Vector Bundles
A vector bundle is a family of vector spaces that locally looks like a product. Local trivializations supply bases, and matrix-valued transition functions describe how those bases change on overlaps. The tangent bundle is the first example; normal bundles and bundles of covectors follow the same pattern.
We develop the definitions used in Section Connections, Lie Brackets, and Curvature. As a recap, a triple \((\mathcal E, \Pi, M)\) is called a differentiable vector bundle of rank \(n\) if \(\mathcal E\) and \(M\) are differentiable manifolds, \(\Pi : \mathcal E \to M\) is a differentiable projection map, and each fibre \(\mathcal E_X \triangleq \Pi^{-1} (X)\) for \(X \in M\) carries the structure of a \(n\)-dimensional vector space. Furthermore, the following local triviality condition holds: for all \(X \in M\), there exists a neighbourhood \(U\) and a diffeomorphism \(\varphi : \Pi^{-1} (U) \to U \times \mathbb R^n\) with the property that for all \(y \in U\), \(\varphi_y \triangleq \varphi\big|_{\mathcal E_y} : \mathcal E_y \to \{y\} \times \mathbb R^n\) is a vector space isomorphism. By way of nomenclature, \((\varphi, U)\) is a bundle chart.
Throughout the smooth discussion, manifolds are Hausdorff and second countable. A smooth vector bundle requires both a smooth projection and smooth local trivializations; smooth structures on the two spaces alone do not guarantee these properties. The same definition works over a manifold with boundary, using compatible boundary charts.
Example 1 (Product bundle). A rank \(n\) vector bundle of any space \(M\) is the product space, \(\mathcal E = M \times \mathbb R^n\), with \(\pi = \pi_1 : M \times \mathbb R^n \to M\) as the projection map. This is the product bundle, which is trivial.
The coordinate description of tangent vectors gives the local trivializations required by the definition.
Proposition 2 (Tangent bundle is a vector bundle). Let \(M\) be a smooth manifold of dimension \(n\), and let \(TM\) be its tangent bundle. With its standard projection map, its natural vector space structure on each fibre, and the topology and smooth structure constructed in Theorem 5, \(TM\) is a smooth vector bundle of rank \(n\) over \(M\).
Proof. Let \((U,\varphi)\) be a smooth chart for \(M\). Let \(\{x^i\}\) denote its coordinate functions. Then, define a map \(\Phi : \pi^{-1} (U) \to U \times \mathbb R^n\) by the formula
This clearly satisfies \(\pi_1 \circ \Phi = \pi\). The map \(\Phi\) is also linear on fibres. The coordinate map \(\widetilde \varphi\) constructed in Theorem 5 is clearly equivalent to the composition
Furthermore, \(\Phi\) is a diffeomorphism, as a composition of diffeomorphisms. Therefore, \(\Phi\) satisfies all the conditions for a smooth local trivialization. ◻
Just as in the case for charts, there is an analogue for the transition function between smooth local trivializations on a differentiable manifold \(M\). The composition of two smooth local trivializations has a simple form on sets where they overlap.
Lemma 3. Let \(\pi : \mathcal E \to M\) be a smooth vector bundle of rank \(k\) over \(M\). Suppose \(\Phi: \pi^{-1}(U) \to U \times \mathbb R^k\) and \(\Psi: \pi^{-1}(V) \to V \times \mathbb R^k\) are two smooth local trivializations of \(\mathcal E\) with \(U \cap V \neq \emptyset\). There exists a smooth map, denoted \(\tau : U \cap V \to \operatorname{GL}_k (\mathbb R)\) such that the composition \(\Phi \circ \Psi^{-1} : (U \cap V) \times \mathbb R^k \to (U \cap V) \times \mathbb R^k\) is characterized by the formula
where \(\tau (p) v\) denotes the action of \(\tau(p) \in \mathbb R^{k \times k}\) on the vector \(v \in \mathbb R^k\).
The transition fixes the base point and is linear on each fibre. Its matrix entries can be obtained by evaluating the transition on the standard basis, which also verifies their smooth dependence on the base point.
Note that if \(M\) is a smooth manifold, and \(\Phi, \Psi\) are smooth local trivializations of \(TM\) associated with two different smooth charts, then it follows from Lemma 3 that the transition function, \(\tau\), is the Jacobian matrix of the coordinate transition map. Furthermore, like the tangent bundle, vector bundles are often most easily described by giving a collection of vector spaces, one for each point of the base manifold. In order to make such a set into a smooth vector bundle, we would first have to construct a manifold topology and a smooth structure on the disjoint union of all the vector spaces, and then construct the local trivializations and show that they have the requisite properties.
Lemma 4 (Vector bundle chart lemma). Let \(M\) be a differentiable manifold, and suppose that for each \(p \in M\), there associates a real vector space \(E_p\) of dimension \(k\) (which is fixed). Let \(E = \bigsqcup_{p \in M} E_p\), and let \(\pi: E \to M\) be the map that takes each element of \(E_p\) to \(p\). Suppose we have
an open cover \(\{U_\alpha\}_{\alpha \in A}\) of \(M\);
for each \(\alpha \in A\), there is a bijection \(\Phi_\alpha : \pi^{-1} (U_\alpha) \to U_\alpha \times \mathbb R^k\), whose restriction to \(E_p\) is a vector space isomorphism from \(E_p \to \{p\} \times \mathbb R^k \cong \mathbb R^k\); and
for each \(\alpha,\beta \in A\) such that \(U_\alpha \cap U_\beta \neq \emptyset\), there exists a smooth map \(\tau_{\alpha\beta} : U_\alpha \cap U_\beta \to \operatorname{GL}_k (\mathbb R)\) such that \(\Phi_\alpha \circ \Phi_\beta^{-1} : (U_\alpha \cap U_\beta) \times \mathbb R^k \to (U_\alpha \cap U_\beta) \times \mathbb R^k\) is given by the formula
\[\Phi_\alpha \circ \Phi_\beta^{-1} (p,v) = (p, \tau_{\alpha\beta} (p) v ) ~.\]
Then, \(\mathcal E\) has a unique topology and smooth structure, making it into a differentiable manifold and a differentiable rank \(k\) vector bundle over \(M\), with \(\pi\) as its projection map, and \(\{(U_\alpha, \Phi_\alpha)\}_{\alpha \in A}\) as smooth local trivializations.
Proof of the chart lemma. Declare a subset \(O\subset E\) open exactly when every \(\Phi_\alpha(O\cap\pi^{-1}(U_\alpha))\) is open in \(U_\alpha\times\mathbb R^k\). The transition maps and their inverses are continuous, so each \(\pi^{-1}(U_\alpha)\) is open and \(\Phi_\alpha\) is a homeomorphism. If two points of \(E\) lie over different base points, disjoint base neighbourhoods separate them. If they lie over the same base point, one trivialization separates them in the Hausdorff product. Thus \(E\) is Hausdorff.
The base is second countable and hence Lindelöf, so a countable subcollection of the \(U_\alpha\) covers it. Countable product bases in the corresponding trivializations give a countable basis for \(E\). Composing the \(\Phi_\alpha\) with base coordinate charts gives Euclidean charts. On overlaps their transition maps are smooth, since they consist of a smooth base change and multiplication by the smooth matrix \(\tau_{\alpha\beta}\). These charts make \(E\) a smooth manifold of dimension \(\dim M+k\), with smooth projection and fibre-linear trivializations. Any other structure making these same maps trivializations has the same open sets and compatible charts, so it coincides with this one. \(\square\)
A section chooses one vector in every fibre of its domain. Let \(\pi : \mathcal E \to M\) be a vector bundle.
Definition 5. A section of \(\mathcal E\) is a section of the map \(\pi\), i.e. a continuous map \(\sigma: M \to \mathcal E\) satisfying \(\pi \circ \sigma = \operatorname{id}_M\). A local section of \(\mathcal E\) is a continuous map \(\sigma : U \to E\) defined on some open subset \(U \subset M\) satisfying \(\pi \circ \sigma = \operatorname{id}_U\). If \(M\) is a differentiable manifold and \(\mathcal E\) is a smooth vector bundle, a smooth section of \(\mathcal E\) is one that is a smooth map from its domain to \(\mathcal E\). Finally, the zero section of \(\mathcal E\) is the section \(\sigma : M \to \mathcal E\) given by \(\sigma(p) = 0 \in E_p\) for each \(p \in M\). The support of a section \(\sigma\), denoted \(\operatorname{spt} (\sigma)\), is defined as \(\operatorname{spt} (\sigma) \triangleq \overline{\{p \in M : \sigma(p) \neq 0\}}\).
Note that vector fields are smooth sections of the tangent bundle, and tangent bundles are examples of vector bundles. Therefore, we can analogously define local and global frames for (smooth) vector bundles, \(\pi : \mathcal E \to M\). If \(\Phi : \pi^{-1}(U) \to U \times \mathbb R^k\) is a smooth local trivialization of \(\mathcal E\), it follows that we can define \(\sigma_1,\cdots,\sigma_k : U \to \mathcal E\) by the formula \(\sigma_i(p) = \Phi^{-1} (p,e_i) = \Phi^{-1} \circ \widetilde e_i (p)\). This corresponds to the following commutative diagram:
Note that each \(\sigma_i\) is smooth because \(\Phi\) is a diffeomorphism, and \(\pi_1 \circ \Phi = \pi\). It follows that \(\pi \circ \sigma_i (p) = \pi \circ \Phi^{-1} (p,e_i) = \pi_1 (p,e_i) = p\), i.e. \(\sigma_i\) is a section. Finally, observe that \(\{\sigma_i(p)\}\) forms a basis for \(E_p\): \(\Phi\) restricted to \(E_p\) is an isomorphism onto \(\{p\} \times \mathbb R^k\), and \(\Phi(\sigma_i(p)) = (p,e_i)\), i.e. \(\Phi\) takes \(\{\sigma_i(p)\}\) to the standard basis for \(\{p\} \times \mathbb R^k \cong \mathbb R^k\). As such, we say \(\{\sigma_i(p)\}\) is the local frame associated with \(\Phi\). In fact, this is true generally.
Proposition 6. Every smooth local frame for a smooth vector bundle is associated with a smooth local trivialization, as above.
Proof of Proposition 6. In an existing trivialization over a smaller neighbourhood, the columns representing a smooth frame \(\sigma_1,\ldots,\sigma_k\) form a smooth invertible matrix \(A(p)\). The map
is represented there by \((p,c)\mapsto(p,A(p)c)\), whose inverse is \((p,v)\mapsto(p,A(p)^{-1}v)\). Matrix inversion is smooth on \(\operatorname{GL}_k(\mathbb R)\), so these are local diffeomorphisms. They agree on overlaps because coefficients in a basis are unique. Consequently \(S:U\times\mathbb R^k\to E|_U\) is a diffeomorphism and \(S^{-1}\) is the required trivialization. \(\square\)
Frames reduce smoothness questions to scalar functions. We use them to characterize smooth sections and the differentiable structure on \(TM\).
Proposition 7 (Smoothness of sections of vector bundles is related to smoothness of local frames). Let \(\pi : \mathcal E \to M\) be a smooth vector bundle, and \(\tau : M \to \mathcal E\) denote a section of the vector bundle. Let \(\{\sigma_i\}\) be a smooth local frame for \(\mathcal E\) over an open subset \(U \subset M\). Then, \(\tau\) is smooth on the frame domain if and only if its component functions with respect to \(\sigma_i\) are smooth.
Proof of Proposition 7. In the frame trivialization from Proposition 6, the section is \(p\mapsto(p,\tau^1(p),\ldots,\tau^k(p))\). It is smooth on \(U\) exactly when these components are smooth on \(U\). Global smoothness requires this condition on a cover of \(M\), not just one local frame domain. \(\square\)
Remark 8. In the notations of Proposition 7, at any \(p \in U \subset M\), \(\tau(p) = \tau^i(p) \sigma_i(p)\) for some uniquely determined coefficients \(\{\tau^1(p),\cdots,\tau^n(p)\}\). These functions \(\tau^i : U\to\mathbb R\) are called the component functions of \(\tau\), with respect to the given local frame \(\sigma_i\).
Proposition 9 (Uniqueness of differentiable structure on \(TM\)). With the qualifications and constructions in Theorem 5, the topology and differentiable structure constructed are the unique ones with respect to which \(\pi : TM \to M\) is a smooth vector bundle with the given vector space structure on the fibres, and such that all coordinate vector fields are smooth local sections.
Proof of Proposition 9. Any vector-bundle structure in which the coordinate fields are smooth admits, by Proposition 6, the trivialization taking \(\sum_i v^i\partial_i|_p\) to \((p,v^1,\ldots,v^n)\). These are exactly the canonical tangent-bundle trivializations. They therefore force both the topology and the smooth atlas by the chart lemma. \(\square\)
1. Bundle Homomorphisms
Definition 10. Let \(\pi : \mathcal E \to M\) and \(\pi' : \mathcal E' \to M'\) be vector bundles. A continuous map \(F: \mathcal E \to \mathcal E'\) is a bundle homomorphism if there exists a map \(f : M \to M'\) satisfying \(\pi' \circ F = f \circ \pi\), i.e. such that the following diagram commutes:
Furthermore, for each \(p \in M\), the restricted map \(F|_{\mathcal E_p} : \mathcal E_p \to \mathcal E'_{f(p)}\) is linear. Here, we say \(F\) covers \(f\).
Lemma 11. Under the qualifications in Definition 10, if \(F\) is a bundle homomorphism that covers \(f\), then \(f\) is continuous, and uniquely determined by \(F\). If the bundles and \(F\) are smooth, then so is \(f\).
The verification and proof is routine by noting that \(f = \pi' \circ F \circ \sigma_0\), where \(\sigma_0 : M \to \mathcal E\) is the zero section.
We note that a lot of the nomenclature carries over from the study of vector fields and vector bundles. For instance, a bundle isomorphism is a bijective bundle homomorphism whose inverse is also a bundle homomorphism. If, additionally, \(F\) is a diffeomorphism, then we say it is a smooth bundle isomorphism. We say two bundles are isomorphic if there exists a bundle isomorphism between the total spaces. As always, we can further qualify the last definition with “smooth”, if the requisite bundle isomorphism is smooth.
Finally, a comment about a more useful definition of bundle homomorphism. When we define vector bundles over the same base space \(M\), we use bundle homomorphism over \(M\) to mean a bundle homomorphism covering the identity map of \(M\)—in other words, a continuous map \(F : \mathcal E \to \mathcal E'\) such that \(\pi' \circ F = \pi\), i.e. such that the following diagram commutes:
As before, we require that the restriction of \(F\) to each fibre of \(\mathcal E\) is linear. The same nomenclature carries over from before, where we speak of bundle homomorphisms between vector bundles on different base spaces.
Lemma 12. Suppose \(\pi: \mathcal E \to M\) and \(\pi': \mathcal E' \to M\) are differentiable vector bundles over a differentiable manifold \(M\), and \(F: \mathcal E \to \mathcal E'\) is a bijective smooth bundle homomorphism over \(M\). Then, \(F\) is a smooth bundle isomorphism.
Proof of Lemma 12. In local trivializations over the same base neighbourhood, \(F\) has the form \((p,v)\mapsto(p,A(p)v)\). Its matrix is smooth: its columns are obtained by applying \(F\) to the constant frame sections. Bijectivity implies that \(A(p)\) is a square invertible matrix for every \(p\). Thus \((p,w)\mapsto(p,A(p)^{-1}w)\) is a smooth local formula for \(F^{-1}\). These inverses agree on overlaps, so \(F^{-1}\) is a smooth bundle homomorphism. \(\square\)
Remark 13. Vector bundles give rise to the induced structure. Let \(\pi: \mathcal E \to M\) and \(\pi': \mathcal E' \to M\) are differentiable vector bundles over a differentiable manifold \(M\), and \(F: \mathcal E \to \mathcal E'\) be a smooth bundle homomorphism over \(M\). Let \(\Gamma(\mathcal E)\) and \(\Gamma(\mathcal E')\) denote spaces of smooth global sections. Then, the induced map between spaces of global sections is given by
with the (pointwise) formula \(\widetilde F(\sigma)(p) = F(\sigma(p))\). Furthermore, \(\widetilde F(\sigma)\) is a section of \(\mathcal E'\), and this is smooth by composition. Clearly, induced map \(\widetilde F\) on sections is linear over \(\mathbb R\), because bundle homomorphism is linear on fibres. But more is true: \(\widetilde F\) is linear over \(\mathcal C^\infty (M)\), in the sense that for any \(a,b \in \mathscr C^\infty (M)\) and smooth sections \(\sigma_1,\sigma_2 \in \Gamma(\mathcal E)\),
Linearity over smooth functions expresses the pointwise character of the bundle map. The next lemma shows that this property also characterizes maps on sections that come from bundle homomorphisms.
Lemma 14. Let \(E,E'\) be smooth vector bundles over \(M\). A map \(\mathscr F:\Gamma(E)\to\Gamma(E')\) is \(C^\infty(M)\)-linear if and only if there exists a unique smooth bundle homomorphism \(F:E\to E'\) over \(M\) such that \(\mathscr F(\sigma)=F\circ\sigma\) for every smooth global section \(\sigma\).
Proof. A bundle homomorphism gives such a module map by fibrewise linearity. Conversely, first observe locality: if \(\sigma\) vanishes near \(p\), choose a smooth cutoff \(\chi=1\) near \(p\) supported where \(\sigma=0\). Then \(\chi\sigma=0\), so \(\chi\mathscr F(\sigma)=0\) and \(\mathscr F(\sigma)(p)=0\).
Choose a local frame \(e_i\) near \(p\), extend cutoff multiples to global sections \(\widetilde e_i\) agreeing with \(e_i\) near \(p\), and write \(\sigma=\sum_i a_i e_i\) there. Cutoff extensions of the \(a_i\) and locality give
In particular, this is zero if \(\sigma(p)=0\). Every fibre vector \(v\in E_p\) is the value of a global smooth section, obtained by extending a local constant-coefficient section with a cutoff. Define \(F_p(v)=\mathscr F(\sigma)(p)\) for any such section with \(\sigma(p)=v\). The preceding observation makes this independent of the choice, and module linearity makes \(F_p\) linear. Near \(p\), the columns of its matrix are the smooth sections \(\mathscr F(\widetilde e_i)\) in a target trivialization, so \(F\) is smooth. The defining identity and uniqueness now hold at every fibre. \(\square\)
2. Subbundles
The definition of a subbundle is contained in Definition 11. We switch notation to use \(\mathcal D \subset \mathcal E\) as the topological subspace of \(\mathcal E\), and use \(\mathcal D_p = \mathcal D \cap \mathcal E_p\) as the linear subspace of \(\mathcal E_p\). Since \(\mathcal D\) is a vector bundle over \(M\), it follows that all fibres \(\mathcal D_p\) must be nonempty, and have the same dimension. It follows from these definitions that we can also characterize smoothness of the vector subbundle via imposing conditions on local sections.
Proposition 15 (Characterization of smoothness of vector subbundles). Let \(\pi : \mathcal E \to M\) be a smooth vector bundle, and for each \(p \in M\), define a \(k\)-dimensional linear subspace \(\mathcal D_p \subset \mathcal E_p\). Then, \(\mathcal D = \bigcup_{p \in M} \mathcal D_p\) is a smooth subbundle of \(\mathcal E\) if and only if, for each \(p \in M\), there exists a neighbourhood \(U = U_p\) such that there are smooth local sections \(\sigma_1,\cdots,\sigma_k : U \to \mathcal E\), with the property that \(\{\sigma_1(q),\cdots,\sigma_k(q)\}\) form a basis for \(\mathcal D_q\) at each \(q \in U\).
Proof of the subbundle criterion. If \(D\) is a smooth subbundle, a bundle trivialization of \(D\) gives a smooth local frame, whose inclusion in \(E\) is smooth. Conversely, suppose the indicated sections \(\sigma_1,\ldots,\sigma_k\) exist. In a trivialization of \(E\), complete their values at one point to a basis by constant vectors. The determinant remains nonzero on a smaller neighbourhood, so this gives a smooth frame of \(E\) whose first \(k\) members span \(D\). In its associated trivialization, \(D\) is precisely \(U\times(\mathbb R^k\times\{0\})\). This proves the required subspace topology, smooth structure, and smooth inclusion. \(\square\)
Example 16. Suppose \(\pi : \mathcal E \to M\) is any trivial bundle. Let \(\{E_1,\cdots,E_k\}\) be a smooth global frame for \(\mathcal E\). For \(0 \leq m \leq k\), the subset \(\mathcal D \subset \mathcal E\), defined by \(\mathcal D_p = \operatorname{span} (\{E_1|_p, \cdots, E_m|_p\})\) for each \(p \in M\) is a smooth subbundle of \(\mathcal E\).
Kernels and images of bundle homomorphisms supply further examples, provided their dimensions remain constant. Suppose \(\pi : \mathcal E \to M\) and \(\pi' : \mathcal E' \to M\) are vector bundles, and \(F : \mathcal E \to \mathcal E'\) is a bundle homomorphism over \(M\). For each \(p \in M\), the rank of the linear map \(F|_{E_p}\) is called the rank of \(F\) at \(p\). We say \(F\) has constant rank if \(F|_{E_p}\) is of the same rank for all \(p \in M\).
Theorem 17. Let \(F:E\to E'\) be a smooth bundle homomorphism between smooth bundles of fixed ranks over \(M\). Its kernel and image are smooth subbundles of fixed ranks if and only if \(F\) has constant rank. If the convention allows ranks to vary between connected components, replace “constant” by “locally constant.”
Proof. In local frames let \(A(p)\) be the smooth \(\ell\times k\) matrix of \(F\). Suppose its rank is the constant \(r\). Near any point choose an invertible \(r\times r\) minor and reorder rows and columns to write
Rank \(r\) forces \(E=DB^{-1}C\): subtract \(DB^{-1}\) times the first block row from the second, leaving a lower-right block that must vanish. The first \(r\) columns of \(A\) are a smooth basis of its image, while the columns of
are a smooth basis of its kernel. Proposition 15 gives both subbundles; the cases \(r=0\) and \(r=k\) have the evident zero or empty frames. Conversely, the rank of an image subbundle is the rank of \(F\), so it is constant under the fixed-rank convention. Without smoothness of \(F\), constant rank alone yields no smooth frame. \(\square\)
A consequence of the construction above is clear on \(\mathbb R^n\).
Lemma 18. Let \(i:M\to\mathbb R^n\) be an immersion and \(D\) a smooth rank-\(k\) subbundle of the pullback bundle \(i^*T\mathbb R^n\cong M\times\mathbb R^n\). The fibrewise Euclidean orthogonal complement \(D^\perp\) is a smooth rank-\((n-k)\) subbundle and has smooth local orthonormal frames.
Proof. Fix a smooth local frame \(X_1,\ldots,X_k\) for \(D\). Complete its values at a point \(p\) to a basis of \(\mathbb R^n\) by constant vectors \(c_{k+1},\ldots,c_n\). On a smaller neighbourhood the determinant remains nonzero. Gram–Schmidt applied to this smooth frame gives smooth orthonormal fields: each subtraction is smooth and every normalizing denominator is positive. The first \(k\) span \(D\), so the last \(n-k\) span \(D^\perp\). Proposition 15 proves the assertion. No extension to an ambient closed slice is required. \(\square\)
Lemma 19. For an immersion \(i:M^m\to\mathbb R^n\), its normal bundle is the smooth rank-\((n-m)\) subbundle
Proof. The differential has constant rank \(m\), so its image is a smooth subbundle by Theorem 17. Apply Lemma 18. When the immersion is an inclusion, this is the usual notation \(TM\subset T\mathbb R^n|_M\). For a noninjective immersion, the base point \(p\) is retained even when two points have the same image. \(\square\)
Remark 20. A smooth distribution \(D\subset TM\) defines an accessibility relation: \(p\sim q\) when a smooth curve from \(p\) to \(q\) has velocity in \(D\) at every time. Constant curves and time reversal give reflexivity and symmetry. To concatenate two curves smoothly, first reparametrize them by smooth nondecreasing functions constant near both endpoints; the new velocities remain scalar multiples of the old ones. This proves transitivity.
If \(D\) is integrable, Frobenius coordinates identify its accessibility classes with connected maximal integral manifolds, the leaves of its foliation. Leaves are immersed submanifolds; they need not be embedded. For instance, on \(\mathbb T^2\subset\mathbb C^2\) take
For \((\omega_1,\omega_2)\ne(0,0)\) this is a rank-one integrable distribution, with leaf
Indeed a tangent curve satisfies \(\gamma'=a(t)V(\gamma)\), and integrating \(a\) gives this same orbit. If \(\omega_2=0\) or \(\omega_1/\omega_2\in\mathbb Q\), the orbit is a periodic embedded circle. If \(\omega_1/\omega_2\) is irrational, the parametrization is injective and immersive. At the times returning the second coordinate to a fixed value, the first coordinate is an irrational rotation orbit and is dense in its circle. To see the latter fact, the pigeonhole principle supplies nonzero integer multiples of the irrational angle arbitrarily close to zero modulo one; their successive multiples approximate any prescribed point with error at most that small angle. Allowing an additional fixed time shift then proves density in the full torus.
Such a dense leaf is not embedded: a local embedded one-dimensional slice would leave a nonempty open part of a coordinate neighbourhood disjoint from the leaf, contradicting density. Its intrinsic manifold topology is therefore finer than the subspace topology. If both \(\omega_i\) vanish, the distribution has rank zero and the leaves are single points.
3. Fibre Bundles
The local-product definition also makes sense when the fibres are arbitrary topological spaces. Removing the linear structure leads to fibre bundles.
Definition 21. Let \(M\) and \(F\) be topological spaces. A fibre bundle over \(M\), with model fibre \(F\), is a topological space \(\mathcal E\) with surjective, continuous map \(\pi : \mathcal E \to M\) such that for each \(x \in M\), there exists a neighbourhood \(U=U_x \subset M\) and a homeomorphism \(\Phi : \pi^{-1} (U) \to U \times F\) such that the following diagram commutes:
By way of nomenclature, \(U\) and \(\Phi\), together, is called a local trivialization of \(\mathcal E\) over \(U\). \(\mathcal E\) is called the total space of the bundle, \(M\) is the base space, and \(\pi\) is the projection map (associated with the fibre bundle). If \(E,M,F\) are differentiable manifolds, \(\pi\) is smooth, and local trivializations can be chosen to be diffeomorphisms, then we refer to the definition above as smooth fibre bundles.
Definition 22. Trivial fibre bundle is one that admits a local trivialization over the entire base space. If the fibre bundle is a smooth bundle, and the global trivialization is a diffeomorphism, then we say the fibre bundle is smoothly trivial.
Example 23. Every rank \(k\) vector bundle is a fibre bundle, with model fibre \(\mathbb R^k\).
Example 24. The product space \(M \times F\) is a fibre bundle with the projection map \(\pi_1 : M \times F \to M\). This is called the product fibre bundle.
Example 25. A covering map whose fibres all have the same cardinality is a fibre bundle with a discrete model fibre \(S\) of that cardinality. This condition holds, in particular, for a covering of a connected manifold. For an evenly covered open set \(U\), label the sheets of \(\pi^{-1}(U)\) by \(S\) and define \(\Phi(x)=(\pi(x),s)\) on sheet \(s\). Each sheet maps homeomorphically to \(U\), and \(S\) is discrete, so \(\Phi\) and its inverse are continuous. For disconnected bases, fibre cardinalities can differ between components; in that case there is no single model fibre for the entire covering. Sheets should not be confused with connected components when \(U\) itself is disconnected.
Reading: Carmo (1992), Riemannian geometry.