In
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, a branch of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, local flatness is smoothness condition that can be imposed on topological
submanifolds. In the
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
of topological manifolds, locally flat submanifolds play a role similar to that of
embedded submanifolds in the category of
smooth manifolds. Violations of local flatness describe ridge networks and
crumpled structures, with applications to materials processing and
mechanical engineering
Mechanical engineering is the study of physical machines that may involve force and movement. It is an engineering branch that combines engineering physics and mathematics principles with materials science, to design, analyze, manufacture, ...
.
Definition
Suppose a ''d'' dimensional manifold ''N'' is embedded into an ''n'' dimensional manifold ''M'' (where ''d'' < ''n''). If
we say ''N'' is locally flat at ''x'' if there is a neighborhood
of ''x'' such that the
topological pair is
homeomorphic
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 isomor ...
to the pair
, with the standard inclusion of
That is, there exists a homeomorphism
such that the
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
of
coincides with
. In diagrammatic terms, the following
square must commute:
We call ''N'' locally flat in ''M'' if ''N'' is locally flat at every point. Similarly, a map
is called locally flat, even if it is not an embedding, if every ''x'' in ''N'' has a neighborhood ''U'' whose image
is locally flat in ''M''.
In manifolds with boundary
The above definition assumes that, if ''M'' has a
boundary, ''x'' is not a boundary point of ''M''. If ''x'' is a point on the boundary of ''M'' then the definition is modified as follows. We say that ''N'' is locally flat at a boundary point ''x'' of ''M'' if there is a neighborhood
of ''x'' such that the topological pair
is homeomorphic to the pair
, where
is a standard
half-space and
is included as a standard subspace of its boundary.
Consequences
Local flatness of an embedding implies strong properties not shared by all embeddings. Brown (1962) proved that if ''d'' = ''n'' − 1, then ''N'' is collared; that is, it has a neighborhood which is homeomorphic to ''N'' ×
,1with ''N'' itself corresponding to ''N'' × 1/2 (if ''N'' is in the interior of ''M'') or ''N'' × 0 (if ''N'' is in the boundary of ''M'').
See also
*
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
*
Neat submanifold
References
* Brown, Morton (1962), Locally flat imbeddings{{sic of topological manifolds. ''Annals of Mathematics'', Second series, Vol. 75 (1962), pp. 331–341.
* Mazur, Barry. On embeddings of spheres. ''Bulletin of the American Mathematical Society'', Vol. 65 (1959), no. 2, pp. 59–65. http://projecteuclid.org/euclid.bams/1183523034.
Topology
Geometric topology