Inverse function theorem for manifolds - Imported Knowlpedia Content

Statement

Let $F:M\to N$ be a smooth map between finite-dimensional smooth manifolds without boundary, and let $p\in M$ . If the differential

dF_p:T_pM\longrightarrow T_{F(p)}N

is a linear isomorphism, then there are open neighborhoods $U$ of $p$ and $V$ of $F(p)$ such that $F|_U:U\to V$ is a diffeomorphism. Equivalently, $F$ is a local diffeomorphism at $p$ . Conversely, any smooth map that is a local diffeomorphism at $p$ has invertible differential there.

Why the manifold statement follows

Choose smooth charts around $p$ and $F(p)$ . In coordinates, the differential of the representative of $F$ is invertible at the coordinate of $p$ . The Euclidean inverse function theorem supplies mutually inverse smooth maps on smaller open sets; conjugating them by the charts gives the required manifold neighborhoods. This also shows that the statement is independent of the chosen charts.

Consequences

Invertibility of $dF_p$ forces $\dim M=\dim N$ . Since invertible linear maps form an open subset, $dF_q$ remains invertible for all $q$ sufficiently near $p$ . Thus the set on which $F$ is a local diffeomorphism is open. A bijective local diffeomorphism is a diffeomorphism, because its locally defined smooth inverses agree with the set-theoretic inverse.

Scope and near misses

The theorem is local and does not imply global injectivity: the exponential map $\mathbb{R}\to S^1$ , $t\mapsto e^{it}$ , has invertible differential everywhere but is not one-to-one. For manifolds with boundary, invertibility of the tangent map alone does not give this conclusion at boundary points without additional hypotheses controlling the boundary.

References

J. M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2013. DOI record. Relevant: Chapter 4.
L. W. Tu, An Introduction to Manifolds, 2nd ed., Springer, 2011. DOI record. Relevant: §6.7 and Remark 8.12.