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 to the pair , with the standard inclusion of That is, there exists a homeomorphism such that the image of coincides with . In diagrammatic terms, the following square must commute:
We call Nlocally 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 boundaryEdit
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.
ConsequencesEdit
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 × [0,1] with 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).
local, flatness, topology, branch, mathematics, local, flatness, smoothness, condition, that, imposed, topological, submanifolds, category, topological, manifolds, locally, flat, submanifolds, play, role, similar, that, embedded, submanifolds, category, smooth. In topology a branch of mathematics local flatness is a smoothness condition that can be imposed on topological submanifolds In the category 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 Contents 1 Definition 1 1 In manifolds with boundary 2 Consequences 3 See also 4 ReferencesDefinition EditSuppose a d dimensional manifold N is embedded into an n dimensional manifold M where d lt n If x N displaystyle x in N nbsp we say N is locally flat at x if there is a neighborhood U M displaystyle U subset M nbsp of x such that the topological pair U U N displaystyle U U cap N nbsp is homeomorphic to the pair R n R d displaystyle mathbb R n mathbb R d nbsp with the standard inclusion of R d R n displaystyle mathbb R d to mathbb R n nbsp That is there exists a homeomorphism U R n displaystyle U to mathbb R n nbsp such that the image of U N displaystyle U cap N nbsp coincides with R d displaystyle mathbb R d nbsp In diagrammatic terms the following square must commute nbsp We call N locally flat in M if N is locally flat at every point Similarly a map x N M displaystyle chi colon N to M nbsp is called locally flat even if it is not an embedding if every x in N has a neighborhood U whose image x U displaystyle chi U nbsp is locally flat in M In manifolds with boundary Edit 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 U M displaystyle U subset M nbsp of x such that the topological pair U U N displaystyle U U cap N nbsp is homeomorphic to the pair R n R d displaystyle mathbb R n mathbb R d nbsp where R n displaystyle mathbb R n nbsp is a standard half space and R d displaystyle mathbb R d nbsp is included as a standard subspace of its boundary Consequences EditLocal 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 0 1 with 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 EditEuclidean space Neat submanifoldReferences EditBrown 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 Retrieved from https en wikipedia org w index php title Local flatness amp oldid 1174941948, wikipedia, wiki, book, books, library,