In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of...

In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common...

between a subspace and its ambient space is known as its codimension. A hyperplane has codimension 1. In geometry, a hyperplane of an n-dimensional space...

linear algebra and geometry, relative dimension is the dual notion to codimension. In linear algebra, given a quotient map V → Q {\displaystyle V\to Q}...

bundle, so it cannot immerse in codimension 0 (in ⁠ R 2 {\displaystyle \mathbb {R} ^{2}} ⁠), though it embeds in codimension 1 (in ⁠ R 3 {\displaystyle \mathbb...

is called the dimension of the foliation and q = n − p is called its codimension. In some papers on general relativity by mathematical physicists, the...

the middle dimension has codimension more than 2: when the codimension is 2, one encounters knot theory, but when the codimension is more than 2, embedding...

stable manifolds of the saddle. In three or more dimensions, higher codimension bifurcations can occur, producing complicated, possibly chaotic dynamics...

tautness is a rigidity property of foliations. A taut foliation is a codimension 1 foliation of a closed manifold with the property that every leaf meets...

something happens, it happens in a particular codimension". For example, ramification is a phenomenon of codimension 1 (in the geometry of complex manifolds...

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

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...

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

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...

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...

2000), states that the singular set of a mass-minimizing surface has codimension at least 2. Almgren's proof of this was 955 pages long. Within the proof...

or those of all prime ideals. The height is also sometimes called the codimension, rank, or altitude of a prime ideal. In a Noetherian ring, every prime...

numbers Cayley–Dickson construction Dimensions by number Zero One Two Three Four Five Six Seven Eight n-dimensions See also Hyperspace Codimension Category...

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

numbers Cayley–Dickson construction Dimensions by number Zero One Two Three Four Five Six Seven Eight n-dimensions See also Hyperspace Codimension Category...

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

numbers Cayley–Dickson construction Dimensions by number Zero One Two Three Four Five Six Seven Eight n-dimensions See also Hyperspace Codimension Category...

surface). The second fundamental form can be generalized to arbitrary codimension. In that case it is a quadratic form on the tangent space with values...

point, (0, 0, 0), which is singular. The same considerations apply to codimension. For example a smooth complex hypersurface in complex projective space...

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...

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;...

isomorphism HiX ≅ Hn−iX. As a result, a closed oriented submanifold S of codimension i in X determines a cohomology class in HiX, called [S]. In these terms...

numbers Cayley–Dickson construction Dimensions by number Zero One Two Three Four Five Six Seven Eight n-dimensions See also Hyperspace Codimension Category...

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...