Smooth Functions and Maps

Smoothness

Smooth Map

Let $M$ be a differentiable manifold with dimension $n$, an atlas $\mathcal{A}$ representing the differentiable structure on $M$. A function $f:M\to {\mathbb{R}}^n$ is smooth if for every chart $\phi :U\to V$ in $\mathcal{A}$, the map $f\circ {\phi }^{-1}$ is smooth on $V\subset {\mathbb{R}}^n$. Let $N$ be another differentiable manifold with dimension $k$, and differentiable atlas $\mathcal{B}$. Let $F$ be a map from $M$ to $N$. Then $F$ is smooth if for all $x\in M$, and every chart $\phi :U\to V$ in $\mathcal{A}$ with $x\in U$ and $\eta :W\to Z$ in $\mathcal{B}$ with $F(x)\in W$, the map $\eta \circ F\circ {\phi }^{-1}$ is smooth on $Z\subset {\mathbb{R}}^k$.

e.g.

• The inclusion map $S^n\hookrightarrow {\mathbb{R}}^{n+1}$ is smooth. We can take
• The quotient map \pi\colon \newcommand{\P}{\mathbb{P}}\R^{n+1}\setminus \{0\}\to\R\P^{n} is smooth. Note that $\mathbb{R}{\P }^n\cong S^n/(-x\sim x)$

Proposition

Although the definition requires that the map $f\circ {\phi }^{-1}$ is smooth for every chart $\phi :U\to V$ in the atlas, it is enough to check the smoothness for a single chart around each point. Similar argument applies to check that a map between manifolds is smooth.

Proof If $\phi :U\to V$ is a chart with $f\circ {\phi }^{-1}$ smooth, and $\eta :W\to Z$ is another chart with $W$ overlapping $U$, then $f\circ {\eta }^{-1}=f\circ {\phi }^{-1}\circ \phi \circ {\eta }^{-1}$ is smooth at the overlapping region. $\square$

Proposition

A map $f:M\to {\mathbb{R}}^n$ is smooth if and only if each component function $f_i$ is smooth.

Proof If $f$ is smooth, then $f_i\circ {\phi }^{-1}$ is smooth for every chart $\phi :U\to V$. Conversely, if each $f_i$ is smooth, then $f\circ {\phi }^{-1}=(f_1\circ {\phi }^{-1},f_2\circ {\phi }^{-1},\dots ,f_n\circ {\phi }^{-1})$ is smooth. $\square$

Proposition

A map $\chi$ between differentiable manifolds $M$ and $N$ is smooth if and only if $f\circ \chi$ is a smooth function on $M$ whenever $f$ is a smooth function on $N$.

Proof Clearly if $\chi$ is smooth, then $f\circ \chi$ is smooth. Conversely, if $f\circ \chi$ is smooth on $M$, then for every chart $\phi :U\to V$ in the atlas of $M$, the composition $f\circ \chi \circ {\phi }^{-1}$ is smooth on $V$. Hence $\chi$ is smooth by definition. $\square$

Further Classification of Maps

Diffeomorphism

A map $F:M\to N$ between smooth manifolds is a diffeomorphism if it is smooth, and has a smooth inverse. $F$ is a local diffeomorphism about $x\in M$ if for every $x\in M$ there is a neighborhood $U$ of $x$ such that $F{|}_U$ is a diffeomorphism to an open subset of $N$.

Immersion

$F:M\to N$ is an immersion if for every $x$ in $M$ there exist charts $\phi :U\to V$ for $M$ and $\eta :W\to Z$ for $N$ with $x\in U$ and $F(x)\in Z$, such that the map $\eta \circ F\circ {\phi }^{-1}$ has derivative which is injective at $\phi (x)$.

Submersion

$F$ is a submersion if for each $x\in M$ there are charts $\phi$ and $\eta$ such that the derivative of $\eta \circ F\circ {\phi }^{-1}$ at $\phi (x)$ is surjective.

Smooth Embedding

A smooth map $F:M\to N$ is an embedding if $F$ is a homeomorphism onto its image with the subspace topology, and for any charts $\phi$ and $\eta$ for $M$ and $N$ respectively, $\eta \circ F\circ {\phi }^{-1}$ has derivative of full rank.

Remark

A smooth embedding is a topological embedding that is also an immersion.