In mathematics, a tubular neighborhood of a

submanifold 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 satisfies certain properties. There are different types of submanifolds depending on exactly which ...

of a smooth manifold is an

open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are ...

around it resembling the normal bundle. The idea behind a tubular neighborhood can be explained in a simple example. Consider a

smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebraic ...

curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

in the plane without self-intersections. On each point on the curve draw a line

perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...

to the curve. Unless the curve is straight, these lines will intersect among themselves in a rather complicated fashion. However, if one looks only in a narrow band around the curve, the portions of the lines in that band will not intersect, and will cover the entire band without gaps. This band is a tubular neighborhood. In general, let ''S'' be a

of a

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 ...

''M'', and let ''N'' be the normal bundle of ''S'' in ''M''. Here ''S'' plays the role of the curve and ''M'' the role of the plane containing the curve. Consider the natural map :

i : N_0 \to S

which establishes a

bijective In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...

correspondence between the

zero section In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...

N_0

of ''N'' and the submanifold ''S'' of ''M''. An extension ''j'' of this map to the entire normal bundle ''N'' with values in ''M'' such that

j(N)

is an open set in ''M'' and ''j'' is a

homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorph ...

between ''N'' and

j(N)

is called a tubular neighbourhood. Often one calls the open set

T = j(N),

rather than ''j'' itself, a tubular neighbourhood of ''S'', it is assumed implicitly that the homeomorphism ''j'' mapping ''N'' to ''T'' exists.

Normal tube

A normal tube to a

is a

defined as the

union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''U ...

of all discs such that * all the discs have the same fixed radius; * the center of each disc lies on the curve; and * each disc lies in a plane normal to the curve where the curve passes through that disc's center.

Formal definition

Let

S \subseteq M

be smooth manifolds. A tubular neighborhood of

S

M

is a vector bundle

\pi: E \to S

together with a smooth map

J : E \to M

such that *

J \circ 0_E = i

where

i

is the embedding

S \hookrightarrow M

and

0_E

the zero section * there exists some

U \subseteq E

and some

V \subseteq M

with

\subseteq U

and

S \subseteq V

such that

J\vert_U : U \to V

is a diffeomorphism. The normal bundle is a tubular neighborhood and because of the diffeomorphism condition in the second point, all tubular neighborhood have the same dimension, namely (the dimension of the vector bundle considered as a manifold is) that of

M.

Generalizations

Generalizations of smooth manifolds yield generalizations of tubular neighborhoods, such as regular neighborhoods, or spherical fibrations for

Poincaré space In algebraic topology, a Poincaré space is an ''n''-dimensional topological space with a distinguished element ''µ'' of its ''n''th homology group such that taking the cap product with an element of the ''k''th cohomology group yields an isomorphi ...

s. These generalizations are used to produce analogs to the normal bundle, or rather to the

stable normal bundle In surgery theory, a branch of mathematics, the stable normal bundle of a differentiable manifold is an invariant which encodes the stable normal (dually, tangential) data. There are analogs for generalizations of manifold, notably PL-manifolds ...

, which are replacements for the tangent bundle (which does not admit a direct description for these spaces).

References

* * * {{commons category, Tubular neighborhood Manifolds Geometric topology Smooth manifolds

Normal tube

Formal definition

Generalizations

See also

References