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Concept in differential geometry
In differential geometry, the holonomy of a connection on a smooth manifold is the extent to which parallel transport around closed loops fails to preserve
Holonomy
Riemannian manifold with SU(n) holonomy
complex structure. M {\displaystyle M} has a Kähler metric with global holonomy contained in S U ( n ) {\displaystyle \mathrm {SU} (n)} . These conditions
Calabi–Yau_manifold
Type of optimization problem
holonomy. Similarly, the concept of non/holonomy in mechanics is related to geometric phases, since those are also described by nontrivial holonomy in
Nonholonomic_system
Seven-dimensional Riemannian manifold
manifold or Joyce manifold is a seven-dimensional Riemannian manifold with holonomy group contained in G2. The group G 2 {\displaystyle G_{2}} is one of the
G2_manifold
Type of Riemannian manifold
classification of Riemannian holonomy groups first raised the issue of the existence of non-symmetric manifolds with holonomy Sp(n)·Sp(1). Interesting results
Hyperkähler_manifold
Type of geometry in mathematics
restricted holonomy group is contained in the symplectic group. A G2 manifold or Spin(7) manifold is a Riemannian manifold whose holonomy group is contained
Ricci-flat_manifold
Topics referred to by the same term
Look up holonomic or holonomy in Wiktionary, the free dictionary. Holonomic (introduced by Heinrich Hertz in 1894 from the Greek ὅλος meaning "whole",
Holonomic
quaternionic Kähler manifold) is a Riemannian 4n-manifold whose Riemannian holonomy group is a subgroup of Sp(n)·Sp(1) for some n ≥ 2 {\displaystyle n\geq
Quaternion-Kähler_manifold
Quantum interpretation of neuroscience
within the larger workings of the brain. This patch holography is called holonomy or windowed Fourier transformations. A holographic model can also account
Holonomic_brain_theory
Closed flat 3-manifold
It is the only closed flat 3-manifold with first Betti number zero. Its holonomy group is Z 2 2 {\displaystyle \mathbb {Z} _{2}^{2}} . It has been suggested
Hantzsche–Wendt_manifold
Eight-dimensional Riemannian manifold
a Spin(7)-manifold is an eight-dimensional Riemannian manifold whose holonomy group is contained in Spin(7). Spin(7)-manifolds are Ricci-flat and admit
Spin(7)-manifold
American mathematician and billionaire (1938–2024)
of Bertram Kostant, gave a new proof of Berger's classification of the holonomy groups of Riemannian manifolds. He subsequently began to work with Shiing-Shen
Jim_Simons
Mathematical measure of how much a curve or surface deviates from flatness
phenomenon is known as holonomy. Various generalizations capture in an abstract form this idea of curvature as a measure of holonomy; see curvature form
Curvature
Chinese-American mathematician (born 1949)
closed manifolds of special holonomy; any simply-connected closed Kähler manifold which is Ricci flat must have its holonomy group contained in the special
Shing-Tung_Yau
Theory of quantum gravity merging quantum mechanics and general relativity
gauge theories and quantum gravity. LQG includes the concept of a holonomy. A holonomy is a measure of how much the initial and final values of a spinor
Loop_quantum_gravity
Defines a notion of parallel transport on a bundle
to x {\displaystyle x} . Parallel transport can be used to define the holonomy group of the connection ∇ {\displaystyle \nabla } based at a point x {\displaystyle
Connection_(vector_bundle)
Simple Lie group; the automorphism group of the octonions
possible special groups that can appear as the holonomy group of a Riemannian metric. The manifolds of G2 holonomy are also called G2-manifolds. G2 is the automorphism
G2_(mathematics)
In mathematics, a partition of a manifold into submanifolds
is called the holonomy of the foliation. Holonomy is implemented on foliated manifolds in various specific ways: the total holonomy group of foliated
Foliation
British mathematician
Compact Manifolds with special holonomy. Oxford University Press. 2000. ISBN 978-0-19-850601-0. Riemannian holonomy groups and calibrated geometry. Oxford
Dominic_Joyce
American mathematician (born 1953)
exceptional holonomy (i.e. whose holonomy groups are G2 or Spin(7)); this showed that every group in Marcel Berger's classification can arise as a holonomy group
Robert_Bryant_(mathematician)
(pseudo-)Riemannian manifold whose geodesics are reversible
tools of Riemannian geometry, leading to consequences in the theory of holonomy; or algebraically through Lie theory, which allowed Cartan to give a complete
Symmetric_space
Concept in differential geometry
weaker than the existence of a metric of holonomy G 2 {\displaystyle G_{2}} , because a compact 7-manifold of holonomy G 2 {\displaystyle G_{2}} must also
G2-structure
Swiss mathematician
a Riemannian product automatically implies a product structure of the holonomy groups. In 1952 De Rham considered the converse, proving that, if there
Georges_de_Rham
Smooth manifold with an inner product on each tangent space
curvature. These spaces are important from the perspective of Riemannian holonomy. As found in the 1950s by Marcel Berger, any Riemannian manifold which
Riemannian_manifold
Description of gauge theories using loop operators
gauge theories in terms of extended objects such as Wilson loops and holonomies. The loop representation is a quantum hamiltonian representation of gauge
Loop representation in gauge theories and quantum gravity
Loop_representation_in_gauge_theories_and_quantum_gravity
American mathematician
Alma mater University of Illinois at Urbana–Champaign Known for Ambrose–Singer holonomy theorem Awards John Simon Guggenheim Fellow (1947) Scientific career Fields
Warren_Ambrose
(principal bundle) Ehresmann connection curvature curvature form holonomy, local holonomy Chern–Weil homomorphism Curvature vector Curvature form Curvature
List of differential geometry topics
List_of_differential_geometry_topics
Concept in mathematics and theoretical physics
Euclidean (ALE) metric on the cotangent bundle of the 2-sphere T*S2. The holonomy group of this 4-real-dimensional manifold is SU(2). The metric is generally
Eguchi–Hanson_space
French mathematician
is a French mathematician, known particularly for his work on special holonomy. Although not a single example of G2 manifold or Spin(7) manifold had been
Edmond_Bonan
Study of vector bundles, principal bundles, and fibre bundles
observed that since the Yang–Mills connections are projectively flat, their holonomy gives projective unitary representations of the fundamental group of the
Gauge_theory_(mathematics)
Differential geometry construct on fiber bundles
78[page needed]. Holonomy for Ehresmann connections in fiber bundles is sometimes called the Ehresmann-Reeb holonomy or leaf holonomy in reference to the
Ehresmann_connection
System of moving vectors in differential geometry
the curvature known as holonomy. The Ambrose–Singer theorem makes explicit this relationship between the curvature and holonomy. Other notions of connection
Parallel_transport
Function in mathematics
theory concerns itself primarily with notions of parallel transport and holonomy. The infinitesimal theory concerns itself with the differentiation of geometric
Connection_(mathematics)
Metric on a determinant line bundle
by Bismut and Freed to compute the holonomy of certain determinant line bundles of Dirac operators, and this holonomy is associated to certain anomaly cancellations
Quillen_metric
(holikós) catholic, holiatry, holism, holistic, holography, holomorphic, holonomy hom- same Greek ὁμός (homós) homiletic, homily, homogeneous, homograph
List of Greek and Latin roots in English/H–O
List_of_Greek_and_Latin_roots_in_English/H–O
French mathematical physicist (1915–1998)
He made pioneering contributions to the theory of the scalar curvature, holonomy groups, Kähler geometry, and the mathematical study of Einstein's equations
André_Lichnerowicz
Tensor in differential geometry
Kähler manifolds already possess holonomy in U ( n ) {\displaystyle \mathrm {U} (n)} , and so the (restricted) holonomy of a Ricci-flat Kähler manifold
Ricci_curvature
Actions in the present are dependent on previous decisions or experiences
Bryant, Robert L. (2006). "Geometry of manifolds with special holonomy: '100 years of holonomy'". 150 years of mathematics at Washington University in St
Path_dependence
Topological quantum field theory
viewed as the Lagrangian integral of the Chern–Simons form and Wilson loop, holonomy of vector bundle on M. These explain why the Chern–Simons theory is closely
Chern–Simons_theory
Space of possible positions for all objects in a physical system
end-effector. This definition, however, leads to complexities described by the holonomy: that is, there may be several different ways of arranging a robot arm
Configuration_space_(physics)
Lowest possible energy of a quantum system or field
March 2009). "Electromagnetic to Gravitational wave Conversion via Nuclear Holonomy". AIP Conference Proceedings. 1103 (1): 524–531. Bibcode:2009AIPC.1103
Zero-point_energy
Generalization of the Levi-Civita connection
a Weyl connection if and only if its holonomy group is a subgroup of the conformal group. The possible holonomy algebras in Lorentzian signature were
Weyl_connection
Intrinsic geometric structures in mathematics
favoured by French authors. Lifts of loops about a point give rise to the holonomy group at that point. The Gaussian curvature at a point can be recovered
Riemannian connection on a surface
Riemannian_connection_on_a_surface
6-manifolds for which the corresponding 7-dimensional Riemannian cone has holonomy G2. The only compact simply connected 6-manifolds known to admit strict
Nearly_Kähler_manifold
Field theory involving topological effects in physics
Borel–Weil–Bott theorem. The Lagrangian of these theories is the classical action (holonomy of the line bundle). Thus topological QFT's with d = 0 relate naturally
Topological quantum field theory
Topological_quantum_field_theory
English mathematician (born 1957)
Imperial College "Some recent developments in Kähler geometry and exceptional holonomy – Simon Donaldson – ICM2018". YouTube. 19 September 2018. (Plenary Lecture
Simon_Donaldson
Mathematical theory
M {\displaystyle M} and L {\displaystyle L} a compact leaf with finite holonomy group. There exists a neighborhood U {\displaystyle U} of L {\displaystyle
Reeb_stability_theorem
Causal relationships between points in a manifold
everywhere future-directed null (or everywhere past-directed null). The holonomy of the ratio of the rate of change of the affine parameter around a closed
Causal_structure
Hypothetical particle with one magnetic pole
charge. During the lassoing, the holonomy changes by the amount of magnetic flux through the sphere. Since the holonomy at the beginning and at the end
Magnetic_monopole
Measure of curvature in differential geometry
strongly scalar-flat, M must be a product of Riemannian manifolds with holonomy group SU(n) (Calabi–Yau manifolds), Sp(n) (hyperkähler manifolds), or Spin(7)
Scalar_curvature
Tensor field in Riemannian geometry
this in a general Riemannian manifold. This failure is known as the non-holonomy of the manifold. Let x t {\displaystyle x_{t}} be a curve in a Riemannian
Riemann_curvature_tensor
Gauge field loop operator
known as the holonomy, which describes a mapping of the fiber into itself upon horizontal lift along a closed loop. The set of all holonomies itself forms
Wilson_loop
Electromagnetic quantum-mechanical effect in regions of zero magnetic and electric field
derivation on the Minkowski space. The monodromy is the holonomy of the flat connection. The holonomy of a connection, flat or non flat, around a closed loop
Aharonov–Bohm_effect
Theory in theoretical physics
has attracted much interest is topological M-theory on a space with G2 holonomy and the A-model on a Calabi–Yau. In this case, the M2-branes wrap associative
Topological_string_theory
Metric space
only one natural connection. The concept of holonomy is strikingly simple in this case. The restricted holonomy group is trivial, and so there is a homomorphism
Polyhedral_space
Type of mathematical space
2002. Jürgen Berndt, Sergio Console and Carlos Olmos, Submanifolds and Holonomy, Chapman & Hall/CRC Press, 2003. Michel Brion, Lectures on the geometry
Generalized_flag_variety
Branch of both theoretical and applied psychology
the psychology of stress. Mit Press. ISBN 0262570270. Stamps, J (1980). Holonomy: A Human Systems Theory. Intersystyems publications. ISBN 0914105175. Tapu
Systems_psychology
Nonlinear generalizations of Maxwell electrodynamics
formulations also consider nonlocal extensions involving fractional U(1) holonomies on twistor space,[citation needed] though these remain speculative. Sorokin
Nonlinear_electrodynamics
Mathematical group formed from the automorphisms of an object
automorphism group Level structure, a technique to remove an automorphism group Holonomy group First, if G is simply connected, the automorphism group of G is that
Automorphism_group
Branch of mathematics that studies abstract algebraic structures
link representation theory and invariant theory to areas as diverse as holonomy, differential operators and the theory of several complex variables. Automorphic
Representation_theory
Approach to the study of finite semigroups and automata
Zeiger (1967) proved an important variant called the holonomy decomposition (Eilenberg 1976). The holonomy method appears to be relatively efficient and has
Krohn–Rhodes_theory
American mathematician
Q-curvatures". Conformal Differential Geometry: Q-Curvature and Conformal Holonomy. Oberwolfach Seminars. Vol. 40. Springer. pp. 21ff. ISBN 9783764399092
C._Robin_Graham
dimension Hausdorff distance Hausdorff measure Hilbert space Hölder map Holonomy group is the subgroup of isometries of the tangent space obtained as parallel
Glossary of Riemannian and metric geometry
Glossary_of_Riemannian_and_metric_geometry
Topologically stable solution of a partial differential equation
behavior of the objects themselves are often described in terms of the holonomy and the monodromy. In abstract settings such as string theory, solitons
Topological_defect
German mathematical physicist
began in 2016. In 2020 she joined the Simons Collaboration on Special Holonomy in Geometry, Analysis, and Physics as one of its Principal Investigators
Sakura_Schafer-Nameki
Manifold with Riemannian, complex and symplectic structure
manifold X {\displaystyle X} of even dimension 2 n {\displaystyle 2n} whose holonomy group is contained in the unitary group U ( n ) {\displaystyle \operatorname
Kähler_manifold
Electromagnetism in general relativity
tensor is the holonomy of the Levi-Civita connection along an infinitesimal closed curve, the curvature of the connection is the holonomy of the U(1)-connection
Maxwell's equations in curved spacetime
Maxwell's_equations_in_curved_spacetime
Partial differential equations whose solutions are instantons
irreducible connections, that is, connections A {\displaystyle A} whose holonomy group is given by all of G {\displaystyle G} , one does obtain Hausdorff
Yang–Mills_equations
Independent video game
straight lines which never cross. There is also one land that relies on the holonomy of hyperbolic geometry: when the player returns to a tile after making
HyperRogue
American mathematician (1931–2019)
he stayed until his retirement in 1997. His thesis, On Imbeddings and Holonomy, was supervised by Isadore Singer. At UIUC, his doctoral students included
Richard_L._Bishop
theorem of Riemannian geometry Riemannian graph Riemannian group Riemannian holonomy Riemannian manifold also called Riemannian space Riemannian metric tensor
List of things named after Bernhard Riemann
List_of_things_named_after_Bernhard_Riemann
American annual mathematics conference
normal bundle Robert Bryant, The construction of metrics with exceptional holonomy Francis Bonahon, Hyperbolic 3-manifolds with arbitrarily short geodesics
Geometry_Festival
Phase of a cycle
nonsingular states will not be simply connected, or there will be a nonzero holonomy. Waves are characterized by amplitude and phase, and may vary as a function
Geometric_phase
Procedure of ordering a product operators
path-ordered exponential to guarantee that the Wilson loop encodes the holonomy of the gauge connection. The parameter σ that determines the ordering is
Path-ordering
Basic question in geometry and topology
cobordism have been classified by Matthias Kreck The Berger classification of holonomy groups. Frolík, Zdeněk (1962). "On the classification of 1-dimensional
Classification_of_manifolds
American mathematician (1924–2021)
other notable contributions in mathematics include the Ambrose–Singer holonomy theorem and the McKean–Singer theorem. Singer was a member of the National
Isadore_Singer
Introduction and Reference on Differential Geometry
analysis, connections in bundles, and the role of Lie groups. It also covers holonomy, the de Rham decomposition theorem and the Hopf–Rinow theorem. According
Foundations of Differential Geometry
Foundations_of_Differential_Geometry
First-order method for approximating parallel transport of a vector along a curve
Levi-Civita parallelogramoid. In a curved space, the error is given by holonomy around the triangle A 1 A 0 X 0 , {\displaystyle A_{1}A_{0}X_{0},} which
Schild's_ladder
French mathematician (1869–1951)
Generalised spaces with structure groups and connections, Cartan connection, holonomy, Weyl tensor Geometry and topology of Lie groups Riemannian geometry Symmetric
Élie_Cartan
Affine connection on the tangent bundle of a manifold
parallel transport with the curvature, thus developing the modern notion of holonomy. In 1869, Christoffel discovered that the components of the intrinsic derivative
Levi-Civita_connection
German mathematician
contributions to differential geometry, in particular manifolds with special holonomy and on non-integrable geometric structures and for service to the mathematical
Ilka_Agricola
polycyclic subgroup of finite index. Known in dimensions up to 6, and when the holonomy of the flat connection preserves a Lorentz metric. Since every virtually
Affine_manifold
Fringe hypothesis
Organization, Vol. 2, pp. 33–60. PRIBRAM, K. H. (1991) Brain and Perception: Holonomy and Structure in Figural Processing. New Jersey: Lawrence Erlbaum Associates
Quantum_mind
Mathematical behavior near singularities
effect when applied to loops based at m {\displaystyle m} is to define a holonomy group of translations of the fiber at m {\displaystyle m} ; if the structure
Monodromy
Mathematical concept in differential geometry
symmetric space. Berndt, J; Olmos, C; Console, S. (2003). "Submanifolds and holonomy", Chapman & Hall/CRC, Research Notes in Mathematics, 434, ISBN 1-58488-371-5
Polar_action
compact affine manifold admits a parallel volume form (i.e., with linear holonomy in SL ( n , R ) {\displaystyle (n,\mathbb {R} )} ; it was shown by Bruno
Chern's conjecture (affine geometry)
Chern's_conjecture_(affine_geometry)
Austrian neuroscientist (1919–2015)
hierarchical relationships within the group. Pribram's work Brain and Perception: Holonomy and Structure in Figural Processing (1991) conveyed his theory, based on
Karl_H._Pribram
Russian physicist
Alma mater MIPT, Princeton Known for quantum topology, string theory, special holonomy manifolds, categorification of quantum group invariants, exact solutions
Sergei_Gukov
Topics referred to by the same term
blocking in computer networking Higher-order logic in mathematics and logic Holonomy, mathematical symbol Hol, in differential geometry Hol IL, a sports club
Hol
Concept in geometry
classification of Riemannian holonomy groups first raised the issue of the existence of non-symmetric manifolds with holonomy Sp(n)·Sp(1).Interesting results
Quaternionic_manifold
Aspect of theoretical physics
392 (1802): 45–57. doi:10.1098/rspa.1984.0023. Simon, Barry (1983). "Holonomy, the Quantum Adiabatic Theorem, and Berry's Phase". Physical Review Letters
Quantum geometry (condensed matter)
Quantum_geometry_(condensed_matter)
American Jewish mathematician
to new branches of mathematics and physics.” Kostant, Bertram (1955). "Holonomy and the Lie algebra of infinitesimal motions of a Riemannian manifold"
Bertram_Kostant
Variables used in general relativity
quantum general relativity and in turn loop quantum gravity and quantum holonomy theory. Let us introduce a set of three vector fields E j a , {\displaystyle
Ashtekar_variables
Coordinates system in an accelerating, "at rest" setting
ISBN 978-2-86332-168-3. Bini, D., & Jantzen, R. T. (2003). "Circular holonomy, clock effects and gravitoelectromagnetism: Still going around in circles
Proper reference frame (flat spacetime)
Proper_reference_frame_(flat_spacetime)
Topological space
deeply related to the theory of holonomy groups. Actually, any isoparametric submanifold is foliated by the holonomy tubes of a submanifold with constant
Isoparametric_manifold
American mathematician
Michael; Witten, Edward (2003), "M-theory dynamics on a manifold of G2 holonomy", Advances in Theoretical and Mathematical Physics, 6 (1): 1–106, arXiv:hep-th/0107177
Claude_LeBrun
South Korean American mathematician
vol. 209, 2017, pp. 425–461 with Dale Winter: Prime number theorems and holonomies for hyperbolic rational maps, Inventiones mathematicae, vol. 208, 2017
Hee_Oh
Constraint in diffeomorphism invariant theories
T, Rohail (4 November 2025). "Diffeomorphism And Gauss Constraints Are Holonomy Corrected, Maintaining A First-class Algebra In Modified Gravity". quantumzeitgeist
Diffeomorphism_constraint
Concept in differential geometry
its leaf spaces. It corresponds to the Morita equivalence class of the holonomy groupoid H o l ( F ) ⇉ M {\displaystyle \mathrm {Hol} ({\mathcal {F}})\rightrightarrows
Differentiable_stack
Subbundle of the tangent bundle
1571–1652. doi:10.25537/dm.2020v25.1571-1652. Debord, Claire (2001-07-01). "Holonomy Groupoids of Singular Foliations". Journal of Differential Geometry. 58
Distribution (differential geometry)
Distribution_(differential_geometry)
HOLONOMY
HOLONOMY
HOLONOMY
HOLONOMY
Boy/Male
Arabic
Calm; Gentle; Noble; Composed
Girl/Female
Gaelic American Scottish Irish Teutonic
warrior.
Girl/Female
Arabic, Muslim
Hope
Girl/Female
Greek
Refused to kill her husband on their wedding night.
Boy/Male
Arabic, Muslim
A Garnet; Ruby; A Precious Stone
Girl/Female
German
Mighty with a spear.
Boy/Male
Australian
Number One
Boy/Male
Hawaiian
Form of James and Jim.
Boy/Male
Muslim
Victory, Mars
Boy/Male
Bengali, Hindu, Indian
Advice
HOLONOMY
HOLONOMY
HOLONOMY
HOLONOMY
HOLONOMY