View source for Locally flat imbedding

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" />.

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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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.C. Cantrell (ed.)  C.H. Edwards jr. (ed.) , ''Topology of manifolds'' , Markham  (1970)</TD></TR></table>