Submanifold, the Glossary
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.[1]
Table of Contents
36 relations: Alexander's theorem, Atlas (topology), Closed set, Compact space, Comparison of topologies, Diffeomorphism, Differentiable manifold, Differential structure, Embedding, Foliation, Frobenius theorem (differential topology), Graduate Texts in Mathematics, Image (mathematics), Immersion (mathematics), Inclusion map, Injective function, Lie group, Linear subspace, Manifold, Mathematics, Neat submanifold, Open and closed maps, Open set, Plane (mathematics), Proper map, Real coordinate space, Schoenflies problem, Second-countable space, Smoothness, Subset, Subspace topology, Tangent space, Topological space, Whitney embedding theorem, Wild arc, Wild knot.
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.
See Submanifold and Alexander's theorem
Atlas (topology)
In mathematics, particularly topology, an atlas is a concept used to describe a manifold. Submanifold and atlas (topology) are manifolds.
See Submanifold and Atlas (topology)
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.
See Submanifold and Closed set
Compact space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space.
See Submanifold and Compact space
Comparison of topologies
In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set.
See Submanifold and Comparison of topologies
Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds.
See Submanifold and Diffeomorphism
Differentiable 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.
See Submanifold and Differentiable manifold
Differential structure
In mathematics, an n-dimensional differential structure (or differentiable structure) on a set M makes M into an n-dimensional differential manifold, which is a topological manifold with some additional structure that allows for differential calculus on the manifold.
See Submanifold and Differential structure
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. Submanifold and embedding are differential topology.
Foliation
In mathematics (differential geometry), a foliation is an equivalence relation on an ''n''-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension p, modeled on the decomposition of the real coordinate space Rn into the cosets x + Rp of the standardly embedded subspace Rp.
Frobenius theorem (differential topology)
In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system of first-order homogeneous linear partial differential equations.
See Submanifold and Frobenius theorem (differential topology)
Graduate Texts in Mathematics
Graduate Texts in Mathematics (GTM) is a series of graduate-level textbooks in mathematics published by Springer-Verlag.
See Submanifold and Graduate Texts in Mathematics
Image (mathematics)
In mathematics, for a function f: X \to Y, the image of an input value x is the single output value produced by f when passed x. The preimage of an output value y is the set of input values that produce y. More generally, evaluating f at each element of a given subset A of its domain X produces a set, called the "image of A under (or through) f".
See Submanifold and Image (mathematics)
Immersion (mathematics)
In mathematics, an immersion is a differentiable function between differentiable manifolds whose differential pushforward is everywhere injective. Submanifold and immersion (mathematics) are differential topology.
See Submanifold and Immersion (mathematics)
Inclusion map
In mathematics, if A is a subset of B, then the inclusion map is the function \iota that sends each element x of A to x, treated as an element of B: \iota: A\rightarrow B, \qquad \iota(x).
See Submanifold and Inclusion map
Injective function
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies.
See Submanifold and Injective function
Lie group
In mathematics, a Lie group (pronounced) is a group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. Submanifold and Lie group are manifolds.
Linear subspace
In mathematics, and more specifically in linear algebra, a linear subspace or vector subspaceThe term linear subspace is sometimes used for referring to flats and affine subspaces.
See Submanifold and Linear subspace
Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. Submanifold and manifold are manifolds.
Mathematics
Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.
See Submanifold and Mathematics
Neat submanifold
In differential topology, an area of mathematics, a neat submanifold of a manifold with boundary is a kind of "well-behaved" submanifold. Submanifold and neat submanifold are differential topology.
See Submanifold and Neat submanifold
Open and closed maps
In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets.
See Submanifold and Open and closed maps
Open set
In mathematics, an open set is a generalization of an open interval in the real line.
Plane (mathematics)
In mathematics, a plane is a two-dimensional space or flat surface that extends indefinitely.
See Submanifold and Plane (mathematics)
Proper map
In mathematics, a function between topological spaces is called proper if inverse images of compact subsets are compact.
See Submanifold and Proper map
Real coordinate space
In mathematics, the real coordinate space or real coordinate n-space, of dimension, denoted or, is the set of all ordered n-tuples of real numbers, that is the set of all sequences of real numbers, also known as coordinate vectors.
See Submanifold and Real coordinate space
Schoenflies problem
In mathematics, the Schoenflies problem or Schoenflies theorem, of geometric topology is a sharpening of the Jordan curve theorem by Arthur Schoenflies. Submanifold and Schoenflies problem are differential topology.
See Submanifold and Schoenflies problem
Second-countable space
In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base.
See Submanifold and Second-countable space
Smoothness
In mathematical analysis, the smoothness of a function is a property measured by the number, called differentiability class, of continuous derivatives it has over its domain.
See Submanifold and Smoothness
Subset
In mathematics, a set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B. The relationship of one set being a subset of another is called inclusion (or sometimes containment).
Subspace topology
In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a topology induced from that of X called the subspace topology (or the relative topology, or the induced topology, or the trace topology).
See Submanifold and Subspace topology
Tangent space
In mathematics, the tangent space of a manifold is a generalization of to curves in two-dimensional space and to surfaces in three-dimensional space in higher dimensions. Submanifold and tangent space are differential topology.
See Submanifold and Tangent space
Topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance.
See Submanifold and Topological space
Whitney embedding theorem
In mathematics, particularly in differential topology, there are two Whitney embedding theorems, named after Hassler Whitney.
See Submanifold and Whitney embedding theorem
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.
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.
References
[1] https://en.wikipedia.org/wiki/Submanifold
Also known as Embedded submanifold, Immersed submanifold, Regular submanifold, Slice chart, Slice coordinate system.