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Slice Chart
In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S \rightarrow M satisfies certain properties. There are different types of submanifolds depending on exactly which properties are required. Different authors often have different definitions. Formal definition In the following we assume all manifolds are differentiable manifolds of class C^r for a fixed r\geq 1, and all morphisms are differentiable of class C^r. Immersed submanifolds An immersed submanifold of a manifold M is the image S of an immersion map f: N\rightarrow M; in general this image will not be a submanifold as a subset, and an immersion map need not even be injective (one-to-one) – it can have self-intersections. More narrowly, one can require that the map f: N\rightarrow M be an injection (one-to-one), in which we call it an injective immersion, and define an immersed submanifold to be the image subset S together with a to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (to allow division), or equivalently, the concept of addition and subtraction. Combining these two ideas, one obtains a continuous group where multiplying points and their inverses is continuous. If the multiplication and taking of inverses are smoothness, smooth (differentiable) as well, one obtains a Lie group. Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the circle group. Rotating a circle is an example of a continuous symmetry. For an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wild Knot
In the mathematical theory of knots, a knot is tame if it can be "thickened", that is, if there exists an extension to an embedding of the solid torus S^1\times D^2 into the 3-sphere. A knot is tame if and only if it can be represented as a finite closed polygonal chain. In knot theory and 3-manifold theory, often the adjective "tame" is omitted. Smooth knots, for example, are always tame. Knots that are not tame are called wild and can have pathological behavior. Every closed curve containing a wild arc is a wild knot. It has been conjectured that every wild knot has infinitely many quadrisecants. As well as their mathematical study, wild knots have also been studied for their potential for decorative purposes in Celtic-style ornamental knotwork. See also * Wild arc In geometric topology, a wild arc is an embedding of the unit interval into 3-dimensional space not equivalent to the usual one in the sense that there does not exist an ambient isotopy taking the arc to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wild Arc
In geometric topology, a wild arc is an embedding of the unit interval into 3-dimensional space not equivalent to the usual one in the sense that there does not exist an ambient isotopy taking the arc to a straight line segment. found the first example of a wild arc. found another example, called the Fox-Artin arc, whose complement is not simply connected. Fox-Artin arcs Two very similar wild arcs appear in the article. Example 1.1 (page 981) is most generally referred to as the Fox-Artin wild arc. The crossings have the regular sequence over/over/under/over/under/under when following the curve from left to right. The left end-point 0 of the closed unit interval ,1/math> is mapped by the arc to the left limit point of the curve, and 1 is mapped to the right limit point. The range of the arc lies in the Euclidean space \mathbb^3 or the 3-sphere S^3. Fox-Artin arc variant Example 1.1* has the crossing sequence over/under/over/under/over/under. According to , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neat Submanifold
In differential topology, an area of mathematics, a neat submanifold of a manifold with boundary is a kind of "well-behaved" submanifold. To define this more precisely, first let :M be a manifold with boundary, and :A be a submanifold of M. Then A is said to be a neat submanifold of M if it meets the following two conditions:. *The boundary of A is a subset of the boundary of M. That is, \partial A \subset \partial M. *Each point of A has a neighborhood within which A's embedding in M is equivalent to the embedding of a hyperplane in a higher-dimensional Euclidean space. More formally, A must be covered by charts (U, \phi) of M such that A \cap U = \phi^(\mathbb^m) where m is the dimension For instance, in the category of smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schoenflies Theorem
In mathematics, the Schoenflies problem or Schoenflies theorem, of geometric topology is a sharpening of the Jordan curve theorem by Arthur Schoenflies. For Jordan curves in the plane it is often referred to as the Jordan–Schoenflies theorem. Original formulation The original formulation of the Schoenflies problem states that not only does every simple closed curve in the plane separate the plane into two regions, one (the "inside") bounded and the other (the "outside") unbounded; but also that these two regions are homeomorphic to the inside and outside of a standard circle in the plane. An alternative statement is that if C \subset \mathbb R^2 is a simple closed curve, then there is a homeomorphism f : \mathbb R^2 \to \mathbb R^2 such that f(C) is the unit circle in the plane. Elementary proofs can be found in , , and . The result can first be proved for polygons when the homeomorphism can be taken to be piecewise linear and the identity map off some compact set; the case o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander's Theorem
In mathematics Alexander's theorem states that every knot or link can be represented as a closed braid; that is, a braid in which the corresponding ends of the strings are connected in pairs. The theorem is named after James Waddell Alexander II, who published a proof in 1923. Braids were first considered as a tool of knot theory In topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be und ... by Alexander. His theorem gives a positive answer to the question ''Is it always possible to transform a given knot into a closed braid?'' A good construction example is found in Colin Adams's book. However, the correspondence between knots and braids is clearly not one-to-one: a knot may have many braid representations. For example, conjugate braids yield equivalent knots. This leads to a second fund ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atlas (topology)
In mathematics, particularly topology, an atlas is a concept used to describe a manifold. An atlas consists of individual ''charts'' that, roughly speaking, describe individual regions of the manifold. In general, the notion of atlas underlies the formal definition of a manifold and related structures such as vector bundles and other fiber bundles. Charts The definition of an atlas depends on the notion of a ''chart''. A chart for a topological space ''M'' is a homeomorphism \varphi from an open subset ''U'' of ''M'' to an open subset of a Euclidean space. The chart is traditionally recorded as the ordered pair (U, \varphi). When a coordinate system is chosen in the Euclidean space, this defines coordinates on U: the coordinates of a point P of U are defined as the coordinates of \varphi(P). The pair formed by a chart and such a coordinate system is called a local coordinate system, coordinate chart, coordinate patch, coordinate map, or local frame. Formal definition of at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plane (mathematics)
In mathematics, a plane is a two-dimensional space or flat surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. When working exclusively in two-dimensional Euclidean space, the definite article is used, so ''the'' Euclidean plane refers to the whole space. Several notions of a plane may be defined. The Euclidean plane follows Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ..., and in particular the parallel postulate. A projective plane may be constructed by adding "points at infinity" where two otherwise parallel lines would intersect, so that every pair of lines intersects in exactly one point. The elliptic plane may be further defined by adding a metr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chart (topology)
In mathematics, particularly topology, an atlas is a concept used to describe a manifold. An atlas consists of individual ''charts'' that, roughly speaking, describe individual regions of the manifold. In general, the notion of atlas underlies the formal definition of a manifold and related structures such as vector bundles and other fiber bundles. Charts The definition of an atlas depends on the notion of a ''chart''. A chart for a topological space ''M'' is a homeomorphism \varphi from an open subset ''U'' of ''M'' to an open subset of a Euclidean space. The chart is traditionally recorded as the ordered pair (U, \varphi). When a coordinate system is chosen in the Euclidean space, this defines coordinates on U: the coordinates of a point P of U are defined as the coordinates of \varphi(P). The pair formed by a chart and such a coordinate system is called a local coordinate system, coordinate chart, coordinate patch, coordinate map, or local frame. Formal definition of atl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group (mathematics), group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is given by some Injective function, injective and structure-preserving map f:X\rightarrow Y. The precise meaning of "structure-preserving" depends on the kind of mathematical structure of which X and Y are instances. In the terminology of category theory, a structure-preserving map is called a morphism. The fact that a map f:X\rightarrow Y is an embedding is often indicated by the use of a "hooked arrow" (); thus: f : X \hookrightarrow Y. (On the other hand, this notation is sometimes reserved for inclusion maps.) Given X and Y, several different embeddings of X in Y may be possible. In many cases of interest there is a standard (or "canonical") embedding, like those of the natural numbers in the integers, the integers i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is given by some injective and structure-preserving map f:X\rightarrow Y. The precise meaning of "structure-preserving" depends on the kind of mathematical structure of which X and Y are instances. In the terminology of category theory, a structure-preserving map is called a morphism. The fact that a map f:X\rightarrow Y is an embedding is often indicated by the use of a "hooked arrow" (); thus: f : X \hookrightarrow Y. (On the other hand, this notation is sometimes reserved for inclusion maps.) Given X and Y, several different embeddings of X in Y may be possible. In many cases of interest there is a standard (or "canonical") embedding, like those of the natural numbers in the integers, the integers in the rational numbers, the rational num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
