Submanifold chart - Imported Knowlpedia Content

Definition

Let $S$ be a $k$ -dimensional embedded submanifold of an $n$ -dimensional smooth manifold $M$ , and let $p\in S$ . A submanifold chart at $p$ is a smooth chart $(U,\varphi)$ of $M$ , with $p\in U$ , for which

\varphi(U\cap S)=\varphi(U)\cap(\mathbb R^k\times\{0\})\subseteq\mathbb R^k\times\mathbb R^{n-k}.

Thus the first $k$ coordinate functions restrict to local coordinates on $S$ , while the last $n-k$ coordinates vanish on $S$ . Reordering the ambient coordinates gives the same notion. The defining local property of embedded submanifolds guarantees such a chart around every point of $S$ .

Tangent-space description

In a submanifold chart, the coordinate vectors in the first $k$ directions span $T_qS$ for every $q\in U\cap S$ . The remaining coordinate directions supply a local complement inside $T_qM$ , although that complement depends on the chart and is not an intrinsic normal subspace.

Examples and non-examples

For the coordinate plane $\mathbb R^k\times\{0\}\subset\mathbb R^n$ , the identity chart is a submanifold chart. A smooth ambient chart that sends $S$ to a curved graph is not adapted in this strict sense, even though its restriction may still be a valid chart on $S$ .

Conventions and scope

“Slice chart” also appears in other contexts, notably proper group actions. Here it means only an ambient chart adapted to an embedded submanifold. Some sources permit an arbitrary coordinate $k$ -plane rather than the first $k$ axes; coordinate reordering makes the definitions equivalent Lee, Chapter 5.

References

J. M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. Springer DOI record. Relevant: Chapter 5, embedded submanifolds and slice charts.