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An imbedding (cf. [[Immersion|Immersion]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l0604301.png" /> of one topological manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l0604302.png" /> into another <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l0604303.png" /> such that for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l0604304.png" /> there are charts in a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l0604305.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l0604306.png" /> and in a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l0604307.png" /> of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l0604308.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l0604309.png" /> in which the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l06043010.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l06043011.png" /> linearly maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l06043012.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l06043013.png" />. In other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l06043014.png" /> is locally linear in suitable coordinate systems. Equivalently: There are neighbourhoods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l06043015.png" /> of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l06043016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l06043017.png" /> of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l06043018.png" /> such that the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l06043019.png" /> can be mapped homeomorphically onto a standard pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l06043020.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l06043021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l06043022.png" /> is the unit ball of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l06043023.png" /> with centre at the origin and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l06043024.png" /> is the intersection of this ball with the half-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l06043025.png" />.
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An imbedding (cf. [[Immersion|Immersion]]) $q$ of one topological manifold $M=M^m$ into another $N=N^n$ such that for any point $x\in M$ there are charts in a neighbourhood $U$ of $x$ and in a neighbourhood $V$ of the point $qx$ in $N$ in which the restriction of $q$ to $U$ linearly maps $U$ to $V$. In other words, $q$ is locally linear in suitable coordinate systems. Equivalently: There are neighbourhoods $U$ of a point $x\in M$ and $V$ of the point $qx\in N$ such that the pair $(V,qU)$ can be mapped homeomorphically onto a standard pair $(D^n,D^m)$ or $(D^n,D_+^m)$, where $D^k$ is the unit ball of the space $\mathbf R^k$ with centre at the origin and $D_+^k$ is the intersection of this ball with the half-space $x_k\geq0$.
  
Any imbedding of a circle and an arc into a plane is locally flat; however, a circle or an arc can be imbedded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l06043026.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l06043027.png" /> in a manner that is not locally flat (see [[Wild imbedding|Wild imbedding]]; [[Wild sphere|Wild sphere]]). Any smooth imbedding is locally flat in the smooth sense (that is, in the definition the coordinates can be chosen to be smooth). A piecewise-linear imbedding need not be locally flat, not only in the piecewise-linear sense, but even not in the topological sense; for example, a cone with vertex in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l06043028.png" /> over a closed polygon knotted in the bounding plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l06043029.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l06043030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l06043031.png" /> there is a homotopy criterion for an imbedding to be locally flat: For every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l06043032.png" /> and neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l06043033.png" /> of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l06043034.png" /> there is a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l06043035.png" /> such that any loop in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l06043036.png" /> is homotopic to zero in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l06043037.png" /> (local simple connectedness). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l06043038.png" />, then such a criterion holds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l06043039.png" />, but is essentially more complicated. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l06043040.png" /> the question remains unsettled (1989). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l06043041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060430/l06043042.png" /> a locally flat imbedding has a topological [[Normal bundle|normal bundle]].
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Any imbedding of a circle and an arc into a plane is locally flat; however, a circle or an arc can be imbedded in $\mathbf R^k$ with $k\geq3$ in a manner that is not locally flat (see [[Wild imbedding|Wild imbedding]]; [[Wild sphere|Wild sphere]]). Any smooth imbedding is locally flat in the smooth sense (that is, in the definition the coordinates can be chosen to be smooth). A piecewise-linear imbedding need not be locally flat, not only in the piecewise-linear sense, but even not in the topological sense; for example, a cone with vertex in $\mathbf R_+^4$ over a closed polygon knotted in the bounding plane $\mathbf R^3$. For $n\neq4$ and $m\neq n-2$ there is a homotopy criterion for an imbedding to be locally flat: For every point $x\in M$ and neighbourhood $U$ of the point $qx$ there is a neighbourhood $V\subset U$ such that any loop in $V\setminus qM$ is homotopic to zero in $U\setminus qM$ (local simple connectedness). If $m=n-2$, then such a criterion holds for $n\neq4$, but is essentially more complicated. For $m=4$ the question remains unsettled (1989). For $m=n-1$ and $m=n-2$ a locally flat imbedding has a topological [[Normal bundle|normal bundle]].
  
  

Latest revision as of 08:32, 19 April 2014

An imbedding (cf. Immersion) $q$ of one topological manifold $M=M^m$ into another $N=N^n$ such that for any point $x\in M$ there are charts in a neighbourhood $U$ of $x$ and in a neighbourhood $V$ of the point $qx$ in $N$ in which the restriction of $q$ to $U$ linearly maps $U$ to $V$. In other words, $q$ is locally linear in suitable coordinate systems. Equivalently: There are neighbourhoods $U$ of a point $x\in M$ and $V$ of the point $qx\in N$ such that the pair $(V,qU)$ can be mapped homeomorphically onto a standard pair $(D^n,D^m)$ or $(D^n,D_+^m)$, where $D^k$ is the unit ball of the space $\mathbf R^k$ with centre at the origin and $D_+^k$ is the intersection of this ball with the half-space $x_k\geq0$.

Any imbedding of a circle and an arc into a plane is locally flat; however, a circle or an arc can be imbedded in $\mathbf R^k$ with $k\geq3$ in a manner that is not locally flat (see Wild imbedding; Wild sphere). Any smooth imbedding is locally flat in the smooth sense (that is, in the definition the coordinates can be chosen to be smooth). A piecewise-linear imbedding need not be locally flat, not only in the piecewise-linear sense, but even not in the topological sense; for example, a cone with vertex in $\mathbf R_+^4$ over a closed polygon knotted in the bounding plane $\mathbf R^3$. For $n\neq4$ and $m\neq n-2$ there is a homotopy criterion for an imbedding to be locally flat: For every point $x\in M$ and neighbourhood $U$ of the point $qx$ there is a neighbourhood $V\subset U$ such that any loop in $V\setminus qM$ is homotopic to zero in $U\setminus qM$ (local simple connectedness). If $m=n-2$, then such a criterion holds for $n\neq4$, but is essentially more complicated. For $m=4$ the question remains unsettled (1989). For $m=n-1$ and $m=n-2$ a locally flat imbedding has a topological normal bundle.


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References

[a1] J.C. Cantrell (ed.) C.H. Edwards jr. (ed.) , Topology of manifolds , Markham (1970)
How to Cite This Entry:
Locally flat imbedding. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Locally_flat_imbedding&oldid=15755
This article was adapted from an original article by A.V. Chernavskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article