mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 S \rightarrow M satisfies certain properties. There are different types of submanifolds depending on exactly ...

of a smooth manifold is an

open set In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line. In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...

around it resembling the

normal bundle In differential geometry, a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding (or immersion). Definition Riemannian manifold Let (M,g) be a Riemannian ...

. The idea behind a tubular neighborhood can be explained in a simple example. Consider a smooth curve in the plane without self-intersections. On each point on the curve draw a line

perpendicular In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', � ...

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 N ...

''M'', and let ''N'' be the

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 correspondence between the zero section

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 mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...

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 smooth curve is a

defined as the union 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 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 eve ...

\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 spaces. These generalizations are used to produce analogs to the normal bundle, or rather to the stable normal bundle, 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