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Study of angle-preserving transformations of a geometric space
conformal geometry is the study of the set of angle-preserving (conformal) transformations on a space. In a real two dimensional space, conformal geometry
Conformal_geometry
Mathematical function that preserves angles
orientation. Conformal maps preserve both angles and the shapes of infinitesimally small figures, but not necessarily their size or curvature. The conformal property
Conformal_map
Causal relationships between points in a manifold
make a conformal rescaling of the metric with a conformal factor which falls off sufficiently fast to 0 as we approach infinity to get the conformal boundary
Causal_structure
Topics referred to by the same term
Conformal geometry, in mathematics Conformal group, in mathematics Conformal map, in mathematics Conformal map projection, in cartography Conformal prediction
Conformal
Gravity theories that are invariant under Weyl transformations
Conformal gravity refers to gravity theories that are invariant under conformal transformations in the Riemannian geometry sense; more accurately, they
Conformal_gravity
Concept in mathematical group theory
the conformal group of a space is the group of one-to-one transformations from the space to itself that preserve angles, that is, the conformal transformations
Conformal_group
Branch of mathematics
example, in Riemannian geometry distances and angles are specified, in symplectic geometry volumes may be computed, in conformal geometry only angles are specified
Differential_geometry
American mathematician
rules. I. Conformal Geometry and Dynamics, vol. 10 (2006), pp. 63–99. M. Bourdon and H. Pajot, Quasi-conformal geometry and hyperbolic geometry. In: Rigidity
James_W._Cannon
Contact structure Contact geometry Hamiltonian system Sasakian manifold Poisson manifold Möbius transformation Conformal map conformal connection tractor bundle
List of differential geometry topics
List_of_differential_geometry_topics
well-suited to use in complex analysis, in particular in conformal maps and conformal geometry. A closely related notion is the transfinite diameter or
Conformal_radius
Elliptic differential operators in geometry mathematics
manifold of dimension n ≥ 3, one can define the conformal Laplacian to generalize these properties. The conformal Laplacian, denoted L, acts on a smooth function
Laplace operators in differential geometry
Laplace_operators_in_differential_geometry
Type of geometric algebra
Graphics using Conformal Geometric Algebra, PhD thesis, University of Cambridge, pp. 14–26, 31—67 Bromborsky, A. (2008), Conformal Geometry via Geometric
Conformal_geometric_algebra
Cosmological model
past conformal boundary of one copy of FLRW spacetime can be "attached" to the future conformal boundary of another, after an appropriate conformal rescaling
Conformal_cyclic_cosmology
flat metric times the conformal factor 1 − 2 G M / r {\displaystyle \textstyle 1-{2GM}/{r}} . Weyl–Schouten theorem Conformal geometry Yamabe problem Ray
Conformally_flat_manifold
Study of angle-preserving transformations
these are conformal maps, and in fact, where the space has three or more dimensions, the mappings generated by inversion are the only conformal mappings
Inversive_geometry
Homeomorphism between plane domains
and medical imaging. Computational quasi-conformal geometry has been developed, which extends the quasi-conformal theory into a discrete setting. It has
Quasiconformal_mapping
Type of non-Euclidean geometry
mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate
Hyperbolic_geometry
CLASS LECTURE: HYPERBOLIC GEOMETRY" (PDF). Retrieved August 12, 2018. Models of hyperbolic geometry Conformal Models of the Hyperbolic Geometry v t e
Band_model
American mathematician (born 1950)
constant scalar curvature metrics in a conformal class". In Lawson, Blaine; Tenenblat, Keti (eds.). Differential geometry. A symposium in honor of Manfredo
Richard_Schoen
Generalization of the Levi-Civita connection
differential geometry, a Weyl connection (also called a Weyl structure) is a generalization of the Levi-Civita connection that makes sense on a conformal manifold
Weyl_connection
In conformal differential geometry, a conformal connection is a Cartan connection on an n-dimensional manifold M arising as a deformation of the Klein
Conformal_connection
In conformal geometry, the tractor bundle is a particular vector bundle constructed on a conformal manifold whose fibres form an effective representation
Tractor_bundle
Special class of linear fractional transformations
In projective geometry, a special conformal transformation is a linear fractional transformation that is not an affine transformation. Thus the generation
Special conformal transformation
Special_conformal_transformation
Model of the extended complex plane plus a point at infinity
The Riemann surface's conformal structure does, however, determine a class of metrics: all those whose subordinate conformal structure is the given one
Riemann_sphere
Subfield of conformal geometry
In mathematics, Möbius geometry is a branch of geometry — specifically a subfield of conformal geometry — which deals with the study of geometric objects
Möbius_geometry
Two geometries based on axioms closely related to those specifying Euclidean geometry
non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the
Non-Euclidean_geometry
Theory of gravity
replacement of the spacetime picture with a picture of evolving spatial conformal geometry opens the door for a number of new approaches to quantum gravity.
Shape_dynamics
Rational function of the form (az + b)/(cz + d)
bijective conformal orientation-preserving maps from the n-sphere to the n-sphere. Such a transformation is the most general form of conformal mapping of
Möbius_transformation
An Einstein–Weyl geometry is a smooth conformal manifold, together with a compatible Weyl connection that satisfies an appropriate version of the Einstein
Einstein–Weyl_geometry
Infinitely detailed mathematical structure
J. W. Cannon, W. J. Floyd, W. R. Parry. Finite subdivision rules. Conformal Geometry and Dynamics, vol. 5 (2001), pp. 153–196. Carbone, Alessandra; Gromov
Fractal
as the (conformal) obstruction tensor. It is, along with the Weyl tensor, one of the two primitive invariants in conformal differential geometry. Aside
Ambient_construction
In differential geometry and mathematical physics (especially string theory), the Polyakov formula expresses the conformal variation of the zeta functional
Polyakov_formula
Overview of and topical guide to geometry
Absolute geometry Affine geometry Algebraic geometry Analytic geometry Birational geometry Complex geometry Computational geometry Conformal geometry Constructive
Outline_of_geometry
Finnish mathematician (1907–1996)
and Conformal invariants (1973). He made decisive contributions to meromorphic curves, value distribution theory, Riemann surfaces, conformal geometry, quasiconformal
Lars_Ahlfors
important in conformal geometry, because in a suitable sense it depends only on the conformal structure. Another operator of this kind is the conformal Laplacian
Paneitz_operator
Low-rank isomorphisms in mathematics
group of the four-dimensional conformal group with the bosonic symmetry group appearing in many formulations of conformal field theory. In the best-known
Exceptional isomorphisms of classical groups
Exceptional_isomorphisms_of_classical_groups
Coating used on PCBs
Conformal coating is a protective, breathable coating of thin polymeric film applied to printed circuit boards (PCBs). Conformal coatings are typically
Conformal_coating
Equation in differential geometry
arises in differential geometry when studying surfaces of constant curvature. It plays a central role in the theory of conformal geometry, where one seeks to
Liouville's_equation
Indian mathematician (born 1942)
S2CID 13568444. with Ulrich Pinkall: Conformal geometry. Aspects of Mathematics (proceedings of a seminar on conformal geometry at the Max-Planck Institute in
Ravindra_Shripad_Kulkarni
Diagram of different points in spacetime
readable introduction to the concept of conformal infinity plus examples. Frauendiener, Jörg (2004). "Conformal Infinity". Living Reviews in Relativity
Penrose_diagram
Taiwanese-American mathematician
specializing in differential geometry, partial differential equations and CR manifolds. He is best known for his work in Conformal geometry for his study of extremal
Paul_C._Yang
Homogeneous quotient space of a semisimple Lie group by a parabolic subgroup
modeled on the conformal sphere. Here the associated Cartan connection is the conformal connection. Other examples include: CR geometry, the study of manifolds
Parabolic geometry (differential geometry)
Parabolic_geometry_(differential_geometry)
Upper-half plane model of hyperbolic non-Euclidean geometry
In non-Euclidean geometry, the Poincaré half-plane model is a way of representing the hyperbolic plane using points in the familiar Euclidean plane. Specifically
Poincaré_half-plane_model
Type of geometry
In mathematics, a Klein geometry is a type of geometry motivated by Felix Klein in his influential Erlangen program. More specifically, it is a homogeneous
Klein_geometry
2002 book on fractal geometry
in 2002 and 2015. The book explores the patterns created by iterating conformal maps of the complex plane called Möbius transformations, and their connections
Indra's_Pearls_(book)
Israeli mathematician
contributions to several topics: The circle packing theorem and discrete conformal geometry. Embeddings of Gromov hyperbolic spaces. Percolation, uniform and
Oded_Schramm
Branch of mathematics
September 2019. Xianfeng David Gu; Shing-Tung Yau (2008). Computational Conformal Geometry. International Press. ISBN 978-1-57146-171-1. Archived from the original
Geometry
Theory proposed by Roger Penrose
spinors for the conformal group SO(4,2)/Z2 of Minkowski space; it is the fundamental representation of the spin group SU(2,2) of the conformal group. This
Twistor_theory
Geometry founded on spheres
Lie sphere geometry is a geometrical theory of planar or spatial geometry in which the fundamental concept is the circle or sphere. It was introduced by
Lie_sphere_geometry
Quantum field theory enjoying conformal symmetry
A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional
Conformal_field_theory
Geometry problem about finding touching circles
In Euclidean plane geometry, the problem of Apollonius (also called Apollonius's problem or the Apollonian problem) is to construct circles that are tangent
Problem_of_Apollonius
Chinese-American mathematician (born 1949)
genus-zero surfaces, a map is conformal if and only if it is harmonic, and so Gu and Yau are able to compute conformal maps by direct minimization of
Shing-Tung_Yau
American mathematician
Washington, known for a number of contributions to the field of conformal geometry and CR geometry; his collaboration with Charles Fefferman on the ambient construction
C._Robin_Graham
Vector field in conformal geometry
called a conformal Killing vector, CKV, or conformal colineation), is a vector field X {\displaystyle X} whose (locally defined) flow defines conformal transformations
Conformal Killing vector field
Conformal_Killing_vector_field
American mathematician
mathematicians. The book The Theory and Practice of Conformal Geometry is a study of classical conformal geometry in the complex plane, and is the first Dover
Steven_G._Krantz
Self-intersecting compact surface, an immersion of the real projective plane
Sciences. Série A (in French). 287: 879–882. Kusner, Rob (1987). "Conformal geometry and complete minimal surfaces" (PDF). Bulletin of the American Mathematical
Boy's_surface
Möbius transformation generalized to rings other than the complex numbers
generators are all conformal. The translation z → z + b is a change of origin and makes no difference to angle. To see that z → az is conformal, consider the
Linear fractional transformation
Linear_fractional_transformation
Book by Roger Penrose
aeon becomes the low-entropy Big Bang state of the next aeon cycle. Conformal geometry preserves the angles but not the distances of the previous aeon, allowing
Cycles_of_Time
Metric tensor describing constant negative (hyperbolic) curvature
on the unit disk. The disk and the upper half plane are related by a conformal map, and isometries are given by Möbius transformations. A third representation
Poincaré_metric
Open-source parametric aircraft geometry tool
available geometries. Advanced components like body of revolution, duct, conformal geometry and such are also available. Besides the geometry modeler,
OpenVSP
Lorentz surface is a two-dimensional oriented smooth manifold with a conformal equivalence class of Lorentzian metrics. It is the analogue of a Riemann
Lorentz_surface
Computational synthetic geometry Computational topology Computer algebra see symbolic computation Conformal geometry the study of conformal transformations on
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Particular mapping that projects a sphere onto a plane
hypersurface. This construction plays a role in algebraic geometry and conformal geometry. The first stereographic projection defined in the preceding
Stereographic_projection
German mathematician
differential geometry". Doc. Math. (Bielefeld) Extra Vol. ICM Berlin, 1998, vol. II. pp. 389–400. Goldman, William M. (1990). "Book Review: Conformal geometry".
Ulrich_Pinkall
Surface of revolution of a catenary
Fraser, Ailana; Schoen, Richard (2011). "The first Steklov eigenvalue, conformal geometry, and minimal surfaces". Advances in Mathematics. 226 (5): 4011–4030
Catenoid
Breakdown of conformal symmetry at the quantum level
A conformal anomaly, scale anomaly, trace anomaly or Weyl anomaly is an anomaly, i.e. a quantum phenomenon that breaks the conformal symmetry of the classical
Conformal_anomaly
Branch of mathematics studying functions of a complex variable
orientation. Conformal maps preserve both angles and the shapes of infinitesimally small figures, but not necessarily their size or curvature. The conformal property
Complex_analysis
Way to divide polygon into smaller parts
\ldots } of tilings of a surface is conformal ( K {\displaystyle K} ) in the above sense, then there is a conformal structure on the surface and a constant
Finite_subdivision_rule
Relates 2 second-order elliptic operators on a manifold with the same principal symbol
{j}}_{2}\dots {\bar {k}}\dots {\bar {j}}_{q}}} with k in the s-th place. In conformal geometry there is a Weitzenböck formula relating a particular pair of differential
Weitzenböck_identity
Mathematical method to constrain and solve conformal field theories
The conformal bootstrap is a non-perturbative mathematical method to constrain and solve conformal field theories, i.e. models of particle physics or statistical
Conformal_bootstrap
Approach to quantum gravity using discrete spacetime
that preserves their causal structure then the map is a conformal isomorphism. The conformal factor that is left undetermined is related to the volume
Causal_sets
they depend only on the conformal structure of the manifold. The GJMS operators generalize the Paneitz operator and the conformal Laplacian. The initials
GJMS_operator
Second-order tensor
Springer-Verlag, 2007. See Ch.1 §J "Conformal Changes of Riemannian Metrics". Spyros Alexakis, The Decomposition of Global Conformal Invariants. Princeton University
Schouten_tensor
Theorem limiting types of conformal mappings in Euclidean space of dimension > 2
in 1850, is a rigidity theorem about conformal mappings in Euclidean space. It states that every smooth conformal mapping on a domain of Rn, where n >
Liouville's theorem (conformal mappings)
Liouville's_theorem_(conformal_mappings)
Finnish mathematician (born 1948)
xix+502 pp.{{cite book}}: CS1 maint: postscript (link) M. Vuorinen: Conformal geometry and quasiregular mappings. Lecture Notes in Mathematics. Vol. 1319
Matti_Vuorinen
Local rescaling of a metric tensor
symmetry in conformal field theory. It is, for example, a symmetry of the Polyakov action. When quantum mechanical effects break the conformal invariance
Weyl_transformation
Russian mathematician (1937–2023)
expert in various fields of mathematical analysis, conformal geometry, and the theory of quasi-conformal mappings. He graduated from the Faculty of Mechanics
Vladimir_A._Zorich
Equation for radii of tangent circles
in Li, Hongbo; Hestenes, David; Rockwood, Alyn (2001), "Spherical conformal geometry with geometric algebra" (PDF), Geometric Computing with Clifford Algebras
Descartes's_theorem
Polygon associated with a compact Riemann surface
fundamental domain can be defined canonically using the conformal structure of C. Note that the group of conformal transformations of C is given by complex affine
Fundamental_polygon
Topics referred to by the same term
Regional Airport Conformal Killing vector field, sometimes shortened to conformal Killing vector or just CKV, a vector field in conformal geometry This disambiguation
CKV
Nonlinear second-order partial differential equation of special kind
naturally in several problems in Riemannian geometry, conformal geometry, affine geometry, and CR geometry. Given a twice-differentiable real-valued function
Monge–Ampère_equation
American physicist (1939–2023)
his Ph.D. in 1966 from North Carolina State University. York used conformal geometry in the initial value problem, and introduced concepts now called the
James_W._York
Area in mathematics devoted to the study of finitely generated groups
Marc Bourdon and Hervé Pajot. Quasi-conformal geometry and hyperbolic geometry. Rigidity in dynamics and geometry (Cambridge, 2000), pp. 1–17, Springer
Geometric_group_theory
Generalization of affine connections
seen as a deformation of Minkowski space; a conformal manifold can be seen as a deformation of the conformal sphere; a manifold equipped with an affine
Cartan_connection
Brazilian mathematician
Mathematicians (ICM) of 2010 in Hyderabad (on "Scalar curvature, conformal geometry, and the Ricci flow with surgery"), and a plenary speaker at the ICM
Fernando_Codá_Marques
Branch of mathematics
Noncommutative geometry (NCG) is a branch of mathematics that studies geometric ideas through noncommutative algebras. In ordinary geometry, a space can
Noncommutative_geometry
Association of professional mathematicians
Prize Norbert Wiener Prize in Applied Mathematics Oswald Veblen Prize in Geometry The AMS is led by the president, who is elected for a two-year term, and
American_Mathematical_Society
Notion in geometry
role in the conformal geometry of Riemannian manifolds. In particular, it can be used to show that if the metric is rescaled by a conformal factor of
Curvature of Riemannian manifolds
Curvature_of_Riemannian_manifolds
One-dimensional complex manifold
{\displaystyle X} is the additional datum of the conformal structure. A complex structure gives rise to a conformal structure by choosing the standard Euclidean
Riemann_surface
German mathematician (1885–1955)
spacetime. The Weyl tensor in Riemannian geometry is of major importance in understanding the nature of conformal geometry. His overall approach in physics was
Hermann_Weyl
classes of connections arise naturally in differential geometry, for example: in conformal geometry an equivalence class of connections is given by the Levi
Invariant differential operator
Invariant_differential_operator
German mathematician
and became reader there in 2016. Leschke is a coauthor of the book Conformal Geometry of Surfaces in S 4 {\displaystyle S^{4}} and Quaternions (Springer
Katrin_Leschke
2D surface which extends indefinitely
even a conformal map. The plane itself is homeomorphic (and diffeomorphic) to an open disk. For the hyperbolic plane such diffeomorphism is conformal, but
Plane_(mathematics)
German mathematician
Brendle has solved major open problems regarding the Yamabe equation in conformal geometry. This includes his counterexamples to the compactness conjecture for
Simon_Brendle
Irreducible representation of the Virasoro algebra
Varieties: Arithmetic, Geometry and Physics: Lecture Notes on Concentrated Graduate Courses. Springer. p. 239. ISBN 978-1-4939-2830-9. Conformal field theory v
Conformal_family
Lie group of Lorentz transformations
homogeneous space SO+(1, 3) / Sim(2) is the Kleinian geometry that represents conformal geometry on the sphere S2. The (identity component of the) Euclidean
Lorentz_group
Generalized sphere of dimension n (mathematics)
square and each point in the third pair; this gives a octahedron. Conformal geometry – Study of angle-preserving transformations of a geometric space Exotic
N-sphere
Special functions used to build correlation functions in 2D CFTs
In two-dimensional conformal field theory, Virasoro conformal blocks (named after Miguel Ángel Virasoro) are special functions that serve as building blocks
Virasoro_conformal_block
American mathematician
of the National Academy of Sciences 11, no. 4 (1925): 199–203. On conformal geometry. Proceedings of the National Academy of Sciences 12, no. 5 (1926):
Tracy_Yerkes_Thomas
Straight figure with zero width and depth
In geometry, a straight line, usually abbreviated line, is an infinitely long object with no width, depth, or curvature. It is a special case of a curve
Line_(geometry)
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CONFORMAL GEOMETRY
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CONFORMAL GEOMETRY
CONFORMAL GEOMETRY
CONFORMAL GEOMETRY
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