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Rigidity theorem for convex polyhedra
Cauchy's theorem is a theorem in geometry, named after Augustin Cauchy. It states that convex polytopes in three dimensions with congruent corresponding
Cauchy's_theorem_(geometry)
Topics referred to by the same term
mean value theorem Cauchy's theorem (group theory) Cauchy's theorem (geometry) on rigidity of convex polytopes The Cauchy–Kovalevskaya theorem concerning
Cauchy_theorem
Existence and uniqueness of solutions to initial value problems
known as Picard's existence theorem, the Cauchy–Lipschitz theorem, or the existence and uniqueness theorem. The theorem is named after Émile Picard,
Picard–Lindelöf_theorem
Every Riemannian manifold can be isometrically embedded into some Euclidean space
The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash Jr., state that every Riemannian manifold can be isometrically embedded
Nash_embedding_theorems
Sequence of points that get progressively closer to each other
Cauchy convergence can simplify both definitions and theorems in constructive analysis. Regular Cauchy sequences were used by Bishop (2012) and by Bridges
Cauchy_sequence
French mathematician (1789–1857)
equations Cauchy–Schwarz inequality Cauchy sequence Cauchy surface Cauchy's theorem (geometry) Cauchy's theorem (group theory) Maclaurin–Cauchy test French
Augustin-Louis_Cauchy
(convex geometry) Cauchy's theorem (geometry) Classification of Platonic solids (geometry) de Bruijn's theorem (discrete geometry) Descartes's theorem on total
List_of_theorems
Branch of mathematics
Atiyah–Singer index theorem. The development of complex geometry was spurred on by parallel results in algebraic geometry, and results in the geometry and global
Differential_geometry
Characteristic property of holomorphic functions
Katsumi (1969). Foundations of differential geometry, volume 2. Wiley. Proposition IX.2.2. Rudin 1966, Theorem 11.2. Dieudonné, Jean Alexandre (1969). Foundations
Cauchy–Riemann_equations
Branch of geometry that studies combinatorial properties and constructive methods
had been studied for many years by people such as Kepler and Cauchy, modern discrete geometry has its origins in the late 19th century. Early topics studied
Discrete_geometry
Polyhedra are determined by surface distance
Alexandrov's theorem on polyhedra is a rigidity theorem in mathematics, describing three-dimensional convex polyhedra in terms of the distances between
Alexandrov's theorem on polyhedra
Alexandrov's_theorem_on_polyhedra
Sumset of a field subject to a specific polynomial restriction
"The Cauchy-Davenport Theorem for Finite Groups". arXiv:1202.1816 [math.CO]. DeVos, Matt (2016). "On a Generalization of the Cauchy-Davenport Theorem". Integers
Restricted_sumset
Length of a line segment
calculated from the Cartesian coordinates of the points using the Pythagorean theorem, and therefore is occasionally called the Pythagorean distance. These names
Euclidean_distance
Mathematical inequality relating inner products and norms
extracting a metric from quantum fidelity. The Cauchy-Schwarz inequality can be used to prove the spectral theorem for self-adjoint operators in the finite-dimensional
Cauchy–Schwarz_inequality
theorem (geometry) Cauchy's theorem (group theory) Cauchy's two-line notation Binet–Cauchy identity (or Cauchy–Binet equation) Cauchy bounds Cauchy completeness
List of things named after Augustin-Louis Cauchy
List_of_things_named_after_Augustin-Louis_Cauchy
On topological spaces where the intersection of countably many dense open sets is dense
of analysis and geometry, including some of the fundamental theorems of functional analysis. Versions of the Baire category theorem were first proved
Baire_category_theorem
Theorem in topology
courses on differential geometry. It appears in unlikely fields such as game theory. In economics, Brouwer's fixed-point theorem and its extension, the
Brouwer_fixed-point_theorem
Property of geometry, also used to generalize the notion of "distance" in metric spaces
triangle with zero area. In Euclidean geometry and some other geometries, the triangle inequality is a theorem about vectors and vector lengths (norms):
Triangle_inequality
Representation of mechanical stress at every point within a deformed 3D object
through that point. However, according to Cauchy's fundamental theorem, also called Cauchy's stress theorem, merely by knowing the stress vectors on three
Cauchy_stress_tensor
Every rigid motion is a screw displacement
In kinematics, Chasles' theorem, or Mozzi–Chasles' theorem, says that the most general rigid body displacement can be produced by a screw displacement
Chasles'_theorem_(kinematics)
Statement about integration on manifolds
differential geometry the generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called the Stokes–Cartan theorem, is
Generalized_Stokes_theorem
Differentiable manifold
volume comparison theorems on CR manifolds with zero Webster torsion akin to the H.E. Rauch comparison theorem in Riemannian Geometry. In recent years
CR_manifold
Mathematics of smooth surfaces
the Gauss–Codazzi equations. A major theorem, often called the fundamental theorem of the differential geometry of surfaces, asserts that whenever two
Differential geometry of surfaces
Differential_geometry_of_surfaces
French mathematician and lawyer (1601–1665)
analytic geometry, probability, and optics. He is best known for his Fermat's principle for light propagation and his Fermat's Last Theorem in number
Pierre_de_Fermat
Field of knowledge
inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It is in Babylonian
Mathematics
Theorem in mathematics
In mathematical analysis, the inverse function theorem gives sufficient conditions for a function to have an inverse function. The essential idea is that
Inverse_function_theorem
Metric geometry
mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively
Complete_metric_space
Conjecture on zeros of the zeta function
hypothesis is true, then the theorem is true. If the generalized Riemann hypothesis is false, then the theorem is true. Thus, the theorem is true!! Care should
Riemann_hypothesis
Area of mathematics
dynamics, minimal surfaces, and complex hyperbolic geometry, which deals with generalizations of Picard's theorem to higher dimensions. A substantial part of
Nevanlinna_theory
Mathematical problem
Erdős–Ginzburg–Ziv theorem after its discoverers. It may also be deduced from the Cauchy–Davenport theorem. More general results than this theorem exist, such
Zero-sum_problem
German mathematician (1826–1866)
made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rigorous
Bernhard_Riemann
Branch of mathematics
curves. These two branches are related to each other by the fundamental theorem of calculus. Calculus uses convergence of infinite sequences and infinite
Calculus
Key result in Hamiltonian mechanics and statistical mechanics
coordinates is available in the mathematical setting of symplectic geometry. Liouville's theorem ignores the possibility of chemical reactions, where the total
Liouville's theorem (Hamiltonian)
Liouville's_theorem_(Hamiltonian)
Fundamental theorem in probability theory and statistics
In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the distribution of a normalized version of the sample
Central_limit_theorem
On converting relations to functions of several real variables
In multivariable calculus, the implicit function theorem is a theorem that provides sufficient conditions under which a planar curve specified by F ( x
Implicit_function_theorem
Key results in general relativity on gravitational singularities
gravitation produces singularities. The Penrose singularity theorem is a theorem in semi-Riemannian geometry and its general relativistic interpretation predicts
Penrose–Hawking singularity theorems
Penrose–Hawking_singularity_theorems
the Pythagorean theorem seems to be the most ancient and widespread mathematical development, after basic arithmetic and geometry. The study of mathematics
History_of_mathematics
American mathematician and Nobel Laureate (1928–2015)
proved the Nash embedding theorems by solving a system of nonlinear partial differential equations arising in Riemannian geometry. This work, also introducing
John_Forbes_Nash_Jr.
exponentiation Euler's rotation theorem – Movement with a fixed point is rotation Euler's theorem (differential geometry) – Orthogonality of the directions
List of topics named after Leonhard Euler
List_of_topics_named_after_Leonhard_Euler
Branch of mathematics
this analysis as the first theorem, signaling the birth of topology. Further contributions were made by Augustin-Louis Cauchy, Ludwig Schläfli, Johann Benedict
Topology
for set union and intersection Cauchy's integral formula Cauchy integral theorem Computational geometry Fundamental theorem of algebra Lambda calculus Invariance
List_of_mathematical_proofs
Submanifold of Lorentzian manifold
In the mathematical field of Lorentzian geometry, a Cauchy surface, also called more properly Cauchy hypersurface, is a certain kind of submanifold of
Cauchy_surface
Basic framework of mathematics
postulates, definitions, and theorems. Aristotle took a majority of his examples for this from arithmetic and from geometry, and his logic served as the
Foundations_of_mathematics
Statement in complex analysis
metric in the Poincaré disk model for hyperbolic geometry in dimension two. The Schwarz–Pick theorem then essentially states that a holomorphic map of
Schwarz_lemma
Theorem in real analysis
fallacious. The theorem was first proved by Cauchy in 1823 as a corollary of a proof of the mean value theorem. The name "Rolle's theorem" was first used
Rolle's_theorem
geometry Thales of Miletus (c. 624 BC – c. 546 BC) – Euclidean geometry Pythagoras (c. 570 BC – c. 495 BC) – Euclidean geometry, Pythagorean theorem Zeno
List_of_geometers
Type of mathematical functions
number Complex geometry CR manifold Dolbeault cohomology Harmonic maps Harmonic morphisms Infinite-dimensional holomorphy Oka–Weil theorem A name adopted
Function of several complex variables
Function_of_several_complex_variables
Branch of mathematics studying functions of a complex variable
functions. Complex geometry Hypercomplex analysis List of complex analysis topics Monodromy theorem Riemann–Roch theorem Runge's theorem Vector calculus
Complex_analysis
Theorem in complex analysis
In complex analysis, the argument principle (or Cauchy's argument principle) is a theorem relating the difference between the number of zeros and poles
Argument_principle
Theorem about zeros of holomorphic functions
residues when one applies Cauchy's residue theorem. Rouché's theorem can also be used to give a short proof of the fundamental theorem of algebra. Let p ( z
Rouché's_theorem
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 is
Conformal_geometry
Subset of Euclidean space is compact if and only if it is closed and bounded
In real analysis in mathematics, the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states: For a subset S {\displaystyle S} of Euclidean
Heine–Borel_theorem
Distance-preserving mathematical transformation
Euclidean plane isometry Flat (geometry) Homeomorphism group Involution Isometry group Motion (geometry) Myers–Steenrod theorem 3D isometries that leave the
Isometry
Mathematical theorem
In complex analysis, the Riemann mapping theorem states that if U {\displaystyle U} is a non-empty simply connected open subset of the complex number
Riemann_mapping_theorem
previously been discussed by Augustin Cauchy, in 1845, and by Georg Frobenius in 1887. Cayley–Hamilton theorem. The theorem was first proved in the easy special
List_of_misnamed_theorems
this version of the theorem does not generalize to higher dimensions. Alexandrov's uniqueness theorem Cauchy's theorem (geometry) Klain, Daniel A. (2004)
Minkowski problem for polytopes
Minkowski_problem_for_polytopes
Theorem about metric spaces
Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem or Banach–Caccioppoli theorem) is an important
Banach_fixed-point_theorem
Textbook by Augustin-Louis Cauchy (1821)
which one demands from geometry, so that one need never rely on arguments drawn from the generality of algebra." On page 6, Cauchy first discusses variable
Cours_d'analyse
Aleksandrov, developer of CAT(k) space and Alexandrov's uniqueness theorem in geometry Pavel Alexandrov, author of the Alexandroff compactification and
List of Russian mathematicians
List_of_Russian_mathematicians
Mathematical subject
applications to computational geometry. Other important areas include metric geometry of polyhedra, such as the Cauchy theorem on rigidity of convex polytopes
Geometric_combinatorics
Every polynomial has a real or complex root
The fundamental theorem of algebra, also called d'Alembert's theorem or the d'Alembert–Gauss theorem, states that every non-constant single-variable polynomial
Fundamental theorem of algebra
Fundamental_theorem_of_algebra
Mathematical function that preserves angles
S2CID 118752074. Richard M. Timoney (2004), Riemann mapping theorem from Trinity College Dublin Geometry/Unified Angles at Wikibooks Tsurusaburo Takasu (1941)
Conformal_map
Theorem limiting types of conformal mappings in Euclidean space of dimension > 2
Transformation groups in differential geometry, Berlin, New York: Springer-Verlag. Solomentsev, E.D. (2001) [1994], "Liouville theorems", Encyclopedia of Mathematics
Liouville's theorem (conformal mappings)
Liouville's_theorem_(conformal_mappings)
Branch of mathematics that studies the properties of groups
groups and field theory. In geometry, groups first became important in projective geometry and, later, non-Euclidean geometry. Felix Klein's Erlangen program
Group_theory
Branch of discrete mathematics
Metric properties of polytopes play an important role as well, e.g. the Cauchy theorem on the rigidity of convex polytopes. Special polytopes are also considered
Combinatorics
Extension of the domain of an analytic function (mathematics)
{\displaystyle f(z)=\sum _{k=0}^{\infty }(-1)^{k}(z-1)^{k}.} By the Cauchy–Hadamard theorem, its radius of convergence is 1. That is, f {\displaystyle f} is
Analytic_continuation
Calculus of vector-valued functions
differential geometry, of which vector calculus forms a subset. Grad and div generalize immediately to other dimensions, as do the gradient theorem, divergence
Vector_calculus
Mathematical set with some added structure
angles cannot appear in theorems of projective geometry, since these notions are neither mentioned in the axioms of projective geometry nor defined from the
Space_(mathematics)
Tensor in differential geometry
Riemannian geometry and geometric analysis. Bounds on Ricci curvature imply strong global geometric and topological consequences, as in Myers's theorem and related
Ricci_curvature
Projective variety that is also an algebraic group
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a smooth projective algebraic variety
Abelian_variety
German polymath and scholar (1777–1855)
1848 and known as the Gauss–Bonnet theorem. During Gauss's lifetime, the parallel postulate of Euclidean geometry was heavily discussed. Numerous efforts
Carl_Friedrich_Gauss
Algebraic variety with a group structure
group schemes occurring naturally in arithmetic geometry are neither. Chevalley's structure theorem asserts that every connected algebraic group is an
Algebraic_group
positive-semidefinite matrix Pfaffian Projection Spectral theorem Perron–Frobenius theorem List of matrices Diagonal matrix, main diagonal Diagonalizable
Outline_of_linear_algebra
Field in mathematics similar to the real numbers
Algebra and Geometry (2nd ed.). Berkeley: University of California Press. Erdős, P.; Gillman, L.; Henriksen, M. (1955). "An isomorphism theorem for real-closed
Real_closed_field
Function that is holomorphic on the whole complex plane
generalize the factorization into simple fractions (the Mittag-Leffler theorem on the decomposition of a meromorphic function), then for entire functions
Entire_function
Mathematical theorem
for the symmetry to hold are given by Schwarz's theorem, also called Clairaut's theorem or Young's theorem. In the context of partial differential equations
Symmetry of second derivatives
Symmetry_of_second_derivatives
Area of discrete mathematics
(March 2002). "§13.6, Andreev's theorem and generalizations, and §13.7, Constructing patterns of circles". The Geometry and Topology of 3-manifolds. MSRI
Graph_theory
Result in integral geometry
named after Morgan Crofton (1826–1915), (also Cauchy-Crofton formula) is a classic result of integral geometry relating the length of a curve to the expected
Crofton_formula
Group that is a topological space with continuous group operations
Hewitt & Ross 1970, Theorem 27.40. Mackey 1976, section 2.4. Banaszczyk 1983. Hatcher 2001, Theorem 4.66. Hatcher 2001, Theorem 3C.4. Edwards 1995, p
Topological_group
Counterintuitive mathematical object
of the assumptions in a theorem. For example, in statistics, the Cauchy distribution does not satisfy the central limit theorem, even though its symmetric
Pathological_(mathematics)
Number, approximately 3.14
to Convex Geometry with Applications. Birkhäuser. doi:10.1007/978-3-030-03868-7. ISBN 978-3-030-03866-3. MR 3930585. See Barbier's theorem, Corollary
Pi
Mathematical notion of infinitesimal difference
various branches of mathematics such as calculus, differential geometry, algebraic geometry and algebraic topology. The term differential is used nonrigorously
Differential_(mathematics)
Geometric relation between the roots of a polynomial and those of its derivative
polynomial roots Cauchy interlacing theorem Marden 1966, Theorem (6,1). Rüdinger, A. (2014). "Strengthening the Gauss–Lucas theorem for polynomials with
Gauss–Lucas_theorem
3-dimensional geometric figure
changed while keeping the shapes of all of its faces unchanged. The Cauchy rigidity theorem shows that in dimension 3 such a polyhedron cannot be convex (this
Flexible_polyhedron
Mathematical group based upon a finite number of elements
constructed and characterized based on their geometry in the sense of Tits. The belief has now become a theorem – the classification of finite simple groups
Finite_group
Riemannian manifold in which geodesics extend infinitely in all directions
T_{p}M} , the entire tangent space at p {\displaystyle p} . The Hopf–Rinow theorem gives alternative characterizations of completeness. Let ( M , g ) {\displaystyle
Complete_manifold
) Cauchy–Hadamard theorem – a statement in complex analysis describing the radius of convergence of a power series Hadamard factorization theorem – concerning
List of things named after Jacques Hadamard
List_of_things_named_after_Jacques_Hadamard
Subfield of mathematics
mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory. Contemporary
Mathematical_logic
Mathematical connection between field theory and group theory
correspondence using notions of derived algebraic geometry. Galois group for more examples Fundamental theorem of Galois theory Differential Galois theory for
Galois_theory
Canonical connection on a pseudo-Riemannian manifold
(pseudo-)Riemannian metric and is torsion-free. The fundamental theorem of Riemannian geometry states that there is a unique connection that satisfies these
Levi-Civita_connection
Topological invariant in mathematics
characteristic was originally defined for polyhedra and used to prove various theorems about them, including the classification of the Platonic solids. It was
Euler_characteristic
Mathematical theorem
Heine–Cantor theorem states that a continuous function between two metric spaces is uniformly continuous if its domain is compact. The theorem is named after
Heine–Cantor_theorem
Concept in differential geometry
used to study Riemannian geometry in a more general setting. In 1952 Georges de Rham proved the de Rham decomposition theorem, a principle for splitting
Holonomy
Number of times a curve wraps around a point in the plane
famous Cauchy integral formula. Some of the basic properties of the winding number in the complex plane are given by the following theorem: Theorem. Let
Winding_number
Functions in mathematics
principle; a theorem of removal of singularities as well as a Liouville theorem holds for them in analogy to the corresponding theorems in complex functions
Harmonic_function
Study of space and shapes locally given by a convergent power series
analytic functions. A fundamental result in the theory is the Riemann mapping theorem. The following are some of the most important topics in geometric function
Geometric_function_theory
Complex-differentiable (mathematical) function
) {\displaystyle f(x+iy)=u(x,y)+i\,v(x,y)} is holomorphic. Cauchy's integral theorem implies that the contour integral of every holomorphic function
Holomorphic_function
1998 mathematics book by Aigner and Ziegler
parts: number theory, geometry, analysis, combinatorics and graph theory. In most cases, each chapter is devoted to a particular theorem, sometimes with multiple
Proofs_from_THE_BOOK
Riemann singularity theorem Riemann-Kempf singularity theorem Riemann surface Compact Riemann surface Planar Riemann surface Cauchy–Riemann manifold The
List of things named after Bernhard Riemann
List_of_things_named_after_Bernhard_Riemann
quotes as of August 2026[update]. The conjecture terminology may persist: theorems often enough may still be referred to as conjectures, using the anachronistic
List_of_conjectures
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CAUCHYS THEOREM-GEOMETRY
CAUCHYS THEOREM-GEOMETRY
CAUCHYS THEOREM-GEOMETRY
CAUCHYS THEOREM-GEOMETRY
CAUCHYS THEOREM-GEOMETRY
CAUCHYS THEOREM-GEOMETRY
CAUCHYS THEOREM-GEOMETRY
CAUCHYS THEOREM-GEOMETRY
CAUCHYS THEOREM-GEOMETRY
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