Submanifolds and the Inverse Function Theorem

Definition of Submanifolds

Definition 1 Let two manifolds \(M,N\) and a \(C^\infty\) function \(F:N\rightarrow M\) be given.

1. We say \(F\) is an immersion if \(\dd{F_p}\) is injective for every \(p\in N\); similarly, we say \(F\) is a submersion if \(\dd{F_p}\) is surjective for every \(p\in N\).
2. If \(F\) is an immersion and also injective, we call \(F\) a submanifold of \(M\).
3. If \(F\) is not only a submanifold of \(M\) but also defines a homeomorphism between \(N\) and \(F(N)\subseteq M\) equipped with the subspace topology, then we call \(F\) an embedding, or, matching the definition in 2, an embedded submanifold.

To distinguish it more clearly from the embedded submanifold in 3, the notion in 2 is sometimes called an immersed submanifold. We shall use the unqualified term submanifold in the sense of immersed submanifold as defined above, while embedded submanifold will always be spelled out in full.

Intuitively, to say that \(F:N\rightarrow M\) is a submanifold means that \(F\) plays the role of the inclusion \(N\hookrightarrow M\). In this situation, there are two ways to topologize the image \(F(N)\subseteq M\): one is to transport the topology of \(N\) via the bijection \(F:N\rightarrow F(N)\), and the other is to induce the topology from \(M\) via the subspace topology. If these two topologies agree, we call \(F\) an embedded submanifold; otherwise, we simply call it a submanifold.

For example, in the figure above, \(N=\mathbb{R}\), \(M=\mathbb{R}^2\), and (a) is an immersion but not a submanifold, (b) is a submanifold but not an embedded submanifold, and (c) is an embedded submanifold. For convenience, in (b), if we denote by \(F(0)\) the limit point that \(F(t)\) approaches as \(t\rightarrow \infty\), then \((-1,1)\) is open in \(\mathbb{R}\), but \(F\bigl((-1,1)\bigr)\) cannot be open in the subspace topology on \(F(N)\).

Example 2 For a manifold \(M\) and its open submanifold \(U\), the inclusion \(\iota:U\hookrightarrow M\) is an embedded submanifold of \(M\). That \(\dd{\iota_p}\) is injective for every \(p\in U\) follows from §Differentials, ⁋Proposition 8, namely the fact that \(\dd{\iota_p}\) is an isomorphism between \(T_pU\) and \(T_{\iota(p)}M\), and moreover, by the definition of open submanifold, the subspace topology is given on \(\iota(U)\).

Example 3 Consider two manifolds \(M,N\) and their product \(M\times N\). Then for any \(q\in N\), the subset \(M\times\{q\}\) is an embedded submanifold of \(M\times N\) diffeomorphic to \(M\); similarly, for any \(p\in M\) the subset \(\{p\}\times N\) is an embedded submanifold diffeomorphic to \(N\), with embeddings given by \(x\mapsto (x,q)\) and \(y\mapsto (p,y)\) respectively.

More generally, let two manifolds \(M,N\) and a \(C^\infty\) function \(f:U\rightarrow N\) defined on an open submanifold \(U\subseteq M\) be given. Then the graph of \(f\)

\[\graph(f)=\{(x,y)\in M\times N\mid x\in U, y=f(x)\}\]

is also an embedded submanifold, and the embedding is of course given by \(x\mapsto (x,f(x))\).

Inverse Function Theorem and Its Consequences

We now lift the inverse function theorem and the implicit function theorem from Euclidean space to the setting of manifolds. First, the inverse function theorem in Euclidean space is as follows.

Theorem 4 (Inverse function theorem) Let \(U\subseteq\mathbb{R}^m\) be an open set, and let \(f:U\rightarrow\mathbb{R}^m\) be a \(C^\infty\) function. If the Jacobian matrix

\[\begin{pmatrix}\partial(r^1\circ f)/\partial r^1&\partial(r^1\circ f)/\partial r^2&\cdots&\partial(r^1\circ f)/\partial r^m\\\partial(r^2\circ f)/\partial r^1&\partial(r^2\circ f)/\partial r^2&\cdots&\partial(r^2\circ f)/\partial r^m\\\vdots&\vdots&\ddots&\vdots\\\partial(r^m\circ f)/\partial r^1&\partial(r^m\circ f)/\partial r^2&\cdots&\partial(r^m\circ f)/\partial r^m\end{pmatrix}\]

is nonsingular at a point \(p_0\in U\), then there exists an open set \(V\) with \(p_0\in V\subseteq U\) such that \(f\vert_V\) defines a diffeomorphism between \(V\) and \(f(V)\).

From this we can prove theorems for functions between general manifolds.

Corollary 5 Let \(F:M\rightarrow N\) be a \(C^\infty\) function between manifolds and let \(p\in M\). If \(\dd{F_p}:T_pM\rightarrow T_{F(p)}N\) is an isomorphism, then there exists an open set \(U\subseteq M\) with \(p\in U\) such that \(F\vert_U:U\rightarrow F(U)\) defines a diffeomorphism between \(U\) and \(F(U)\).

Proof

First, from the fact that \(\dd{F_p}\) is an isomorphism we obtain \(\dim M=\dim T_pM=\dim T_{F(p)}N=\dim N\). Now take a coordinate system \((W,\tau)\) containing the point \(F(p)\), and a coordinate system \((V,\varphi)\) containing \(p\) such that \(F(V)\subseteq W\). Then the map \((\tau\circ F\circ\varphi^{-1})\vert_{\varphi(V)}\) is a map between Euclidean spaces of the same dimension, and moreover, since \(\dd{F_p}\) is an isomorphism, the Jacobian matrix of this map at the point \(\varphi(p)\) is nonsingular.

Therefore, by the inverse function theorem, there exists an open set \(U'\) with \(\varphi(p)\in U'\subseteq\varphi(V)\) such that \((\tau\circ F\circ\varphi^{-1})\vert_{U'}\) defines a diffeomorphism between \(U'\) and \(\tau\circ F\circ\varphi^{-1}(U')\). Setting \(U=\varphi^{-1}(U')\), the map

\[\tau^{-1}\circ\bigl((\tau\circ F\circ\varphi^{-1})\vert_{U'}\bigr)\circ\varphi\]

defines a diffeomorphism between \(U\) and \(F(U)\).

Let elements \(y^1, \ldots, y^k\) of \(\mathcal{C}_{M,p}^\infty\) be given for a manifold \(M\) and \(p\in M\). If their differentials \(\dd{y}^i\) form a linearly independent subset of \(T_p^\ast M\), we call them independent functions at the point \(p\).

Corollary 6 Consider an \(m\)-dimensional manifold \(M\). If \(y^1, \ldots, y^m\) are independent at a point \(p_0\in M\), then \((y^1, \ldots, y^m)\) forms a coordinate system in a neighborhood of \(p_0\).

Proof

First, considering the dimension of \(T_{p_0}^\ast M\), we see that the differentials of the given functions form a basis of \(T_{p_0}^\ast M\).

Suppose the \(m\) functions \(y^i\) are all defined on an open neighborhood \(U\) of \(p_0\).¹ Define \(\varphi:U\rightarrow\mathbb{R}^m\) by

\[\varphi(p)=(y^1(p),\ldots, y^m(p))\]

as given. Then since each component function \(y^i\) is \(C^\infty\), so is \(\varphi\). Now consider \((\dd{\varphi_{p_0}})^\ast:T_{\varphi(p_0)}^\ast\mathbb{R}^m\rightarrow T_{p_0}^\ast M\). Applying \((\dd{\varphi_{p_0}})^\ast\) to the \(\dd{r}^i\vert_{\varphi(p_0)}\), we have

\[\left(\dd{\varphi_{p_0}}\right)^\ast\left(\dd{r}^i\vert_{\varphi(p_0)}\right)=\left(\dd{r}^i\vert_{\varphi(p_0)}\right)\circ\left(\dd{\varphi_{p_0}}\right)=\dd{(r^i\circ\varphi)_{p_0}}=\dd{y}^i\vert_{p_0}\]

so the basis elements \(\dd{r}^i\vert_{\varphi(p_0)}\) of \(T_{\varphi(p_0)}^\ast\mathbb{R}^m\) are each mapped to a basis of \(T_{p_0}^\ast M\), and thus \((\dd{\varphi_{p_0}})^\ast\) is an isomorphism. Hence \(\dd{\varphi_{p_0}}\) is also an isomorphism, and therefore, applying Corollary 5, we see that there exists a suitable \(V\) with \(p_0\in V\subseteq U\) such that \(\varphi\vert_V:V\rightarrow\varphi(V)\) is a coordinate system.

Obtaining the following two corollaries from the above is essentially undergraduate linear algebra.

Corollary 7 For an \(m\)-dimensional manifold \(M\), a point \(p_0\in M\), and an integer \(0<k<m\), let \(y^1,\ldots, y^k\) be elements of \(\mathcal{C}_{M,p_0}^\infty\) that are independent functions at \(p_0\). Then there exist suitable functions \(x^{k+1},\ldots, x^{m}\) such that \((y^1,\ldots, y^k, x^{k+1}, \ldots, x^m)\) defines a coordinate system in a neighborhood of \(p_0\).

Proof

Let a coordinate system \((U,\varphi)\), \(\varphi=(x^i)_{i=1}^{m}\) about the point \(p_0\) be given. Then the \(\dd{x}^i\) form a basis of \(T_{p_0}^\ast M\). Now, just as in the proof of [Linear Algebra] §Dimension of Vector Spaces, ⁋Lemma 2, we insert the \(\dd{y}^i\) one by one and remove the \(\dd{x}^j\) one by one, adjusting indices appropriately.

Corollary 8 For an \(m\)-dimensional manifold \(M\) and a point \(p_0\in M\), let elements \(y^1,\ldots, y^k\) of \(\mathcal{C}_{M,p_0}^\infty\) be given. If the \(\dd{y}^i\) span \(T_{p_0}^\ast M\), then a suitable subset of \(\{y^1,\ldots, y^k\}\) forms a coordinate system in a neighborhood of \(p_0\).

Proof

Find an appropriate subset of \(\{\dd{y}^1,\ldots, \dd{y}^k\}\) that forms a basis of \(T_{p_0}^\ast M\); this subset must necessarily consist of \(m\) elements. Therefore, applying Corollary 6 suffices.

The following two corollaries will be used frequently in what follows under the name rank theorem.

Corollary 9 (Rank theorem, Submersion case) For an \(m\)-dimensional manifold \(M\), an \(n\)-dimensional manifold \(N\), a \(C^\infty\) function \(F:M\rightarrow N\), and a point \(p\in M\), suppose \(\dd{F_p}\) is surjective. Then for a coordinate system \(\psi=(y^j)_{j=1}^n\) defined in a neighborhood of \(F(p)\), there exist suitable functions \(x^{n+1},\ldots, x^m\) such that the functions

\[x^1=y^1\circ F,\quad x^2=y^2\circ F,\quad\ldots,\quad x^n=y^n\circ F,\qquad x^{n+1},\quad \ldots,\quad x^m\]

form a coordinate system in a neighborhood of \(p\).

Proof

Since \(\dd{F_p}\) is surjective, its dual \((\dd{F_p})^\ast:T_{F(p)}^\ast N\rightarrow T_p^\ast M\) is injective. That is, the elements

\[(\dd{F_p})^\ast(\dd{y}^j\vert_{F(p)})=\dd{y}^j\vert_{F(p)}\circ \dd{F_p}=\dd{(y^j\circ F)_p}=\dd{x}^j\vert_p\]

are linearly independent in \(T_p^\ast M\). Therefore, the desired result follows from Corollary 7.

Corollary 10 (Rank theorem, Immersion case) For a manifold \(M\), an \(n\)-dimensional manifold \(N\), a \(C^\infty\) function \(F:M\rightarrow N\), and a point \(p\in M\), suppose \(\dd{F_p}\) is injective. Then for a coordinate system \(\psi=(y^j)_{j=1}^n\) defined in a neighborhood of \(F(p)\), a subset of

\[\{x^j=y^j\circ F\mid j=1,\ldots, n\}\]

forms a coordinate system on \(M\) in a neighborhood of \(p\).

Proof

Since \(\dd{F_p}\) is injective, its dual \((\dd{F_p})^\ast:T_{F(p)}^\ast N\rightarrow T_p^\ast M\) is surjective. That is, the elements

\[(\dd{F_p})^\ast(\dd{y}^j\vert_{F(p)})=\dd{y}^j\vert_{F(p)}\circ \dd{F_p}=\dd{(y^j\circ F)_p}=\dd{x}^j\vert_p\]

must span \(T_p^\ast M\), and therefore by Corollary 8 a subset of the given set forms a coordinate system on \(M\) in a neighborhood of \(p\).

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References

[War] Frank W. Warner. Foundations of Differentiable Manifolds and Lie Groups, Graduate texts in mathematics, Springer, 2013
[Lee] John M. Lee. Introduction to Smooth Manifolds, Graduate texts in mathematics, Springer, 2012

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1. This is possible because there are finitely many \(y^i\). That is, if each \(y^i\) is defined on \(U^i\), we may set \(U=\bigcap U^i\). ↩