Suppose the immersed subset is not embedded. Using the assumed embedded neighborhoods, there are $p\in N$, a relatively small coordinate neighborhood $U\ni p$, and points $p_j\in N\setminus U$ with $p_j\to p$ in $M$. Choose a <smooth bump function>[bump function] $g$ on $N$, supported in $U$, with $g(p)=1$. Then $g(p_j)=0$.

If $g=f|_N$ for some smooth $f$ on $M$, continuity gives both $f(p_j)\to f(p)=1$ and $f(p_j)=0$, a contradiction. Thus the extension hypothesis forces the subspace and manifold topologies to agree locally, and the immersion is an embedding. This proves the <smooth extension criterion for an immersed submanifold>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 115 / 2 / c / Solution

An injectively immersed submanifold $N\subset M$ is embedded if every smooth function on $N$ extends to a smooth function on $M$. Otherwise a sequence from a different local sheet can converge in $M$ to a point $p$ while remaining outside a small embedded neighborhood of $p$ in $N$; a bump function supported on that neighborhood cannot have a continuous ambient extension.


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