Vector bundles and characteristic classes

MotivationH

Combing, twisting and curvature

You cannot comb the hair on a ball flat without leaving a cowlick: every continuous tangent vector field on the sphere $S^2$ vanishes somewhere. On a doughnut you can. On the circle $S^1$ there is a nowhere-zero tangent field, and on $S^3$ there are three that are orthonormal at every point. Why these differences?

The tangent spaces of a manifold form a family of vector spaces parametrised by the manifold — a vector bundle. Locally it is a product, but globally it may twist, just as a Möbius band is locally a strip $(a, b) \times \R$ but is not the cylinder $S^1 \times \R$ . Characteristic classes are cohomology classes of the base that measure this twisting. They obstruct sections ("is there a nowhere-zero vector field?"), frames ("is the manifold parallelisable?") and embeddings ("does $\R P^4$ immerse in $\R^6$ ?"), and, through Chern–Weil theory, they are computed by integrating curvature. The prototype is Gauss–Bonnet: for every metric on a closed surface, $\int_M K \, dA = 2\pi \chi(M)$ — geometry on the left, topology on the right.

This lesson builds on algebraic topology (cohomology, degree), smooth manifolds and differential forms.

IntuitionH

Local products, global twists

Take a strip $[0, 1] \times \R$ and glue its ends. Glue $(0, v)$ to $(1, v)$ and you get a cylinder; glue $(0, v)$ to $(1, -v)$ and you get the (infinite) Möbius band. Both are lines over a circle, and both look the same over any proper arc. The difference is global, and a section — a continuous choice of a vector in each fibre — detects it: the cylinder has the nowhere-zero section $v = 1$ , but any section of the Möbius band starting at height $v$ must come back at height $-v$ , so it crosses zero.

Left: the cylinder (sides glued the same way) carries a section that never meets the zero section (dashed). Right: on the Möbius band (sides glued with a flip) a section leaving the left edge at height +h must arrive at the right edge at height −h, so it crosses zero.

Characteristic classes generalise this sign-flip. A bundle can be described by transition functions on overlaps of trivialising open sets, valued in $GL_k$ ; the cohomology of the base records how they fail to be undone. The Möbius band has transition function $-1$ on one overlap: the non-trivial element of $H^1(S^1; \Z/2)$ , its first Stiefel–Whitney class.

DefinitionsH

Definitions

Definition— vector bundle

A real vector bundle of rank $k$ over a space $B$ is a map $p : E \to B$ with a real vector space structure on each fibre $E_b = p^{-1}(b)$ , such that every point of $B$ has an open neighbourhood $U$ and a homeomorphism $\varphi : p^{-1}(U) \to U \times \R^k$ (a local trivialisation) mapping each fibre $E_b$ linearly isomorphically onto $\{b\} \times \R^k$ . Complex bundles are defined with $\C^k$ . A morphism of bundles over $B$ is a continuous map $E \to E'$ over $B$ that is linear on fibres; an isomorphism is a morphism with an inverse morphism.

Definition— trivial bundle; section; frame

The trivial bundle is $\varepsilon^k = B \times \R^k$ ; a bundle is trivial if it is isomorphic to it. A section is a continuous $s : B \to E$ with $p \circ s = \mathrm{id}_B$ ; every bundle has the zero section. A frame over $U$ is a family of sections $s_1, \dots, s_k$ that form a basis of $E_b$ for every $b \in U$ . A manifold is parallelisable if its tangent bundle is trivial.

Definition— pullback; Whitney sum

For a bundle $E \to B$ and a map $f : B' \to B$ , the pullback $f^*E = \{ (b', v) \in B' \times E : f(b') = p(v) \}$ is a bundle over $B'$ with fibre $E_{f(b')}$ over $b'$ . The Whitney sum $E \oplus E'$ of two bundles over $B$ has fibres $E_b \oplus E'_b$ ; it is the pullback of $E \times E' \to B \times B$ along the diagonal. Tensor products, duals, $\operatorname{Hom}(E, E')$ and exterior powers are defined fibrewise in the same way.

Definition— transition functions

If $\varphi_\alpha$ and $\varphi_\beta$ are trivialisations over $U_\alpha$ and $U_\beta$ , then $\varphi_\alpha \circ \varphi_\beta^{-1}(b, v) = (b, g_{\alpha\beta}(b) v)$ for continuous transition functions $g_{\alpha\beta} : U_\alpha \cap U_\beta \to GL_k(\R)$ , satisfying the cocycle condition $g_{\alpha\beta}\, g_{\beta\gamma} = g_{\alpha\gamma}$ on triple overlaps. Conversely, any cocycle defines a bundle by gluing the pieces $U_\alpha \times \R^k$ . A bundle is orientable if the cocycle can be chosen with values in $GL_k^+(\R)$ (positive determinant).

Definition— tangent, normal and tautological bundles

The tangent bundle $TM$ of a smooth $n$ -manifold has fibres $T_xM$ ; its transition functions are the Jacobians of the chart changes. For $M \subset \R^N$ , the normal bundle $\nu$ has fibres $(T_xM)^\perp$ , and $TM \oplus \nu \cong \varepsilon^N$ . The Grassmannian $\mathrm{Gr}_k(\R^n)$ is the space of $k$ -dimensional subspaces of $\R^n$ , and the tautological bundle $\gamma_k$ over it has fibre over a plane the plane itself. For $k = 1$ this is the tautological line bundle $\gamma$ over $\R P^{n-1}$ ; over $\R P^1 = S^1$ it is the Möbius bundle. Letting $n \to \infty$ gives $\mathrm{Gr}_k(\R^\infty) = BO(k)$ , and in the complex case $\mathrm{Gr}_k(\C^\infty) = BU(k)$ .

Worked examplesH

Examples

Example— Spheres

$TS^n = \{ (x, v) \in S^n \times \R^{n+1} : v \perp x \}$ , with normal bundle $\{ (x, tx) \}$ trivialised by the section $x \mapsto x$ . So $TS^n \oplus \varepsilon^1 \cong \varepsilon^{n+1}$ : the tangent bundle of a sphere is stably trivial. Yet $TS^2$ is not trivial (Theorem 4): adding a trivial bundle can untwist a bundle.

Example— Line bundles over the circle

A real line bundle over $S^1$ can be trivialised over two arcs whose intersection has two components; the transition function is a non-zero constant on each, and only the signs matter. Up to isomorphism there are exactly two line bundles: the trivial one and the Möbius bundle. This is the first case of the clutching construction (Theorem 5).

Example— Projective spaces

$T\R P^n \oplus \varepsilon^1 \cong (n+1)\gamma$ , the sum of $n + 1$ copies of the tautological line bundle (Milnor–Stasheff, Lemma 4.4): a tangent vector at a line $L$ is a linear map $L \to L^\perp$ , so $T\R P^n \cong \operatorname{Hom}(\gamma, \gamma^\perp)$ , and adding the trivial bundle $\operatorname{Hom}(\gamma, \gamma) \cong \varepsilon^1$ gives $\operatorname{Hom}(\gamma, \varepsilon^{n+1}) \cong (n+1)\gamma^* \cong (n+1)\gamma$ . Similarly $T\C P^n \oplus \varepsilon^1_\C \cong (n+1)\,\overline{\gamma}$ for the complex tautological bundle.

ProofH

Sections, frames and the hairy ball

Theorem— Proposition 1 (frames trivialise)

A rank- $k$ bundle $E \to B$ is trivial if and only if it has $k$ sections $s_1, \dots, s_k$ that are linearly independent at every point. In particular, a line bundle is trivial if and only if it has a nowhere-zero section.

Proof

A trivialisation $E \cong B \times \R^k$ gives the frame $s_i(b) = (b, e_i)$ . Conversely, given a frame, define $\Phi : B \times \R^k \to E$ by $\Phi(b, t) = \sum_i t_i s_i(b)$ . It is continuous, linear and bijective on each fibre. To see that $\Phi^{-1}$ is continuous, work in a local trivialisation over $U$ : there $\Phi$ becomes $(b, t) \mapsto (b, A(b) t)$ , where the columns of the matrix $A(b)$ are the coordinates of the $s_i(b)$ , so $A : U \to GL_k(\R)$ is continuous, and $(b, v) \mapsto (b, A(b)^{-1} v)$ is continuous because matrix inversion is (Cramer's rule). So $\Phi$ is an isomorphism.

∎

Theorem— Theorem 2 (the Möbius bundle is not trivial)

Every section of the Möbius bundle vanishes somewhere; hence the Möbius bundle is not trivial.

Proof

Realise the Möbius bundle as the quotient of $\R \times \R$ by $(t, v) \sim (t + 1, -v)$ , over $S^1 = \R/\Z$ . A section is the same as a continuous function $s : \R \to \R$ with $s(t + 1) = -s(t)$ (its graph is invariant under the identification). Then $s(0)$ and $s(1) = -s(0)$ are either both zero or of opposite signs, and the intermediate value theorem gives $t \in [0, 1]$ with $s(t) = 0$ . By Proposition 1 the bundle is not trivial.

∎

Theorem— Theorem 3 (odd spheres can be combed; S³ is parallelisable)

(a) For every $m \ge 1$ , $S^{2m-1}$ carries a nowhere-zero tangent vector field. (b) $S^1$ and $S^3$ are parallelisable.

Proof

(a) Regard $S^{2m-1} \subset \C^m$ and set $v(z) = iz$ , i.e. $(x_1, y_1, \dots, x_m, y_m) \mapsto (-y_1, x_1, \dots, -y_m, x_m)$ . Then $\langle z, iz \rangle = \sum_k \operatorname{Re}(\bar z_k \cdot i z_k) = \sum_k \operatorname{Re}(i |z_k|^2) = 0$ , so $v(z)$ is tangent, and $|v(z)| = |z| = 1$ . (b) For $S^1$ this single field is a frame. For $S^3 \subset \mathbb{H} = \R^4$ , the quaternions, let $v_1(x) = ix$ , $v_2(x) = jx$ , $v_3(x) = kx$ . Left multiplication by a unit quaternion is an isometry of $\R^4$ , and $1, i, j, k$ are orthonormal; so $x, ix, jx, kx$ are orthonormal for every unit $x$ (they are the images of $1, i, j, k$ under right multiplication by $x$ , also an isometry). Thus $v_1, v_2, v_3$ form an orthonormal tangent frame, and $TS^3$ is trivial by Proposition 1. The octonions give $S^7$ the same way.

∎

Theorem— Theorem 4 (hairy ball theorem; Poincaré, Brouwer)

If $n$ is even, every continuous tangent vector field on $S^n$ vanishes somewhere. In particular $TS^2$ is not trivial.

Proof

Suppose $v$ is a tangent field on $S^n$ with no zero; replacing $v(x)$ by $v(x)/|v(x)|$ we may assume $|v(x)| = 1$ and $v(x) \perp x$ . Define $H(x, t) = \cos(\pi t)\, x + \sin(\pi t)\, v(x)$ . Since $x$ and $v(x)$ are orthonormal, $|H(x, t)| = 1$ , so $H$ is a homotopy in $S^n$ from the identity ( $t = 0$ ) to the antipodal map $x \mapsto -x$ ( $t = 1$ ). Homotopic maps have equal degree, but $\deg(\mathrm{id}) = 1$ and the antipodal map, a composite of $n + 1$ reflections, has degree $(-1)^{n+1} = -1$ for $n$ even. Contradiction.

∎

So whether $S^n$ can be combed depends only on the parity of $n$ . Parallelisability is much subtler: $S^n$ is parallelisable only for $n = 0, 1, 3, 7$ (Bott–Milnor and Kervaire, 1958), and J. F. Adams (1962) determined exactly how many independent vector fields $S^{n-1}$ carries: $\rho(n) - 1$ , where $\rho(2^{4a+b} \cdot \text{odd}) = 2^b + 8a$ for $0 \le b \le 3$ are the Radon–Hurwitz numbers.

ExplanationH

Clutching and classifying spaces

Theorem— Theorem 5 (clutching)

Every bundle over $S^n$ is trivial over the two closed hemispheres $D^n_\pm$ , and is determined up to isomorphism by the clutching function $f : S^{n-1} \to GL_k$ comparing the two trivialisations along the equator. This gives a bijection between isomorphism classes of rank- $k$ complex bundles over $S^n$ and $[S^{n-1}, GL_k(\C)] \cong \pi_{n-1}(U(k))$ ; for oriented real bundles one gets $[S^{n-1}, SO(k)] \cong \pi_{n-1}(SO(k))$ (for a connected topological group, free and based homotopy classes of maps from a sphere agree).

Bundles over a disc are trivial (the disc is contractible: Theorem 6 below), and homotopic clutching functions give isomorphic bundles (Hatcher, Vector Bundles and K-Theory, Proposition 1.11). Examples: real line bundles over $S^1$ correspond to $\pi_0(GL_1(\R)) = \{\pm 1\}$ — trivial and Möbius; complex line bundles over $S^2$ correspond to $\pi_1(U(1)) = \Z$ , and the integer is the first Chern class; $TS^2$ is clutched by a map of degree $2$ , matching $\chi(S^2) = 2$ .

Theorem— Theorem 6 (classification)

Let $B$ be paracompact. (a) Homotopic maps $f \simeq g : B' \to B$ pull back isomorphic bundles; so bundles over a contractible paracompact space are trivial. (b) The map $[B, \mathrm{Gr}_k(\R^\infty)] \to \operatorname{Vect}^k_\R(B)$ , $f \mapsto f^*\gamma_k$ , is a bijection, and likewise $[B, BU(k)] \cong \operatorname{Vect}^k_\C(B)$ with $BU(k) = \mathrm{Gr}_k(\C^\infty)$ .

See Milnor–Stasheff, Theorems 5.6 and 5.7, and Hatcher, VBKT, Theorem 1.16. Since $\R P^\infty = K(\Z/2, 1)$ and $\C P^\infty = K(\Z, 2)$ are Eilenberg–MacLane spaces (see homotopy theory), it follows that real line bundles over a CW complex correspond to $H^1(B; \Z/2)$ and complex line bundles to $H^2(B; \Z)$ — via the classes $w_1$ and $c_1$ defined next.

Definition— characteristic class

A characteristic class of rank- $k$ real bundles with coefficients in $R$ assigns to each bundle $E \to B$ a class $c(E) \in H^\bullet(B; R)$ depending only on the isomorphism class of $E$ and natural under pullback: $c(f^*E) = f^* c(E)$ . By Theorem 6 these are exactly the elements of $H^\bullet(BO(k); R)$ (evaluate on the tautological bundle); for complex bundles, of $H^\bullet(BU(k); R)$ .

ProofH

Stiefel–Whitney, Chern, Euler and Pontryagin classes

Theorem— Theorem 7 (Stiefel–Whitney classes)

There are unique classes $w_i(E) \in H^i(B; \Z/2)$ , $i \ge 0$ , for real vector bundles over paracompact spaces, such that: (1) $w_0 = 1$ and $w_i(E) = 0$ for $i > \operatorname{rank} E$ ; (2) naturality: $w_i(f^*E) = f^* w_i(E)$ ; (3) Whitney sum formula: $w(E \oplus E') = w(E)\, w(E')$ for the total class $w = 1 + w_1 + w_2 + \cdots$ ; (4) normalisation: $w_1$ of the Möbius bundle over $\R P^1$ is non-zero.

Existence and uniqueness are Milnor–Stasheff, §4 and §8: $H^\bullet(BO(k); \Z/2) = \Z/2[w_1, \dots, w_k]$ , and the classes can be built from the Thom isomorphism and Steenrod squares. Immediate consequences: a trivial bundle has $w = 1$ (it is pulled back from a point), so $w(E \oplus \varepsilon^m) = w(E)$ ; $E$ is orientable if and only if $w_1(E) = 0$ ; if $E$ has $m$ everywhere independent sections then $E \cong E' \oplus \varepsilon^m$ and $w_i(E) = 0$ for $i > k - m$ . The tautological line bundle over $\R P^n$ has $w(\gamma) = 1 + a$ , where $a$ generates $H^1(\R P^n; \Z/2)$ and $H^\bullet(\R P^n; \Z/2) = \Z/2[a]/(a^{n+1})$ .

Theorem— Theorem 8 (Stiefel–Whitney classes of projective space)

$w(T\R P^n) = (1 + a)^{n+1} = \sum_{i=0}^{n} \binom{n+1}{i} a^i$ . Consequently: (a) if $\R P^n$ is parallelisable, then $n + 1$ is a power of $2$ ; (b) if $n = 2^r$ , then $\R P^n$ does not immerse in $\R^{2n-2}$ .

Proof

From $T\R P^n \oplus \varepsilon^1 \cong (n+1)\gamma$ , the Whitney sum formula and $w(\varepsilon^1) = 1$ give $w(T\R P^n) = w(\gamma)^{n+1} = (1 + a)^{n+1}$ .

(a) If $\R P^n$ is parallelisable then $w = 1$ , so $\binom{n+1}{i}$ is even for $1 \le i \le n$ . Write $n + 1 = 2^r m$ with $m$ odd. In characteristic $2$ , $(1 + a)^{2^r} = 1 + a^{2^r}$ , so $(1 + a)^{n+1} = (1 + a^{2^r})^m = 1 + m\, a^{2^r} + \cdots$ . If $m > 1$ then $2^r \le n$ , so the coefficient $m \equiv 1$ of $a^{2^r}$ is a non-zero class: contradiction. Hence $m = 1$ .

(b) If $M^n$ immerses in $\R^{n+k}$ , its normal bundle $\nu$ has rank $k$ and $TM \oplus \nu \cong \varepsilon^{n+k}$ , so $w(\nu) = w(TM)^{-1} =: \bar w(M)$ and $\bar w_i(M) = 0$ for $i > k$ (Whitney duality). For $n = 2^r$ : $(1 + a)^{n+1} = (1 + a)(1 + a^n)$ , so $\bar w = (1 + a)^{-1} (1 + a^n)^{-1} = \big( \sum_{i=0}^{n} a^i \big)(1 + a^n) = \sum_{i=0}^{n-1} a^i$ in $\Z/2[a]/(a^{n+1})$ . Thus $\bar w_{n-1} \neq 0$ , and an immersion into $\R^{n+k}$ needs $k \ge n - 1$ .

∎

Whitney proved that every $n$ -manifold immerses in $\R^{2n-1}$ (for $n \ge 2$ ), so (b) is sharp. Bott–Milnor and Kervaire (1958) showed that only $\R P^1$ , $\R P^3$ and $\R P^7$ are parallelisable; the characteristic-class argument alone leaves $\R P^{15}, \R P^{31}, \dots$ undecided.

Theorem— Theorem 9 (Chern, Euler and Pontryagin classes)

(a) For complex bundles there are unique Chern classes $c_i(E) \in H^{2i}(B; \Z)$ with the analogous axioms: $c_0 = 1$ , $c_i = 0$ for $i > \operatorname{rank}_\C E$ , naturality, $c(E \oplus E') = c(E) c(E')$ , and $c_1$ of the tautological line bundle $\gamma$ over $\C P^1$ satisfying $\langle c_1(\gamma), [\C P^1] \rangle = -1$ (with the standard orientation conventions). Then $c(T\C P^n) = (1 + x)^{n+1}$ , where $x = -c_1(\gamma)$ generates $H^2(\C P^n; \Z)$ . (b) An oriented rank- $k$ real bundle has an Euler class $e(E) \in H^k(B; \Z)$ , natural, multiplicative under Whitney sum, zero if $E$ has a nowhere-zero section, reducing mod $2$ to $w_k$ , and equal to $c_k$ for the underlying real bundle of a complex rank- $k$ bundle. (c) The Pontryagin classes of a real bundle are $p_i(E) = (-1)^i c_{2i}(E \otimes \C) \in H^{4i}(B; \Z)$ .

See Milnor–Stasheff, §9, §14 and §15. The decisive link between the Euler class and the topology of a manifold is the Poincaré–Hopf theorem: for a closed oriented manifold, $\langle e(TM), [M] \rangle = \chi(M)$ , and for a vector field with isolated zeros the sum of their indices is $\chi(M)$ (Milnor–Stasheff, Corollary 11.12). For instance $\langle c_n(T\C P^n), [\C P^n] \rangle = n + 1 = \chi(\C P^n)$ . Pontryagin numbers are oriented-cobordism invariants; Hirzebruch's signature theorem gives $\operatorname{sign}(M^4) = \tfrac13 \langle p_1(TM), [M] \rangle$ , and Milnor (1956) used it to discover exotic smooth structures on $S^7$ .

ExplanationH

Curvature computes characteristic classes

Let $E \to M$ be a smooth complex vector bundle with a connection $\nabla$ (a rule for differentiating sections along vector fields). In a local frame, $\nabla = d + A$ with $A$ a matrix of $1$ -forms, and the curvature $F = dA + A \wedge A$ is a matrix of $2$ -forms which transforms by conjugation, $F \mapsto g^{-1} F g$ , under a change of frame. It satisfies the Bianchi identity $dF = F \wedge A - A \wedge F$ .

Theorem— Theorem 10 (Chern–Weil)

For every polynomial $P$ on $\mathfrak{gl}_k(\C)$ invariant under conjugation, $P(F)$ is a well-defined closed even-degree form on $M$ , and its de Rham class does not depend on the connection. The total Chern class is represented by

c(E) = \Big[ \det\Big( I + \frac{i}{2\pi} F \Big) \Big] \in H^\bullet(M; \R), \qquad c_1(E) = \Big[ \frac{i}{2\pi} \operatorname{tr} F \Big].

Why: invariance under conjugation makes $P(F)$ independent of the frame. Closedness follows from the Bianchi identity, since $d\,\operatorname{tr}(F^j) = j\, \operatorname{tr}(F^{j-1}(F \wedge A - A \wedge F)) = 0$ by cyclicity of the trace. For two connections $\nabla_0, \nabla_1$ , the path $\nabla_t = (1 - t)\nabla_0 + t\nabla_1$ gives an explicit transgression form whose exterior derivative is $P(F_1) - P(F_0)$ . That these de Rham classes are the images of the integral Chern classes is proved by checking the axioms (Milnor–Stasheff, Appendix C; Bott–Tu, §20 for the splitting principle).

Example— The Dirac monopole and the tautological bundle over the sphere

The tautological line bundle over $\C P^1 = S^2$ , with the connection induced from $\C^2$ , has curvature $F$ with $\frac{i}{2\pi} \int_{S^2} F = -1$ . In physics this is the magnetic monopole of unit charge: the connection is the vector potential, $F$ is the field strength, and the integrality of $\int c_1$ is Dirac's quantisation condition. The integral of the curvature of any connection on this bundle is the same — a topological charge.

For an oriented Riemannian $2m$ -manifold with Levi-Civita connection and curvature matrix $\Omega$ in an oriented orthonormal frame, the Pfaffian $\operatorname{Pf}(\Omega)$ (the polynomial with $\operatorname{Pf}(X)^2 = \det X$ on skew-symmetric matrices) is an invariant $2m$ -form representing $(2\pi)^m e(TM)$ .

Theorem— Theorem 11 (Gauss–Bonnet–Chern; Allendoerfer–Weil 1943, Chern 1944)

For a closed oriented Riemannian manifold $M$ of even dimension $n = 2m$ ,

\int_M \operatorname{Pf}(\Omega) = (2\pi)^{m}\, \chi(M).

For $n = 2$ , $\operatorname{Pf}(\Omega) = \Omega_{12} = K \, dA$ (with the usual sign conventions), and this is the Gauss–Bonnet theorem $\int_M K\, dA = 2\pi \chi(M)$ .

Chern's intrinsic proof (1944) transgresses $\operatorname{Pf}(\Omega)$ on the unit sphere bundle and reduces the theorem to Poincaré–Hopf. In dimension two there is also an elementary argument, sketched here: assume the local Gauss–Bonnet formula $\alpha + \beta + \gamma - \pi = \int_T K\, dA$ for a geodesic triangle $T$ with angles $\alpha, \beta, \gamma$ , and a triangulation of $M$ by $F$ geodesic triangles with $E$ edges and $V$ vertices. Summing over triangles, the angles around each vertex add up to $2\pi$ , so

\int_M K \, dA = 2\pi V - \pi F = 2\pi V - 2\pi E + 2\pi F = 2\pi \chi(M),

using $3F = 2E$ . Curvature integrates to an Euler characteristic.

Formal mathematicsF

Sections, frames and Stiefel–Whitney arithmetic in Lean

Mathlib has topological vector bundles (VectorBundle), with the trivial bundle, pullbacks, Whitney sums (fibrewise products E₁ ×ᵇ E₂) and the tangent bundle of a manifold as instances. We cite these to connect the lesson's definitions to the library:

trivial_bundleLean 4Applies a library theoremFormally verified

/-- The product `B × ℝᵏ`, as Mathlib's `Bundle.Trivial B F`, is a vector bundle. This applies
the instance `Bundle.Trivial.vectorBundle`. -/
theorem trivial_bundle (B F : Type*) [TopologicalSpace B] [NormedAddCommGroup F]
    [NormedSpace ℝ F] : VectorBundle ℝ F (Bundle.Trivial B F) :=
  inferInstance