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Contact geometry (2,430 words) [view diff] exact match in snippet view article find links to article

integrability' of a hyperplane distribution, i.e. that it be tangent to a codimension one foliation on the manifold, whose equivalence is the content of the

is codimension zero cycles, which are linear combinations of the irreducible components of the variety. The first non-trivial case is of codimension one

element from H 1 ( V ) {\displaystyle H^{1}(V)} . A connected sum along a codimension-two V {\displaystyle V} can also be carried out in the category of symplectic

Y} ). A codimension-1 subvariety Z ⊂ X {\displaystyle Z\subset X} is said to be exceptional if f ( Z ) {\displaystyle f(Z)} has codimension at least

functorial, i.e. push-forward (with change of codimension) and pull-back of cycles is well-defined. Codimension 1 cycles modulo rational equivalence form

boundary (top dimension and codimension 1 boundary) and manifolds with corners (top dimension, codimension 1 boundary, codimension 2 corners), real or complex

something that happens in codimension two (like knot theory, and monodromy); since real codimension two is complex codimension one, the local complex example

{\displaystyle i:X\hookrightarrow Y} of schemes is a regular embedding of codimension r if each point x in X has an open affine neighborhood U in Y such that

for codimension-one Anosov diffeomorphisms (i.e., those for which the contracting or the expanding subbundle is one-dimensional) and for codimension one

perimeters (De Giorgi) for codimension 1 and the theory of rectifiable currents (Federer and Fleming) for higher codimension have been developed. The theory

singular sets of codimension seven. In 1970, Federer proved that this codimension is optimal: all such singular sets have codimension of at least seven

Novikov's compact leaf theorem, named after Sergei Novikov, states that A codimension-one foliation of a compact 3-manifold whose universal covering space

connected submanifolds of codimension 1 and where n denotes the dimension of M. Note that two submanifolds of codimension 1 are orthogonal if their normal

each point p {\displaystyle p} of M , {\displaystyle M,} one selects a codimension-one linear subspace Q p {\displaystyle Q_{p}} of the tangent space T

In algebraic geometry, flips and flops are codimension-2 surgery operations arising in the minimal model program, given by blowing up along a relative

In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular

embedding codimension c, meaning that c = dimk(m/m2) − dim(R). In geometric terms, this holds for a local ring of a subscheme of codimension c in a regular

some p × q matrix of elements of S. If the codimension (or height) of I is equal to the "expected" codimension (p−r+1)(q−r+1), R is called a determinantal

the initial value problem is well-posed for initial data given on a codimension-one hypersurface. And later, in 2022, a research team at the University

Lawson, Osserman studied the minimal surface problem in the case that the codimension is larger than one. They considered the case of a graphical minimal submanifold

intersection? This condition is fairly hard to check as soon as the codimension n − m ≥ 2. When n − m = 1 then V is automatically a hypersurface and

is a singularity of codimension k, if it has a neighborhood isometric to R^{n-k} with a metric cone. Singularities of codimension 2 are of major importance;

F(U-Y)} is bijective for every open subset U and a closed subset Y of U of codimension at least 2. With this terminology, a coherent sheaf on an integral normal

2 structure. It is the dual of the Segre cubic. It can be given as a codimension 2 variety in P5 by the equations ∑ x i = 0 {\displaystyle \sum x_{i}=0}

diffeomorphism of a sphere of dimension at least 4 can be a knotted sphere of codimension 2. Hilbert–Smith conjecture Eilenberg, Samuel (1949), "On the Problems

F(U-Y)} is bijective for every open subset U and a closed subset Y of U of codimension at least 2. With this terminology, a coherent sheaf on an integral normal

of Codimension Two. American Mathematical Society, 1064–5616/95 (42 pages). 7. A.V. Abrosimov (2003). Linearity of Standard Quadrics of Codimension m in

winning the Van Amringe Mathematical Prize. He is best known for his "codimension one splitting theorem", which is a standard tool in high-dimensional

an irreducible closed subset of X {\displaystyle {\mathfrak {X}}} of codimension 1, k i ∈ Z {\displaystyle k_{i}\in \mathbb {Z} } , and λ ∞ ∈ R {\displaystyle

B {\textstyle W_{B}} with codimension j − 1 {\textstyle j-1} . Now W A ∩ W B {\textstyle W_{A}\cap W_{B}} has codimension ≤ i + j − 2 {\textstyle \leq

invariants of symplectic manifolds, and the proof of the vanishing in codimension at least g of the tautological ring of the moduli space of genus-g curves

Ph.D. in 1969 at Columbia University. His dissertation, A Theory of Codimension One Phenomena with an Application to the Theory of Purely Inseparable

{\displaystyle i:Z\to X} a closed immersion of a regular scheme of pure codimension r, an integer n that is invertible on the base scheme, F {\displaystyle

bundle on V, and V the pullback of any section of Vm vanishes along any codimension 1 component of the exceptional locus of a resolution of its singularities

connected manifold with Euler characteristic zero and dimension > 3 has a C1 codimension-one foliation that has no compact leaf. Schweitzer was raised in New

dimension of A. If Y is an irreducible closed subset of a scheme X, then the codimension of Y in X is the supremum of the lengths ℓ of chains of irreducible closed

> 0, the space of points of Y where the fiber has dimension r is of codimension greater than 2r. Roughly speaking, this means that most fibers are small

a local field, the space of functions with compact support in F* has codimension 0, 1, or 2 in the Kirillov model, depending on whether the irreducible

minimal submanifolds in every Riemannian manifold. In the special case of codimension 1 cycles with mod 2 coefficients Almgren isomorphism theorem implies

positive. Step 1: Find an ideal h {\displaystyle {\mathfrak {h}}} of codimension one in g {\displaystyle {\mathfrak {g}}} . This is the most difficult

X be a complex manifold, D ⊂ X a reduced divisor (a sum of distinct codimension-1 complex subspaces), and ω a holomorphic p-form on X−D. If both ω and

kind of Haefliger structure, which uniquely determines the foliation. A codimension- q {\displaystyle q} Haefliger structure on a topological space X {\displaystyle

be of codimension two in the M i {\displaystyle M_{i}} , as the following argument shows. A symplectic sum along a submanifold of codimension 2 k {\displaystyle

University of California, San Diego with doctoral thesis Link homotopy in codimension 2. As a postdoc Bartels was at the University of Münster, where he habilitated

is also possible to study the Gauss map of minimal surfaces of higher codimension in higher-dimensional Euclidean spaces. There are a number of variants

{\displaystyle n} . If two submanifolds A , B {\displaystyle A,B} of codimension i {\displaystyle i} and j {\displaystyle j} intersect transversely, then

the group Z r ( X ) {\displaystyle Z^{r}(X)} of algebraic cycles of codimension r on X, γ X : Z r ( X ) → H 2 r ( X ) , {\displaystyle \gamma _{X}:Z^{r}(X)\to

1995. Her dissertation The Equations of Rees Algebras of ideals of Low Codimension was supervised by Wolmer Vasconcelos. After receiving her Ph.D., Morey

X^{(p)}} the set of codimension p {\displaystyle p} points, meaning the set of subschemes x : Y → X {\displaystyle x:Y\to X} of codimension p {\displaystyle

2 {\displaystyle \geq 2} . Item (i) is often phrased as "regular in codimension 1". Note (i) implies that the set of associated primes A s s ( A ) {\displaystyle

Fredholm operator is one with finite dimensional kernel and range of finite codimension (equivalent to the kernel of its adjoint being finite dimensional). Note

Riemann curvature tensor. In the even more general setting of arbitrary codimension, the formula involves a complicated polynomial in the second fundamental

{w}})} . Then V , W {\displaystyle V,W} have codimension one, while V ∩ W {\displaystyle V\cap W} has codimension three. Some authors such as Bloch define

spacelike infinity has codimension 2. Anti-de Sitter space: there's no timelike or null infinity, and spacelike infinity has codimension 1. de Sitter space:

{\mathcal {E}}} on a smooth projective variety X {\displaystyle X} and codimension 2 subvarieties Y {\displaystyle Y} using a certain Ext 1 {\displaystyle

Morse foliation F on a manifold M is a singular transversely oriented codimension one foliation of class C 2 {\displaystyle C^{2}} with isolated singularities

Morse foliation F on a manifold M is a singular transversely oriented codimension one foliation of class C 2 {\displaystyle C^{2}} with isolated singularities

regularity theory of minimal submanifolds obtained by Almgren in the case of codimension 1. He showed that when the dimension n of the manifold is between 3 and

{\mathcal {O}}_{g}(\Sigma )} to any symmetry element g {\displaystyle g} and codimension-1 hypersurface Σ {\displaystyle \Sigma } such that O g ( Σ ) {\displaystyle

{R} ^{n}} . Then for any rectifiable surface S {\displaystyle S} of codimension 1, we have area ⁡ ( S ) = C n ∬ n γ ( φ , p ) d φ d p . {\displaystyle

where U ⊂ X {\displaystyle U\subset X} is some open subset with the codimension codim ⁡ ( X ∖ U ) ≥ 2 {\displaystyle \operatorname {codim} (X\backslash

Equivalence relations on algebraic cycles and subvarieties of small codimension, Proceedings of Symposia in Pure Mathematics, volume 29, American Mathematical

is a real analytic submanifold R ⊆ M {\displaystyle R\subseteq M} of codimension r ( P ) {\displaystyle r(P)} containing P {\displaystyle P} and such

hypersurfaces in R 2 d + 1 {\displaystyle \mathbb {R} ^{2d+1}} (Hopf) or codimension two surfaces embedded in R 2 d + 2 {\displaystyle \mathbb {R} ^{2d+2}}

certainly contains an ideal of codimension one and by the ideal correspondence, it corresponds to an ideal of codimension one in g {\displaystyle {\mathfrak

existence and convergence of graphic mean curvature flow in arbitrary codimension", Inventiones Mathematicae 148 (2002), no. 3, 525-543 (with Knut Smoczyk)

construction in the context of an embedding of Hilbert manifolds of finite codimension, which can be used in string topology. M. Kreck, Differential Algebraic

over a Noetherian local ring is its dimension minus its depth. codimension The codimension of a prime ideal is another name for its #height. coefficient

studies and received a Ph.D. in 1973 for his doctoral dissertation titled Codimension-Two Surgery, written under the supervision of William Browder. After

/ c i j {\displaystyle 2\pi /c_{ij}} fixing the subspace Hi ∩ Hj of codimension 2. Thus, viewed as an abstract group, every reflection group is a Coxeter

AD-regular measures with bounded Riesz transform operator: the case of codimension 1", by Fedor Nazarov, Xavier Tolsa, and Alexander Volberg has been selected

University (current), Thesis Unknotting Close Embeddings of Polyhedra in Codimension Greater Than Three  (1969) Doctoral advisor James Milton Kister Website

dimensional manifold and codimension 1 boundary) and manifolds with corners (top dimensional manifold, codimension 1 boundary, codimension 2 corners). Whitney

f|_{U}:U{\overset {i}{\to }}Z{\overset {\pi }{\to }}Y} , a regular embedding of codimension k {\displaystyle k} followed by a smooth morphism of relative dimension

only if the kernel of f on T(V∗) contains a two-sided ideal of finite codimension. Equivalently, C ( V ) = ⋃ { I 0 ⊆ T ^ ( V ) : I ◃ T ( V ∗ ) , c o d

normal intersection is an intersection with the "expected" codimension (given a sum of codimensions). (Semple & Roth 1949, p.16) 4.  Local rings are integrally

Cambridge University Press, 1997. Eccles, Peter John. "Multiple points of codimension one immersions of oriented manifolds." Mathematical Proceedings of the

dimension d that satisfies the zero-tension condition and is connected in codimension one. When d is one, the zero-tension condition means that around each

James W. (1979), "Shrinking cell-like decompositions of manifolds. Codimension three", Annals of Mathematics, Second Series, 110 (1): 83–112, doi:10

most k. If all components of the degeneracy locus have the expected codimension (m – k)(n – k) then Porteous's formula states that its fundamental class

that an integrable module of 1-forms of rank r is the same thing as a codimension-r foliation. The correspondence to the definition in terms of vector

irreducible monodromy group can be realised by a Fuchsian system. The codimension of the variety of monodromy groups of regular systems of size n {\displaystyle

affine variety; cf. Hartogs' extension theorem. The subvarieties of codimension one in the affine space k n {\displaystyle k^{n}} are exactly the hypersurfaces

d-dimensional fiber bundle is flat if it can be endowed with a foliation of codimension d that is transverse to the fibers. The Milnor–Wood inequality is named

affine variety; cf. Hartogs' extension theorem. The subvarieties of codimension one in the affine space k n {\displaystyle k^{n}} are exactly the hypersurfaces

property of being a hypersurface (or certain real submanifolds of higher codimension) in complex space by studying the properties of holomorphic vector fields

dimensions 2, 4, 8, 16. Hartshorne, Robin (1974), "Varieties of small codimension in projective space" (PDF), Bulletin of the American Mathematical Society

corresponds to integration. Let i: X → Y be a (closed) regular embedding of codimension d, Y' → Y a morphism and i': X' = X ×Y Y' → Y' the induced map. Let N

{\displaystyle \mathbb {R} ^{6}} (Haefliger 1962) (Levine 1965). Thus the codimension of a smooth knot can be arbitrarily large when not fixing the dimension

at x. The rank of the fibration in the product is one less than the codimension of X ⊂ P n {\displaystyle X\subset \mathbf {P} ^{n}} , so that the total

are arranged in increasing order with respect to the codimension so that the first l have codimension one and Zi is the zero-set of the irreducible polynomial

existence of triangulations is unknown. Simplices Glue together pairs of codimension-one faces Jaco-Shalen/Johannson torus decomposition Irreducible, orientable

operator S is injective (actually, isometric) and has a closed range of codimension 1, hence S is Fredholm with ind ⁡ ( S ) = − 1 {\displaystyle \operatorname

Štan'ko, M. A. (1971), "Approximation of the imbedding of compacta in a codimension larger than two", Doklady Akademii Nauk SSSR, 198: 783–786, ISSN 0002-3264

cohomological restrictions on the underlying manifold; by Stokes' theorem, every codimension-one complex subvariety is homologically nontrivial. For instance, the

Noetherian regular scheme, the closed parafactorial subsets are those of codimension at least 2. Grothendieck, Alexander; Raynaud, Michele (2005) [1968],

quasi-elliptic fibration.) It follows that having an elliptic fibration is a codimension-1 condition on a K3 surface. So there are 19-dimensional families of

variety), a rational function is regular if and only if it has no poles of codimension one. This is an algebraic analog of Hartogs' extension theorem. There

theorem any complete, full and irreducible isoparametric submanifold of codimension > 2 is an orbit of a s-representation, i.e. an H-orbit as above where

James W. (1979), "Shrinking cell-like decompositions of manifolds. Codimension three.", Annals of Mathematics, Second Series, 110 (1): 83–112, doi:10

S'=\operatorname {span} \{u_{k},u_{k+1},\ldots \}} . The subspace S' has codimension k − 1. By the same dimension count argument as in the matrix case, S'

additional 2003 (FINAL subroutines, GENERIC keyword,) and 2008 (Coarrays, CODIMENSION, SYNC ALL, SYNC IMAGES, SYNC MEMORY, CRITICAL, LOCK, ERROR STOP, ALLOCATE/DEALLOCATE)

This definition generalizes in higher dimensions to hypersurfaces (codimension 1 subvarieties), rather than to sets of points, and regular divisors

occurring in lensing and caustics produced by generic general maps up to codimension five. The invariants hold with a probability of 1 for random lenses and

integral domain). In a normal analytic space, the singular locus has codimension at least two. When X is a local complete intersection at x, then X is

ISBN 9780821850886. Rourke, Colin; Fenn, Roger (1992). "Racks and links in codimension 2". Journal of Knot Theory and Its Ramifications. 1 (4): 343–406. doi:10

for complete intersections the number of equations can be taken as the codimension; but the general projective variety has no defining set of equations

to treat modules M and N as pseudo-isomorphic if M/N has support of codimension at least two. This idea is much used in Iwasawa theory. The derived category

curvature is n+1 times the Fubini–Study symplectic form). Let D be a smooth codimension-1 subvariety in Pn. The adjunction formula implies that KD = (KX + D)|D

singular". The conjectural solution to this problem is the flip, a kind of codimension-2 surgery operation on X i {\displaystyle X_{i}} . It is not clear that

11) showed that every normal variety is regular outside a subset of codimension at least 2, and a similar result is true for schemes. So, for example

soap film regularity of a solution to a universal Plateau's problem for codimension one surfaces using Hausdorff measure to define area. As a graduate student

C* algebras over AdS. Any two distinct null geodesic hypersurfaces of codimension 1 which intersect at more than just a point in AdS divides AdS into four

\ldots ,\alpha _{n}} homology classes in X, such that the sum of the codimensions of β , α 1 , … , α n {\displaystyle \beta ,\alpha _{1},\ldots ,\alpha

bornivores. Suppose M {\displaystyle M} is a vector subspace of finite codimension in a locally convex space X {\displaystyle X} and B ⊆ M . {\displaystyle

Epstein and E. Vogt, A counterexample to the periodic orbit conjecture in codimension 3. Annals of Mathematics (2) 108 (1978), no. 3, 539–552. D.B.A. Epstein

bornivores. Suppose M {\displaystyle M} is a vector subspace of finite codimension in a locally convex space X {\displaystyle X} and B ⊆ M . {\displaystyle

E. Araujo, Carolina; Corrêa, Mauricio; Massarenti, Alex (2018), "Codimension one Fano distributions on Fano manifolds", Communications in Contemporary

fiber over p {\displaystyle p} . More generally, one can blow up any codimension-k complex submanifold Z of Cn. Suppose that Z is the locus of the equations

whenever it is a closed subspace with infinite dimension and infinite codimension. The AIHS problem asks whether every operator admits an AIHS. In the

Combinatorics. – 2011. – V. 33, No.4. – P. 531–542. Dzhumadildaev A.S., Codimension growth and non-Koszulity of Novikov operad // Communications in Algebra

Keyfitz, 'Classification of one state variable bifurcation problems up to codimension seven', Dynamics and Stability of Systems, 1, (1986), 1-41. B. L. Keyfitz

{\displaystyle Y.} Suppose M {\displaystyle M} is a vector subspace of finite codimension in a locally convex space X {\displaystyle X} and B ⊆ M . {\displaystyle

JSTOR 1970726, MR 0246310. Cappell, Sylvain E.; Shaneson, Julius L. (1974), "The codimension two placement problem and homology equivalent manifolds", Annals of Mathematics

singularities since it gives a process for resolving singularities of codimension 1. For example, the integral closure of C [ x , y , z ] / ( x y ) {\displaystyle

S2CID 120702104. Haefliger, André (1966). "Enlacements de sphères en codimension supérieure à 2". Commentarii Mathematici Helvetici. 41: 51–72. doi:10

definitions of divisors, but in general they form an abstraction of a codimension-one subvariety of an algebraic variety, the set of solution points of

{\mathcal {L}}} . The dimension of L {\displaystyle {\mathcal {L}}} is the codimension of V ( F ) {\displaystyle {\mathbf {V} }(F)} , and chosen to intersect

{\text{II}}(v,w)=\langle S(v),w\rangle } It can be also generalized to arbitrary codimension, in which case it is a quadratic form with values in the normal space

Topology 3 (1999), 103-118. Haefliger, André, Enlacements de sphères en codimension supérieure à 2 Munson, Brian (2005), Syllabus for Math 283: Calculus

are very close. For example, they are isomorphic outside subsets of codimension at least 2, and more precisely they are related by a sequence of flops

define the canonical divisor. If the variety is normal, it is smooth in codimension one. In particular, we can define canonical divisor on the smooth locus

thesis Restrictions of the Fourier transform to submanifolds of low codimension. At Princeton University, Christ worked with Elias M. Stein from 1982

made a general study of the Willmore functional for surfaces in general codimension, relating the value of the functional to several geometric quantities

Trotman and E.C.Zeeman, The classification of elementary catastrophes of codimension less than or equal to 5, in Structural stability, the theory of catastrophes

H_{2n-i}(X,\mathbf {Z} (n-j)).} In particular, the Chow group CHi(X) of codimension-i cycles is isomorphic to H2i(X,Z(i)) when X is smooth over k. The motivic

integral and the regularity of area-minimizing rectifiable currents up to codimension 2, World Scientific Monograph Series in Mathematics, vol. 1, River Edge

M., & Thompson, G. (2006). "The inverse problem for six-dimensional codimension two nilradical Lie algebras". Journal of Mathematical Physics. 47 (11):

regular. This is briefly stated as "arithmetic surfaces are regular in codimension one." The theory is developed in Hartshorne's Algebraic Geometry, for

chosen exactly so that the moduli space of multiply covered maps has codimension at least two in the space of non-multiply-covered maps. Then the image

union of locally closed submanifolds of G and using induction on the codimension of the strata. While it is possible for an eigenfunction of a differential

Fenchel in 1940 to a Riemannian submanifold of a Euclidean space of any codimension, for which they used the Lipschitz–Killing curvature (the average of

since the equidistant locus for two points may fail to be subspace of codimension 1, even in the two-dimensional case. A weighted Voronoi diagram is the

Therefore a PU(n) gauge theory with adjoint scalars will have nontrivial codimension 2 vortices in which the expectation values of the scalars wind around

Cauchy surface, in fact, a state can be defined on any surface. Σ is of codimension one, and so one may cut M along Σ. After such a cutting M will be a manifold

conclude that a minimal hypersurface is analytic outside a closed subset of codimension at least two.[citation needed] He also established regularity theory

3, 399–422. Schaps, Mary. Deformations of Cohen–Macaulay schemes of codimension 2 and non-singular deformations of space curves. Amer. J. Math. 99 (1977)

PMID 9120378. S2CID 41903562. Glendinning, P.; Sparrow, C. (1986). "T-points: A codimension two heteroclinic bifurcation". Journal of Statistical Physics. 43 (3–4):

dimension n {\displaystyle n} and a subvariety Y {\displaystyle Y} of codimension k {\displaystyle k} , there is the formula c k ( O Y ) = ( − 1 ) k −

explicitly given as the complement of a set in T {\displaystyle T} , a set of codimension-one subtori corresponding to the root system of G {\displaystyle G}

locally convex bornological space X {\displaystyle X} that has finite codimension in X {\displaystyle X} is bornological. The finest locally convex topology

AD-regular measures with bounded Riesz transform operator: the case of codimension 1". Acta Mathematica. 213 (2): 237–321. arXiv:1212.5229. doi:10.1007/s11511-014-0120-7

Center vortices also exist on higher dimensional spaces. They are always codimension two, and the above construction is generalized by cutting along a tube

i=1,\ldots ,n} , such that their boundaries are smooth submanifolds of codimension one. Let B = M   ( ⋃ i = 1 n Int ⁡ ( B i ) ) {\displaystyle B=M\ (\bigcup

J. Math. Oxford (2), 30 (1979). J.Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44, 229-244 (1969) J.Milnor, Infinite cyclic

commutator subgroup of N. Denote by p the dimension of Z and by q the codimension of Z; i.e. the dimension of N is p+q. It is known (see Raghunathan) that

Equivalence relations on algebraic cycles and subvarieties of small codimension, Proceedings of Symposia in Pure Mathematics, volume 29, American Mathematical

Vortices are points in 2D fields and lines in 3D fields (as they have codimension two). Integrating the phase of the field around a path enclosing a vortex

{\displaystyle X} is semi-positive, this is indeed the case (except for codimension 2 boundaries of the moduli space) if the almost complex structure J {\displaystyle

no. 1079, 116 pp. "The rational homology of spaces of long knots in codimension >2", with P. Lambrechts and V. Turchin, Geometry & Topology, 14 (2010)

Volodymyr; Wiesner, Emilie (2014). "Simple Virasoro modules induced from codimension one subalgebras of the positive part". Proceedings of the American Mathematical

Semmes: Uniform rectifiability and quasiminimizing sets of arbitrary codimension, Memoirs AMS 2000 with Stephen Semmes: Singular integrals and rectifiable

described various classes of Semi-Einstein submanifolds of arbitrary codimension in Euclidean spaces, Has established fundamental interrelation between

difference between Weil divisors (in the free abelian group generated by codimension-one subvarieties), and Cartier divisors coming from sections of invertible

Vilonen, Kari (1 September 2014). "Microdifferential systems and the codimension-three conjecture". Annals of Mathematics. 180 (2): 573–620. arXiv:1209

of positive scalar curvature is preserved by topological surgery in codimension at least three, and in particular is preserved by the connected sum.

homotopy equivalence. In particular, if Y {\displaystyle Y} has pure codimension, then it yields the long exact sequence for higher Chow groups (called

class is closed under the operation of connected sum and of surgery in codimension at least three. Their proof used elementary methods of partial differential

definition is the intersection of hypersurfaces (subvarieties of X of codimension one) that are in general position at x. Specifically, assume we have

follows naturally. In general, if we have a family of submanifolds with codimension c then we need to have at least a c-parameter family of such submanifolds

"Piecewise linear approximations of embeddings of cells and spheres in codimensions higher than two". Mathematics of the USSR-Sbornik. 9 (3): 321–343. Bibcode:1969SbMat

^{2}}}\,d{\vec {x}}^{2}} in which 0 ≤ r {\displaystyle 0\leq r} . The codimension 2 surface r = 0 {\displaystyle r=0} is the Poincaré Killing horizon and

homomorphisms φ: Ck(M) → R, as such a homomorphism φ corresponds to a codimension one ideal in Ck(M) (namely the kernel of φ), which is necessarily a maximal

versal deformation, while the non-generic maps form a subset of infinite codimension in the space of all germs. Varchenko was among creators of the theory

G acts freely and the complement Z = V − U {\displaystyle Z=V-U} has codimension > − dim ⁡ G − p {\displaystyle >-\operatorname {dim} G-p} . Let ∗ × G

step works for varieties X of any dimension (with points replaced by codimension 1 subvarieties), not just curves. The same long exact sequence above

\operatorname {Fix} (r):=\operatorname {ker} (r-\operatorname {Id} _{V})} has codimension 1. A (finite) complex reflection group W ⊆ G L ( V ) {\displaystyle W\subseteq

Viktor K. (2017-07-25). "Fast–Slow Bursters in the Unfolding of a High Codimension Singularity and the Ultra-slow Transitions of Classes". The Journal of

energy, they will converge smoothly on the complement of a subset of codimension at least four, known as the complement of the "singular set". Tian showed

Later, Calabi studied the two-dimensional minimal surfaces (of high codimension) in round spheres.[C67] He proved that the area of topologically spherical

Little, John A.; Pohl, William F. (1971). "On tight immersions of maximal codimension" (PDF). Inventiones Mathematicae. 13 (3): 179–204. Bibcode:1971InMat

with values in a surface is a minimal submanifold of the domain with codimension 2. This gives an attractive method for manufacturing whole families of

Little, John A.; Pohl, William F. (1971). "On tight immersions of maximal codimension" (PDF). Inventiones Mathematicae. 13 (3): 179–204. Bibcode:1971InMat

2, 133–143 (1988). Extreme positive functionals and ideals of finite codimension in commutative Banach * -algebras. Atti Semin. Mat. Fis. Univ. Modena

which deals simultaneously with high-regularity embeddings and low codimension. Nirenberg's work on the Minkowski problem was extended to Riemannian

complete intersection defined by a regular sequence of k polynomials has a codimension of k, and that its degree is the product of the degrees of the polynomials

general, when the last number of the Schläfli symbol becomes ∞, faces of codimension two intersect the Poincaré hyperball only in one ideal point. There are

(1991), 45-68. An invariant subspace of the Bergman space having the codimension two property. Journal für die reine und angewandte Mathematik 443 (1993)

a nontrivial, irreducible, invariant subspace W {\displaystyle W} of codimension one. Let C V {\displaystyle C_{V}} denote the action of C {\displaystyle

{\mathfrak {m}})} be the local ring of a variety X at a closed subvariety V codimension n (for example, V can be a closed point). Then length A ⁡ ( A / m t )

irreducible closed subset of P n {\displaystyle \mathbb {P} ^{n}} of codimension one is a hypersurface; i.e., the zero set of some homogeneous irreducible

nonincreasing.[H90] He later extended his formula to allow for general codimension and general positive solutions of the "backwards" heat equation; the

where A k ( X ) {\displaystyle A^{k}(X)} denotes the Chow-cycles of codimension k. Correspondences are often denoted using the "⊢"-notation, e.g., α

↪ X {\displaystyle Y_{i}\hookrightarrow X} are regular embeddings of codimensions c i {\displaystyle c_{i}} and if W := ⋂ i Y i ↪ X {\textstyle W:=\bigcap

Jeff; Naber, Aaron (2015). "Regularity of Einstein Manifolds and the Codimension 4 Conjecture". Annals of Mathematics. 182 (3): 1093–1165. arXiv:1406

H^{2}(X,{\mathcal {K}}_{2})} is isomorphic to the Chow group CH2(X) of codimension 2 cycles on X. Inspired by this, Gersten conjectured that for a regular

{\displaystyle T_{x}} of | D χ E | {\displaystyle |D\chi _{E}|} , i.e. a codimension-1 subspace T x {\displaystyle T_{x}} of R n {\displaystyle \mathbb {R}

dimension smaller than n {\displaystyle n} In particular, there is no codimension 1 totally geodesic subspace of the complex hyperbolic space. On the unit

is an isomorphism. Weil divisor Another but more standard term for a "codimension-one cycle"; see divisor. Weil reciprocity See Weil reciprocity. Zariski–Riemann

notational convention S λ {\displaystyle S_{\lambda }} indicated above, its codimension in G r ( k , n ) {\displaystyle \mathbf {Gr} (k,n)} is the weight | λ

Inventiones Mathematicae. vol. 140 (2000), no. 3, pp. 605–637 M. Sageev. Codimension-1 subgroups and splittings of groups. Journal of Algebra, vol. 189 (1997)

frame bundle so that its tangent vectors lie in a special subspace of codimension one in the three-dimensional tangent space of the frame bundle. The projection

× C , L ) {\displaystyle (X\times \mathbb {C} ,L)} outside a set of codimension 2. Uniformly K-stable if DF ⁡ ( X , L ) ≥ ε ‖ ( X , L ) ‖ m {\displaystyle

ideal, especially if the number of generators needed is larger than the codimension of the corresponding variety. polarization A method of reducing the degree

we are given Y″ → Y' and suppose i': X' = X ×Y Y' → Y' is regular of codimension d' so that one can define i'! as before. Let F be the excess bundle of

{\displaystyle X=\operatorname {Bl} _{Z}(Y)} of a scheme Y {\displaystyle Y} at a codimension k {\displaystyle k} locally complete intersection subscheme Z {\displaystyle

\lambda _{n}(x_{0}).} Therefore, one expects hyperbolic LCSs to appear as codimension-one local maximizing surfaces (or ridges) of the FTLE field. This expectation

the closure of the image of S contains the image of ½ I – T, which has codimension 1. Its orthogonal complement is given by the kernel of T – ½ I, i.e.

systems Limit cycles Melbourne, Ian (1989-10-01). "Intermittency as a codimension-three phenomenon". Journal of Dynamics and Differential Equations. 1

(projecting the codomain of the intermediate function to the output codimension) operators, respectively. These operators act pointwise on functions