Let us now investigate the manner in which an embedded sub-manifold is embedded in Euclidean space. There is a compatibility between the intrinsic differentiable structure from local parametrisations and the ambient differential structure from the surrounding Euclidean space. We will also see how the regularity condition of injective differential is precisely what is required to allow us to develop a theory of calculus on an embedded sub-manifold. The main tool used is the Inverse Function Theorem and several variants such as the Implicit Function Theorem .
Throughout this section let \(M \subset \mathbb{R}^n\) be an embedded sub-manifold of dimension \(k\), and let \(\{\varphi_{\alpha} : U_{\alpha} \to M\}\) be continuous local parametrisations covering \(M\). That is, each \(\varphi_{\alpha}\) is a homeomorphism onto its image \(V_{\alpha} = \varphi_{\alpha}(U_{\alpha})\), and \[ M = \bigcup_{\alpha} V_{\alpha}. \] In this section we do not assume that \(\varphi_{\alpha}\) is \(C^{\infty}\). Then of course, in this section there is also no requirement that the differential is injective since it may not even be defined!
The basic idea is that since \(\varphi_{\alpha}\) is a homeomorphism, it is in particular, bijective and thus establishes a one-to-one correspondence between \(U_{\alpha}\) and \(V_{\alpha}\). This is useful because \(U_{\alpha}\) is an open subset of \(\mathbb{R}^k\) on which we understand limits and continuity. In comparison, \(V_{\alpha}\) is some (generally) curved subset of \(\mathbb{R}^n\) on which - with a little effort - we can define the notions of open sets, limits and continuity. We will do just that, and then show that the definition is compatible with the corresponding notions via the local parametrisations.
A set \(W \subseteq M\) is open if there is an open set \(\tilde{W} \subseteq \mathbb{R}^n\) such that \(W = \tilde{W} \cap M\).
The collection of open sets of \(M\) is a topology . That is, \(\emptyset, M\) are open sets and the collection of open sets is closed under arbitrary unions and under finite intersections.
Since \(\emptyset \subseteq \mathbb{R}^n\) is open in \(\mathbb{R}^n\), \(\emptyset = \emptyset \cap M \subseteq M\) is open.
Since \(\mathbb{R}^n \subseteq \mathbb{R}^n\) is open in \(\mathbb{R}^n\), \(M = \mathbb{R}^n \cap M \subseteq M\) is open.
Let \(\{W_a : a \in A\}\) be an arbitrary collection of open sets in \(M\). Then there are open sets \(\tilde{W}_{a} \subseteq \mathbb{R}^n\) such that \(W_a = M \cap \tilde{W}_a\). Then \[ \bigcup_a W_a = \bigcup_a M \cap \tilde{W}_a = M \bigcap \bigcup_a \tilde{W}_a \] is open since \(\cup_a \tilde{W}_a\) is open.
Let \(\{W_i : i = 1, \dots, n\}\) be an arbitrary finite collection of open sets in \(M\). Then there are open sets \(\tilde{W}_i \subseteq \mathbb{R}^n\) such that \(W_i = M \cap \tilde{W}_i\). Then \[ \bigcap_i W_i = \bigcap_i M \cap \tilde{W}_i = M \bigcap \bigcap_i \tilde{W}_i \] is open since \(\cup_i \tilde{W}_i\) is open.
The topology of \(M\) is thus induced by the topology of \(\mathbb{R}^n\). It is known as the induced topology . This topology is also the same as the topology generated by the local parametrisations.
A set \(W \subseteq M\) is open if and only if \[ W_{\alpha} := \varphi_{\alpha}^{\ast} (W) \subseteq U_{\alpha} \] is open for every \(\alpha\).
Assuming \(W\) is open, \(W_{\alpha}\) is open since \(\varphi_{\alpha}\) is continuous.
Conversely, suppose each \(W_{\alpha}\) is open. Then \[ Z_{\alpha} := \varphi_{\alpha}(W_{\alpha}) = (\varphi_{\alpha}^{-1})^{\ast} (W_{\alpha}) \] is open for every \(\alpha\) since \(\varphi_{\alpha}^{-1}\) is continuous. Then \(W = \bigcup_{\alpha} Z_{\alpha}\) is a union of open sets, hence open.
Let us define an open set of \(M\) to be any subset of \(M\) of the form \(\cup_{\alpha} Z_{\alpha}\) where \(Z_{\alpha} = \varphi_{\alpha}(W_{\alpha})\) with \(W_{\alpha} \subseteq U_{\alpha}\) and open set. Without using the lemma, show that this defines a topology on \(M\). That is, the collection of open sets contains \(\emptyset, M\) and is closed under arbitrary unions and finite intersections.
Thus we have two natural ways to define a topology on \(M\) - the induced topology from the ambient \(\mathbb{R}^n\) and the topology generated by the local parametrisations. The lemma above shows that these are precisely the same topology. In other words, the local parametrisations provide a sort of compatibility between the topology of the ambient space and the topology of the local parametrisations.
A function \(f : M \to \mathbb{R}^m\) is continuous if and only if \(f_{\alpha} := f \circ \varphi_{\alpha} : U_{\alpha} \to \mathbb{R}^m\) is continuous for every \(\alpha\).
Note that \(f_{\alpha}\) is a function defined on an open set of \(\mathbb{R}^k\), so continuity of \(f_{\alpha}\) is just the usual Euclidean continuity we are used to. On the other hand, continuity of \(f : M \to \mathbb{R}^m\) requires us to use the induced topology on \(M\). This is no great challenge, but it can be comforting to relate the less familiar continuity on \(M\) to the more familiar continuity on Euclidean space. When it comes to differentiability, this perspective will be essential!
If \(f\) is continuous, then \(f_{\alpha} = f \circ \varphi_{\alpha}\) is continuous since compositions of continuous functions are continuous.
Converse, suppose that each \(f_{\alpha}\) is continuous. Then \[ f|_{V_{\alpha}} = f \circ \varphi_{\alpha} \circ \varphi_{\alpha}^{-1} = f_{\alpha} \circ \varphi_{\alpha}^{-1} \] is a composition of continuous functions hence is continuous. Thus \(f\) is locally continuous, hence continuous.
Show that if for every \(\alpha\), \(f|_{V_{\alpha}}\) is continuous, then \(f\) is continuous.
Thus we have established that both the notion of openness and the notion of continuity are local properties. To say a property is local is to say the property holds on each set \(W_a}\) of some open cover of \(M\). Thus a set is open if and only if it is locally open and a function is continuous if and only if it is locally continuous. Moreover both of these local properties are equivalent to the corresponding properties in the local parametrisations by covering \(M\) by \(V_{\alpha} = \varphi_{\alpha}(U_{\alpha})\).
Let \(\{W_a\}\) and \(\{Z_k\}\) be open covers of \(M\). Show that a set \(S \subseteq M\) is locally open with respect to \(\{W_a\}\) (i.e. \(S \cap W_a\) is open for every \(a\)) if and only if it is locally open with respect to \(\{Z_k\}\). Likewise, show that a function \(f : M \to \mathbb{R}^m\) is locally continuous with respect to \(\{W_a\}\) (i.e. \(f|_{W_a}\) is continuous for every \(a\)) if and only if it is locally continuous with respect to \(\{Z_k\}\).
There is one final topological notion to discuss here. It is a so-called separation axiom that allows us to separate points by open sets. It was included in Hausdorff's original definition of topology, but is now a separate axiom due to the presence of interesting non-Hausdorff topologies (e.g. in algebraic geometry). These are not relevant here but it's worth pointing out that, while it is a fairly natural axiom to impose, there is interesting mathematics to be done without the axiom!
A topology is Hausdorff if for every \(p, q \in M\), \(p \neq q\) there are disjoint open neighbourhoods of \(p, q\). That is, there are open sets \(U, V \subseteq M\) such that \(p \in U\), \(q \in V\), and \(U \cap V = \emptyset\).
The topology of \(\mathbb{R}^n\) and hence also of \(M\) is Hausdorff.
Let \(p, q \in \mathbb{R}^n\), \(p \neq q\) and let \(r = d(p, q) = \|p - q\|\) be the distance between \(p\) and \(q\). Let \(U = \mathbb{B}_{r/4} (p)\) and \(V = \mathbb{B}_{r/4} (q)\). Then \(U, V\) satisfy the requirements: by definition \(p \in U\) and \(q \in V\). If \(x \in U\), then by the reverse triangle inequality (first inequality below),
\begin{align*} \|x - q\| &= \|p - q + x - p\| \\ &\geq \big|\|p-q\| - \|x-p\|\big| \\ &\geq \|p-q\| - \|x-p\| \\ &\geq r - r/4 \\ &= 3r/4 > r/4. \end{align*}Thus if \(x \in U\), then \(x \notin V\) hence \(U \cap V = \emptyset\).
As to \(M\), let \(p, q \in M\), \(p \neq q\). Let \(\tilde{U}, \(\tilde{V}\) be disjoin open neighbourhoods of \(p, q\) \(\mathbb{R}^n\) respectively. Then \(U = \tilde{U} \cap M\) and \(V = \tilde{V} \cap M\) are disjoint open neighbourhoods in \(M\) of \(p\) and \(q\) respectively.
As an example of why Hausdorff is a desirable property is that limits in Hausdorff spaces are unique. The same is not true in general in non-Hausdorff spaces.
Let \((x_n)_{n \in \mathbb{N}} \subseteq M\) be a sequence. Show that if \(x = \lim_{n\to\infty} x_n\) and \(y = \lim_{n\to\infty} x_n\) then \(x = y\).
By definition, an embedded sub-manifold \(M\) is covered by local parametrisations, \(\varphi : U \subseteq \mathbb{R}^k \to V \subseteq M \subseteq \mathbb{R}^{n+k}\). Each such local parametrisation can in fact be locally extended to a diffeomorphism \(\Phi : W \subseteq \mathbb{R}^{n+k} \to Z \subseteq \mathbb{R}^{n+k}\) in such a way that locally \(\Phi|_{\mathbb{R}^k} = \varphi\). Thus \(\Phi\) locally identifies \(\mathbb{R}^k \subseteq \mathbb{R}^{n+k}\) with \(M\).
In other words, up to diffeomorphism, submanifolds are just linear subspaces!
Previously we have used the implicit function theorem in coordinates. To keep things manageable we assumed the last \(k\) rows of \(d\varphi\) were invertible and observed that in the case if was some other set of \(k\) rows, we could just reorder the rows. Now we'll look at the situation a little more abstractly which has the advantage of a cleaner presentation without needing to bother with indices and with reordering. The extra layer of abstraction is sometimes a little confusing at first and not to everyone's taste, but it's worth spending some time understanding different approaches and learning a broader context.
Recall that if \(E \subseteq \mathbb{R}^{n+k}\) is a vector subspace of dimension \(k\), then there exists a vector subspace \(F \subseteq \mathbb{R}^{n+k}\) of dimension \(n\) such that \(\mathbb{R}^{n+k} \simeq E \oplus F\). Such a subspace \(F\) is called a complementary subspace to \(E\).
Given \(E\), we construct \(F\) as follows: if \(E = \mathbb{R}^{n+k}\), then \(n=0\) and we take \(F = \{0\}\) and we're done. Otherwise, let \(\{e_1, \dots, e_k\}\) be a basis for \(E\). Then let \(e_{k+1}\) be any vector not in \(E\) so that \(\{e_1, \dots, e_k, e_{k+1}\}\) is linearly dependent. If \(n=1\), then \(\operatorname{span} \{e_1, \dots, e_{k+1}\} = \mathbb{R}^{1+k}\) and we're done by taking \(F = \operatorname{span} \{e_{k+1}\}\). Inductively assume that we have constructed \(\{e_1, \dots, e_k, e_{k+1}, \dots, e_{k+n-1}\}\). Then let \(e_{k+n} \not \in \operatorname{span} \{e_1, \dots, e_k, e_{k+1}, \dots, e_{k+n-1}\}\) in which case, \(\{e_1, \dots, e_{n+k}\}\) spans \(\mathbb{R}^{n+k}\) and taking \(F = \operatorname{span} \{e_{k+1}, \dots, e_{k+n}\}\) suffices.
Let \(E = \{(x, x) : x \in \mathbb{R}\} \subseteq \mathbb{R}^2\). Then we could take \(F = \{(-y, y) : y \in \mathbb{R}\}\) or we could take \(F =\{(0, y) : y \in \mathbb{R}\}\), or we could take \(F = \{(y, 0) : y \in \mathbb{R}\}\), and so on.
In the example, the first choice of \(F\) is the orthogonal complement , \(E^{\perp}\) to \(E\), \[ \{(-y, y) : y \in \mathbb{R}\} = \{Y \in \mathbb{R}^2 : \langle X, Y\rangle = 0 \> \forall X \in E\} \] For any subspace \(E \subseteq \mathbb{R}^n\) we always have \(\mathbb{R}^n \simeq E \oplus E^{\perp}\). But there are infinitely many possibles choices of complementary subspace. Often it doesn't really matter what we choose, but the orthogonal complement has the nice property of orthogonality with the original subspace.
Let \(\varphi : U \subseteq \mathbb{R}^k \to \mathbb{R}^{n+k}\) be a local parametrisation and let \(p \in U\). Then there exists an injective linear map \(\iota : \mathbb{R}^k \to \mathbb{R}^{n+k}\) and a diffeomorphism \(\Phi : W \subseteq \mathbb{R}^{n+k} \to Z \subseteq \mathbb{R}^{n+k}\) with \(\iota(p) \in W\), such that \[ \Phi \circ \iota|_{V} = \varphi|_{V} \] where \(V = \iota|_U^{\ast}(W) \subseteq U \subseteq \mathbb{R}^k\) is the pre-image of \(W\) under \(\iota|_U\).
Abusing language somewhat, since \(\iota\) is injective, we can identify \(\mathbb{R}^k\) with it's image \(E = \iota(\mathbb{R}^k) \subseteq \mathbb{R}^{n+k}\). With this identification, \(\varphi : U \to \mathbb{R}^{n+k}\) with \(U \subseteq E\) and the theorem gives a local extension \(\Phi\) to an open set \(W\) of \(\mathbb{R}^n\). That is, \[ \Phi|_{W \cap E} = \varphi|_{W \cap E} \] where \(V = \iota^{\ast} (W) = W \cap E\).
A local parametrisation \(\varphi\) is a \(C^{\infty}\) homeomorphism such that \(d\varphi_p\) is injective for every \(p \in U\). Let \(E = d\varphi_p(\mathbb{R}^k)\) denote the image of the differential. Since \(d\varphi_p\) is injective, \(E\) is a subspace of dimension \(k\) in \(\mathbb{R}^{n+k}\). Let \(F\) be a complementary subspace to \(E\). Write
\begin{align*} \varphi_E(p) &= \pi_E \circ \varphi \\ \varphi_F(p) &= \pi_F \circ \varphi \end{align*}where \(\pi_E : E \times F \to E \subseteq \mathbb{R}^{n+k}\) is projection onto the first factor, \(\pi_E(p, q) = p\) and \(\pi_F : E \times F \to F \subseteq \mathbb{R}^{n+k}\) is projection onto the second factor, \(\pi_F(p, q) = q\). Then \[ \varphi(p) = \varphi_E(p) + \varphi_F(p) \in E \times F \simeq \mathbb{R}^{n+k} \]
Define \[ \Phi(p, q) = \varphi(p) + q = \varphi_E(p) + \varphi_E(p) + q \] for \((p, q) \in E \times F\). Then
\begin{equation*} d\Phi = \begin{pmatrix} d\varphi_E & 0 \\ d\varphi_F & \operatorname{Id}_n \end{pmatrix} \end{equation*}By the chain rule, \[ d\varphi_E = d(\pi_E \circ \varphi) = d\pi_E \circ d\varphi = \pi_E \circ d\varphi. \] By definition \(E\) is the image of \(d\varphi_p\), hence \(\pi_E \circ d\varphi_p = d\varphi_p\). This is since \(\pi_E\) is the identity on \(E\) and \(d\varphi_p\) maps to \(E\). By assumption, \(d\varphi\) is injective, hence \(d\varphi_E\) is an isomorphism.
Thus at \((p, 0) \in E\), \(d\Phi\) is an isomorphism hence we may apply the inverse function theorem to obtain \(W, Z \subseteq \mathbb{R}^{n+k}\) with \((p, 0) \in W\) such that \(\Phi|_W: W \to Z\) is a diffeomorphism. Letting \(V = \iota^{\ast} (W)\) we have \[ \Phi \circ \iota|_V (p)= \Phi(\iota(p)) = \Phi(p, 0) = \varphi(p). \]
In this section we investigate the dependence of a sub-manifold \(M\) on local parametrisations. That is we study the effect of changing from one parametrisation to another. In particular, although we can already consider the notions of limit and continuity on \(M\), differentiability is a challenge due to the lack of linear structure on \(M\). The local parametrisations allow us to develop a suitable theory of calculus by identifying \(M\) locally with \(\mathbb{R}^k\).
Throughout this section let \(M \subset \mathbb{R}^n\) be an embedded sub-manifold of dimension \(k\), and let \(\{\varphi_{\alpha} : U_{\alpha} \to M\}\) be local parametrisations covering \(M\). That is, each \(\varphi_{\alpha}\) is a \(C^{\infty}\) homeomorphism onto its image \(V_{\alpha} = \varphi_{\alpha}(U_{\alpha})\), with injective differential at all points of \(U_{\alpha}\) and \[ M = \bigcup_{\alpha} V_{\alpha}. \]
Like continuity, differentiability is a local property hence to define differentiability on \(M\) it suffices to define differentiability on each \(V_{\alpha}\). To reiterate a point made above, differentiability on \(V_{\alpha}\) needs some thought due to the lack of linear structure on \(V_{\alpha\). But differentiability on \(U_{\alpha}\) is no problem, we already know how to do that! Thus we define differentiability on \(V_{\alpha}\) in terms of differentiability on \(U_{\alpha}\). Ensuring that we obtain a consistent notion of differentiability across all the sets \(V_{\alpha}\) requires the transition maps which is the focus of this section. In later sections we will see how the transition maps allows us to define a consistent calculus on \(M\).
For each \(\alpha, \beta\) the overlap is \(V_{\beta\alpha} = V_{\beta} \cap V_{\alpha} \subseteq M\). The transition map from \(\alpha\) to \(\beta\) is the map \[ \tau_{\beta\alpha} = \varphi_{\beta}^{-1} \circ \varphi_{\alpha}|_{U_{\beta\alpha}} \] where \(U_{\beta\alpha} = \varphi_{\alpha}^{\ast} (V_{\beta} \cap V_{\alpha})\).
Note here that we need to restrict \(\varphi_{\alpha}\) to \(U_{\beta\alpha} =\varphi_{\alpha}^{\ast} (V_{\beta\alpha}) \subseteq U_{\alpha}\) since \(\varphi_{\beta}^{-1}\) is only defined on \(V_{\beta}\). Of course \(\varphi_{\alpha}\) takes values in it's range \(V_{\alpha}\). Thus to form the composition \(\varphi_{\beta}^{-1} \circ \varphi_{\alpha}\) we must restrict \(\varphi_{\alpha}\) to the points \(U_{\alpha\beta}\) of \(U_{\alpha}\) that are mapped by \(\varphi_{\alpha}\) to \(V_{\alpha\beta} = V_{\alpha} \cap V_{\beta}\).
It could be the case that \(\varphi_{\alpha}\) and \(\varphi_{\beta}\) don't overlap; i.e. \(V_{\beta\alpha} = \emptyset\). In what follows we want to consider in what manner \(\varphi_{\alpha}\) and \(\varphi_{\beta}\) are compatible with each other. If there is no overlap, then there's nothing to say!
The transition maps \(\tau_{\beta\alpha}\) are homeomorphisms \(U_{\beta\alpha} \to U_{\alpha\beta}\).
Since \(\varphi_{\alpha}\) and \(\varphi_{\beta}\) are homeomorphisms, they and their inverses are continuous. Thus \(\tau_{\beta\alpha} = \varphi_{\beta}^{-1} \circ \varphi_{\alpha}\) is continuous.
We also have that \(\varphi_{\alpha}|_{U_{\beta\alpha}}\) is a bijection between \(U_{\beta\alpha}\) and \(V_{\beta\alpha}\) while \(\varphi_{\alpha}|_{U_{\beta\alpha}}\) is a bijection between \(U_{\alpha\beta}\) and \(V_{\alpha\beta}\). But \(V_{\alpha\beta} = V_{\beta\alpha}\) hence \(\tau_{\beta\alpha}\) is a bijection between \(U_{\beta\alpha}\) and \(U_{\alpha\beta}\).
Moreover, we have \[ \tau_{\beta\alpha} \circ \tau_{\alpha\beta} = \varphi_{\beta}^{-1} \circ \varphi_{\alpha} \circ \varphi_{\alpha}^{-1} \circ \varphi_{\beta} = \operatorname{Id} \]. Thus \(\tau_{\beta\alpha}^{-1} = \tau_{\alpha\beta}\) is continuous.
The question of differentiability is more subtle, since \(\varphi_{\beta}^{-1}\) is defined on \(V_{\beta} \subseteq M\) which is generally not an open subset of \(\mathbb{R}^n\). Since the latter is required for differentiability, we must work a little harder. Fortunately we already did the work!
The transition maps \(\tau_{\beta\alpha}\) are diffeomorphisms \(U_{\beta\alpha} \to U_{\alpha\beta}\).
Note that we have already established that \(\tau_{\beta\alpha}^{-1} = \tau_{\alpha\beta}\). Thus if we can prove \(\tau_{\beta\alpha}\) is \(C^{\infty}\) for every \(\beta,\alpha\), then applying that to \(\tau_{\alpha\beta}\) also yields \(\tau_{\beta\alpha}^{-1}\) is \(C^{\infty}\) and hence \(\tau_{\beta\alpha}\) is a diffeomorphism.
Now, recall that we may locally extend \(\varphi_{\beta}\) to a diffeomorphism \(\Phi_{\beta}\) of open subsets of \(\mathbb{R}^n\) such that \(\Phi_{\beta}|_{U_{\beta}} = \varphi_{\beta}\). But this also implies that \(\Phi_{\beta}^{-1}|_{V_{\beta}} = \varphi_{\beta}^{-1}\). More precisely, for every \(p \in U_{\beta}\), there exists an open sets \(W_{\beta}, Z_{\beta} \subseteq \mathbb{R}^n\) with \(p \in W_{\beta}\) and a diffeomorphism \(\Phi_{\beta} : W_{\beta} \to Z_{\beta}\) such that \[ \Phi_{\beta}|_{W_{\beta} \cap U_{\beta}} = \varphi_{\beta}. \] Then \(\Phi_{\beta}^{-1}|_{Z_{\beta} \cap V_{\beta}} = \varphi_{\beta}^{-1}\) and hence \[ \tau_{\beta\alpha} = \varphi_{\beta}^{-1} \circ \varphi_{\alpha}|_{U_{\beta\alpha}} = \Phi_{\beta}|_{Z_{\beta} \cap V_{\beta}}^{-1} \circ \varphi_{\alpha}|_{U_{\beta\alpha}} = \big(\Phi_{\beta}^{-1} \circ \varphi_{\alpha}\big)\big|_{U_{\beta\alpha}} \] is the composition of \(C^{\infty}\) maps hence is \(C^{\infty}\).
Now we have a suitable structure from which we may define calculus on embedded sub-manifolds. We won't pursue that just yet however. Rather we will next develop a theory of manifolds as spaces in their own right (i.e. not necessarily embedded into some larger space) generalising that of sub-manifolds. As part of that theory we will develop calculus on manifolds which in particular will also apply to sub-manifolds.