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 → M satisfies certain properties. There are different types of submanifolds depending on exactly which properties are required. Different authors often have different definitions.
In the following we assume all manifolds are differentiable manifolds of class Cr for a fixed r ≥ 1, and all morphisms are differentiable of class Cr.
An immersed submanifold of a manifold M is the image S of an immersion map f : N → 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 → 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 topology and differential structure such that S is a manifold and the inclusion f is a diffeomorphism: this is just the topology on N, which in general will not agree with the subset topology: in general the subset S is not a submanifold of M, in the subset topology.
Given any injective immersion f : N → M the image of N in M can be uniquely given the structure of an immersed submanifold so that f : N → f(N) is a diffeomorphism. It follows that immersed submanifolds are precisely the images of injective immersions.
The submanifold topology on an immersed submanifold need not be the relative topology inherited from M. In general, it will be finer than the subspace topology (i.e. have more open sets).
Immersed submanifolds occur in the theory of Lie groups where Lie subgroups are naturally immersed submanifolds. They also appear in the study of foliations where immersed submanifolds provide the right context to prove the Frobenius theorem.
An embedded submanifold (also called a regular submanifold), is an immersed submanifold for which the inclusion map is a topological embedding. That is, the submanifold topology on S is the same as the subspace topology.
Given any embedding f : N → M of a manifold N in M the image f(N) naturally has the structure of an embedded submanifold. That is, embedded submanifolds are precisely the images of embeddings.
There is an intrinsic definition of an embedded submanifold which is often useful. Let M be an n-dimensional manifold, and let k be an integer such that 0 ≤ k ≤ n. A k-dimensional embedded submanifold of M is a subset S ⊂ M such that for every point p ∈ S there exists a chart (U ⊂ M, φ : U → Rn) containing p such that φ(S ∩ U) is the intersection of a k-dimensional plane with φ(U). The pairs (S ∩ U, φ|S ∩ U) form an atlas for the differential structure on S.
Alexander's theorem and the Jordan–Schoenflies theorem are good examples of smooth embeddings.
There are some other variations of submanifolds used in the literature. A neat submanifold is a manifold whose boundary agrees with the boundary of the entire manifold.Sharpe (1997) defines a type of submanifold which lies somewhere between an embedded submanifold and an immersed submanifold.
Many authors define topological submanifolds also. These are the same as Cr submanifolds with r = 0. An embedded topological submanifold is not necessarily regular in the sense of the existence of a local chart at each point extending the embedding. Counterexamples include wild arcs and wild knots.
Given any immersed submanifold S of M, the tangent space to a point p in S can naturally be thought of as a linear subspace of the tangent space to p in M. This follows from the fact that the inclusion map is an immersion and provides an injection
Suppose S is an immersed submanifold of M. If the inclusion map i : S → M is closed then S is actually an embedded submanifold of M. Conversely, if S is an embedded submanifold which is also a closed subset then the inclusion map is closed. The inclusion map i : S → M is closed if and only if it is a proper map (i.e. inverse images of compact sets are compact). If i is closed then S is called a closed embedded submanifold of M. Closed embedded submanifolds form the nicest class of submanifolds.
Smooth manifolds are sometimes defined as embedded submanifolds of real coordinate space Rn, for some n. This point of view is equivalent to the usual, abstract approach, because, by the Whitney embedding theorem, any second-countable smooth (abstract) m-manifold can be smoothly embedded in R2m.
In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions of two variables; see the figure at right. However, surfaces can also be defined abstractly, without reference to any ambient space. For example, the Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space.
In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory.
In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution may be given as the kernel of a differential one-form, and the non-integrability condition translates into a maximal non-degeneracy condition on the form. These conditions are opposite to two equivalent conditions for 'complete 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 Frobenius theorem.
This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology.
This is a glossary of terms specific to differential geometry and differential topology. The following three glossaries are closely related:
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.
In mathematics, an n-dimensional differential 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. If M is already a topological manifold, it is required that the new topology be identical to the existing one.
In differential geometry, a Poisson structure on a smooth manifold is a Lie bracket on the algebra of smooth functions on , subject to the Leibniz rule
In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. The idea is to take advantage of the way a differential form restricts to a submanifold, and the fact that this restriction is compatible with the exterior derivative. This is one possible approach to certain over-determined systems, for example, including Lax pairs of integrable systems. A Pfaffian system is specified by 1-forms alone, but the theory includes other types of example of differential system. To elaborate, a Pfaffian system is a set of 1-forms on a smooth manifold.
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or n-manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to the Euclidean space of dimension n.
In mathematics, a differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply. If the charts are suitably compatible, then computations done in one chart are valid in any other differentiable chart.
In topology, a branch of mathematics, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The subspace is then called a retract of the original space. A deformation retraction is a mapping that captures the idea of continuously shrinking a space into a subspace.
In mathematics, a Hilbert manifold is a manifold modeled on Hilbert spaces. Thus it is a separable Hausdorff space in which each point has a neighbourhood homeomorphic to an infinite dimensional Hilbert space. The concept of a Hilbert manifold provides a possibility of extending the theory of manifolds to infinite-dimensional setting. Analogously to the finite-dimensional situation, one can define a differentiable Hilbert manifold by considering a maximal atlas in which the transition maps are differentiable.
In mathematics, an immersion is a differentiable function between differentiable manifolds whose derivative is everywhere injective. Explicitly, f : M → N is an immersion if
In differential topology, a branch of mathematics, the Whitney conditions are conditions on a pair of submanifolds of a manifold introduced by Hassler Whitney in 1965.
In topology, a branch of mathematics, local flatness is a property of a submanifold in a topological manifold of larger dimension. In the category of topological manifolds, locally flat submanifolds play a role similar to that of embedded submanifolds in the category of smooth manifolds. Local flatness and the topology of ridge networks is of importance in the study of crumpled structures with importance in materials processing and mechanical engineering.
In mathematics, especially in the area of mathematical analysis known as dynamical systems theory, a linear flow on the torus is a flow on the n-dimensional torus
In mathematics, the closed-subgroup theorem is a theorem in the theory of Lie groups. It states that if H is a closed subgroup of a Lie group G, then H is an embedded Lie group with the smooth structure agreeing with the embedding. One of several results known as Cartan's theorem, it was first published in 1930 by Élie Cartan, who was inspired by John von Neumann's 1929 proof of a special case for groups of linear transformations.
In functional analysis and related areas of mathematics, a complete topological vector space is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point towards which they all get closer to. The notion of "points that get progressively closer" is made rigorous by Cauchy nets or Cauchy filters, which are generalizations of Cauchy sequences, while "point towards which they all get closer to" means that this net or filter converges to Unlike the notion of completeness for metric spaces, which it generalizes, the notion of completeness for TVSs does not depend on any metric and is defined for all TVSs, including those that are not metrizable or Hausdorff.