
geometry used to describe the physical phenomena of quantum physics Quaternionic analysis Ramsey theory the study of the conditions in which order must
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Metric on a complex projective space endowed with Hermitian form
(2n+1)-sphere. In algebraic geometry, one uses a normalization making CPn a Hodge manifold. The Fubini–Study metric arises naturally in the quotient space construction
Fubini–Study_metric
Regular object in four dimensional geometry
cell ring as a polytope in its own right which fills a three-dimensional manifold (such as the 3-sphere) with its corresponding honeycomb. He found that
24-cell
Connected non-abelian Lie group lacking nontrivial connected normal subgroups
ISBN 978-1-4612-0979-9. Yokota, Ichiro (2009). Exceptional Lie Groups. Besse, Einstein Manifolds. ISBN 0-387-15279-2 Helgason, Differential Geometry, Lie groups, and Symmetric
Simple_Lie_group
Super vector space forming base superspace for supersymmetric field theories
supersymmetric extension of Minkowski space, sometimes used as the base manifold (or rather, supermanifold) for superfields. It is acted on by the super
Super_Minkowski_space
Completion of the usual space with "points at infinity"
naturally to the case where K is a division ring; see, for example, Quaternionic projective space. The notation PG(n, K) is sometimes used for Pn(K).
Projective_space
Moduli space of the Yang–Mills equations
the Donaldson invariants used to study four-dimensional smooth manifolds (short 4-manifolds). A difficulity is, that the Yang–Mills moduli space is usually
Yang–Mills_moduli_space
concretely, in a real projective space, complex projective space or quaternionic projective space K P n {\displaystyle \mathbb {KP} ^{n}} where K {\displaystyle
Stunted_projective_space
Generalized sphere of dimension n (mathematics)
-sphere, Lie group structure Sp(1) = SU(2). 4-sphere Homeomorphic to the quaternionic projective line, H P 1 {\displaystyle \mathbf {HP} ^{1}} . SO
N-sphere
Vector bundle of rank 1
H^{2}(X)} (integral cohomology). There is a further, analogous theory with quaternionic (real dimension four) line bundles. This gives rise to one of the Pontryagin
Line_bundle
Type of topological space
n + 1 . {\displaystyle \mathbb {R} ^{n+1}.} It is a compact, smooth manifold of dimension n, and is a special case G r ( 1 , R n + 1 ) {\displaystyle
Real_projective_space
Geometric model of the physical space
models physical space. More general three-dimensional spaces are called 3-manifolds. The term may refer colloquially to a subset of space, a three-dimensional
Three-dimensional_space
Double cover Lie group of the special orthogonal group
mathematics the spin group, denoted Spin(n), is a Lie group whose underlying manifold is the double cover of the special orthogonal group SO(n) = SO(n, R), such
Spin_group
Topologist
Washington (September 1989). "Nonlinearly Equivalent Representations of Quaternionic 2-Groups" (PDF). Transactions of the American Mathematical Society. 315
Washington_Mio
Geometric concept of a 2D space with "points at infinity" adjoined
pappian planes) serve as fundamental examples in algebraic geometry. The quaternionic projective plane HP2 is also of independent interest. By Wedderburn's
Projective_plane
Hypercomplex number system
basis with signature (− − − −) and is given in terms of the following 7 quaternionic triples (omitting the scalar identity element): ( I , j , k ) , ( i
Octonion
Optimal stable 2-systolic inequality
attained by the symmetric metric of the projective plane. Meanwhile, in the quaternionic case, the symmetric metric on H P 2 {\displaystyle \mathbb {HP} ^{2}}
Gromov's inequality for complex projective space
Gromov's_inequality_for_complex_projective_space
American mathematician (1936–2023)
arXiv:math/0402283 Complex forms of quaternionic symmetric spaces, in Complex, contact and symmetric manifolds, Progress in Mathematics 234, Birkhäuser
Joseph_A._Wolf
function whose domain is the entire complex plane. Quaternionic function: a function whose domain is quaternionic. Hypercomplex function: a function whose domain
List_of_types_of_functions
Element of a unital algebra over the field of real numbers
{\displaystyle \mathbb {H} ^{\otimes 3}=M(4,\mathbb {H} )} yields a quaternionic matrix and its even subalgebra H ⊗ 2 ⊗ R C {\displaystyle \mathbb {H}
Hypercomplex_number
Algebraic structure designed for geometry
analysis, developed out of quaternionic analysis in the late 19th century by Gibbs and Heaviside. The legacy of quaternionic analysis in vector analysis
Geometric_algebra
German mathematician
mathematician specialising in differential geometry and known for her work on quaternionic analysis and Willmore surfaces. She works in England as a reader in mathematics
Katrin_Leschke
Not-necessarily-associative commutative algebra satisfying (xy)(xx) = x(y(xx))
sometimes denoted H(A,σ). 1. The set of self-adjoint real, complex, or quaternionic matrices with multiplication ( x y + y x ) / 2 {\displaystyle (xy+yx)/2}
Jordan_algebra
operators on an infinite-dimensional real, complex or quaternionic Hilbert space. The quaternionic space is defined as all sequences x = (xi) with xi in
Jordan_operator_algebra
Mathematical concept
and Kottwitz (2005) Harry Reimann, The semi-simple zeta function of quaternionic Shimura varieties, Lecture Notes in Mathematics, 1657, Springer, 1997
Shimura_variety
Group of unitary matrices
Classical Mechanics (Second ed.). Springer. p. 225. Baez, John. "Symplectic, Quaternionic, Fermionic". Retrieved 1 February 2012. Grove (2002), Theorem 10.3. Grove
Unitary_group
Lie groups and their associated Lie algebras
of G. Note that a "complex Lie group" is defined as a complex analytic manifold that is also a group whose multiplication and inversion are each given
Table_of_Lie_groups
Generalization of the concept of directional derivative
Generalization of a derivative of a function between two Banach spaces Quaternionic analysis – Function theory with quaternion variable Semi-differentiability –
Gateaux_derivative
Fringe theory of physics
single Lie group geometry—specifically, excitations of the noncompact quaternionic real form of the largest simple exceptional Lie group, E8. A Lie group
An Exceptionally Simple Theory of Everything
An_Exceptionally_Simple_Theory_of_Everything
Italian mathematician (1911–1999)
; Pontecorvo, M., eds. (1999), Proceedings of the Second Meeting on Quaternionic Structures in Mathematics and Physics. Dedicated to the Memory of André
Enzo_Martinelli
Discrete subgroup in a locally compact topological group
differential geometry (through the construction of locally homogeneous manifolds), in number theory (through arithmetic groups), in ergodic theory (through
Lattice_(discrete_subgroup)
Four-dimensional analog of the dodecahedron
S2CID 119288632. Koca, Mehmet; Al-Ajmi, Mudhahir; Ozdes Koca, Nazife (2011). "Quaternionic representation of snub 24-cell and its dual polytope derived from E8
120-cell
Matrix representing a Euclidean rotation
\mathrm {SO} (3).} For a detailed account of the SU(2)-covering and the quaternionic covering, see spin group SO(3). Many features of these cases are the
Rotation_matrix
researcher Katrin Leschke (born 1968), German differential geometer, quaternionic analyst, and minimal surface theorist Nandi Olive Leslie, American industrial
List_of_women_in_mathematics
Four-dimensional associative algebra over the reals
Mohaupt 2006) Manifolds with para-quaternionic structures are studied in differential geometry and string theory. In the para-quaternionic literature, k
Split-quaternion
Four-dimensional analog of the icosahedron
Cartesian coordinate — the 120 vertices of the 600-cell form a group under quaternionic multiplication. This group is often called the binary icosahedral group
600-cell
Development of linear transformations forming the Lorentz group
2}\end{aligned}}\end{matrix}}} Arthur W. Conway in February 1911 explicitly formulated quaternionic Lorentz transformations of various electromagnetic quantities in terms
History of Lorentz transformations
History_of_Lorentz_transformations
American mathematician
MR 0133834. Floyd, E. E. (1971). "Stiefel-Whitney numbers of quaternionic and related manifolds". Trans. Amer. Math. Soc. 155: 77–94. doi:10.1090/s0002-9947-1971-0273632-8
Edwin_E._Floyd
Representation theory
the Weyl group of A. The group G = SL(2,C) acts transitively on the quaternionic upper half space H 3 = { x + y i + t j ∣ t > 0 } {\displaystyle {\mathfrak
Plancherel theorem for spherical functions
Plancherel_theorem_for_spherical_functions
QUATERNIONIC MANIFOLD
QUATERNIONIC MANIFOLD
Boy/Male
Indian, Sanskrit
Manifold; Multiplied
Boy/Male
Hindu, Indian
Manifoldness; Variety
Girl/Female
Hindu, Indian, Marathi, Sanskrit, Tamil
Manifold; Variegated
Biblical
a guard of four soldiers,...and delivered him to four quaternions of soldiers to guard him...
Boy/Male
Indian, Sanskrit
Plenty; Much; Strong; Manifold
Surname or Lastname
English
English : unexplained. It may be a variant of Minnifield, which is likewise unexplained.
QUATERNIONIC MANIFOLD
QUATERNIONIC MANIFOLD
Boy/Male
Hindu, Indian, Sanskrit
This
Boy/Male
Hindu
Girl/Female
Arabic, Muslim
Greenness
Girl/Female
English American Russian
Abbreviation of Antonia and Antoinette.
Boy/Male
Gujarati, Hindu, Indian, Kannada
Shining; Beaming; Radiant
Boy/Male
American, Australian, Chinese, French, German, Latin, Portuguese, Spanish, Swiss, Teutonic
Of the Army; French Form of Herman; Army Man; Soft Air; Gentle Breeze; Similar to Herman; Form of Armand
Boy/Male
Hindu
To rise, Blue lotus
Girl/Female
Greek
Form of Danae; the mythological mother of Perseus by Zeus.
Boy/Male
Indian
Awf guest, Fragrance, Lion
Girl/Female
Tamil
Yuvrani | யà¯à®µà®°à®¾à®¨à¯€
Young queen, Princess
QUATERNIONIC MANIFOLD
QUATERNIONIC MANIFOLD
QUATERNIONIC MANIFOLD
QUATERNIONIC MANIFOLD
QUATERNIONIC MANIFOLD
imp. & p. p.
of Manifold
n.
A generalized concept of magnitude.
v. t.
To divide into quaternions, files, or companies.
n.
The number four.
n.
The turning factor of a quaternion.
n.
A set of four parts, things, or person; four things taken collectively; a group of four words, phrases, circumstances, facts, or the like.
a.
Exhibited at divers times or in various ways; -- used to qualify nouns in the singular number.
v. t.
To take copies of by the process of manifold writing; as, to manifold a letter.
p. pr. & vb. n.
of Manifold
n.
A cylindrical pipe fitting, having a number of lateral outlets, for connecting one pipe with several others.
n.
In the quaternion analysis, a quantity that has magnitude, but not direction; -- distinguished from a vector, which has both magnitude and direction.
adv.
In a manifold manner.
n.
The third stomach of a ruminant animal.
n.
A copy of a writing made by the manifold process.
a.
Having many folds, layers, or plates; as, a manifolded shield.
n.
The quotient of two vectors, or of two directed right lines in space, considered as depending on four geometrical elements, and as expressible by an algebraic symbol of quadrinomial form.
n.
Multiplicity.
a.
Different; diverse; several; manifold; as, men of various names; various occupations; various colors.
n.
A word of four syllables; a quadrisyllable.
n.
The number four; a collection of four things; a quaternion.