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Positive-codimension immersed submanifolds are null
Statement
Every immersed submanifold of positive codimension in a smooth manifold is a null subset of the ambient manifold.
Facts & Assumptions
Given: An immersed -dimensional submanifold of an -manifold with .
An immersed submanifold is locally the image of a smooth immersion from an -manifold (Immersed submanifolds).
The image of a lower-dimensional manifold is null (The image of a lower-dimensional manifold is null).
Proof
By [F1], every point of has a neighbourhood in on which is the image of a smooth immersion from an -manifold.
Because , [L1] makes each such local image null in the ambient neighbourhood. Therefore is locally null, and a countable cover of by such neighbourhoods shows that is null in .
Hence every positive-codimension immersed submanifold is null.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Marco Gualtieri, Topology I: Smooth Manifolds, cumulative notes (standard reference, not scraped)