Local flatness

 ( Topology ) 

In topology, a branch of mathematics, local flatness is a smoothness condition that can be imposed on topological . 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 Crumpling, with applications to materials processing and mechanical engineering.

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Definition Suppose a d dimensional manifold N is embedded into an n dimensional manifold M (where d < n). If $x\; \backslash in\; N,$ we say N is locally flat at x if there is a neighborhood $U\; \backslash subset\; M$ of x such that the topological pair $(U,\; U\backslash cap\; N)$ is homeomorphic to the pair $(\backslash mathbb\{R\}^n,\backslash mathbb\{R\}^d)$, with the standard inclusion of $\backslash mathbb\{R\}^d\backslash to\backslash mathbb\{R\}^n.$ That is, there exists a homeomorphism $U\backslash to\; \backslash mathbb\{R\}^n$ such that the image of $U\backslash cap\; N$ coincides with $\backslash mathbb\{R\}^d$. In diagrammatic terms, the following Commuting square:

We call N locally flat in M if N is locally flat at every point. Similarly, a map $\backslash chi\backslash colon\; N\backslash to\; M$ is called locally flat, even if it is not an embedding, if every x in N has a neighborhood U whose image $\backslash chi(U)$ is locally flat in M.

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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 $U\backslash subset\; M$ of x such that the topological pair $(U,\; U\backslash cap\; N)$ is homeomorphic to the pair $(\backslash mathbb\{R\}^n\_+,\backslash mathbb\{R\}^d)$, where $\backslash mathbb\{R\}^n\_+$ is a standard half-space and $\backslash mathbb\{R\}^d$ is included as a standard subspace of its boundary.

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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 × 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).

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Non-example Let $K$ be a non-trivial knot in $S^3$; that is, a connected, locally flat one-dimensional submanifold of $S^3$ such that the pair $(S^3,\; K)$ is not homeomorphic to $(S^3,\; S^1)$. Then the cone on $K$ from the center $\backslash underline\{0\}$ of $D^4$ is a submanifold of $D^4$, but it is not locally flat at $\backslash underline\{0\}$.

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See also

• Euclidean space
• Neat submanifold

• Brown, Morton (1962), Locally flat imbeddings 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.

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