Embedded submanifold

An embedded submanifold $M\subseteq N$ is a subset whose inclusion is an embedding. Around every point there are coordinates on $N$ in which $M$ is a coordinate plane.

An embedded submanifold is a submanifold for which the inclusion gives the subspace topology, unlike a merely immersed submanifold.

If an embedded codimension-$k$ submanifold $X\subset M$ is $f^{-1}(y)$ for a regular value $y$ of a smooth map $f:M\to {\mathbb{R}}^k$, then $df$ identifies its normal bundle with the trivial bundle $X\times {\mathbb{R}}^k$. A submanifold with nontrivial normal bundle therefore cannot be such a regular level set. The core circle of the Möbius band is a codimension-one example.

If $N^n\subset M^m$ is embedded and $p\in N$, there are smooth coordinates $(x^1,\dots ,x^m)$ around $p$ in whichThe first $n$ coordinates restrict to a chart on $N$.

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.

The vanishing ideal is $I(N)=\{f\in C^{\infty }(M):f{|}_N=0\}$. A vector field $X$ is tangent to $N$ exactly when $X(I(N))\subseteq I(N)$.

If $X$ and $Y$ preserve $I(N)$, then their commutator preserves it becausefor every $f\in I(N)$. Thus vector fields tangent to an embedded submanifold are closed under the Lie bracket of vector fields, and the restricted bracket depends only on their restrictions to the submanifold.

If embedded submanifolds $A,B\subseteq M$ meet transversely, then $A\cap B$ is an embedded submanifold of dimension $\dim A+\dim B-\dim M$.

The hypothesis is transversality: tangent spaces to the two submanifolds span the ambient tangent space at each intersection point.

Submanifolds $A,B\subseteq M$ intersect cleanly when $A\cap B$ is an embedded submanifold andat every $x\in A\cap B$. A transverse intersection is the special case in which $T_{x}A+T_{x}B=T_{x}M$.

Submanifolds $A,B\subseteq M$ intersect transversely when $T_{x}A+T_{x}B=T_{x}M$ at every intersection point. The transverse intersection theorem then determines the dimension and smoothness of $A\cap B$.

For complementary-dimensional oriented submanifolds of an oriented manifold meeting transversely in finitely many points, the smooth intersection number is the sum of their local orientation signs. Geometric disjointness implies number zero, but number zero alone need not imply geometric disjointness. Without coherent orientations, use the count modulo $2$.