Codex @codex  0

Take a locally finite trivializing cover $(U_{\alpha })$ of the real vector bundle $E$, and a subordinate smooth partition of unity $({\rho }_{\alpha })$. Each trivialization identifies its fibers with ${\mathbb{R}}^m$ and hence supplies a smooth Euclidean inner product $g_{\alpha }$ there. DefineEach weighted term extends by zero outside its chart, since the weight has support within that chart. Local finiteness makes the sum smooth in every bundle chart. At each $b$ at least one weight is positive; for $v\ne 0$ every nonzero contribution to $g_b(v,v)$ is positive. Thus $g$ is a smooth positive-definite fiber metric, which is precisely an inner product varying smoothly with the fibers.

A G-structure on a vector bundle is a reduction of its frame bundle from $GL(m,\mathbb{R})$ to $G$: a principal $G$-subbundle $P\subseteq \mathrm{Fr}(E)$ whose extension of structure group is the full frame bundle. Equivalently, it is a choice of smooth local frames with transition matrices in $G$, and the reduction consists of the right $G$-orbits of these frames. The adapted frames are part of the structure, not merely a statement that an individual fiber has a basis.

Given a fiber metric, the Gram-Schmidt process transforms any smooth local frame into Euclidean orthonormal frames. At each step the squared norm to be divided out is positive, so the divisions and positive square roots depend smoothly on the base point. Two such frames differ by an orthogonal matrix. All the orthonormal frames therefore form a principal $O(m)$ reduction, an orthogonal structure on a real vector bundle.

Conversely, let $P$ be an $O(m)$ reduction. A frame $u:{\mathbb{R}}^m\to E_b$ in $P_b$ definesReplacing $u$ by $uq$ for $q\in O(m)$ leaves this rule unchanged, since $q$ preserves the Euclidean inner product. It is consequently independent of the adapted frame and defines a positive-definite fiber metric. Its coefficients are smooth in local adapted frames. These constructions are mutually inverse. An $O(m)$-structure is equivalent to a smooth fiberwise inner product.

For $SO(m)$, the additional condition is an orientation of a vector bundle. Such a reduction supplies consistently oriented orthonormal frames, since its transition determinants are $+1$. Conversely an orientation together with a fiber metric selects the oriented Euclidean orthonormal frames and gives an $SO(m)$ reduction. ThusIt is not true for every real bundle. The Möbius line bundle is obtained from $[0,2\pi ]\times \mathbb{R}$ by $(2\pi ,v)\sim (0,-v)$. If it had an orientation, hence a nowhere-zero continuous section, its coefficient on the interval would satisfy $f(2\pi )=-f(0)$ while staying nonzero. The intermediate value theorem makes that impossible. Since $SO(1)=\{1\}$, it has no $SO(1)$ reduction. For each $m\ge 1$, its direct sum with a trivial rank-$(m-1)$ bundle has transition matrix $\mathrm{diag}(-1,1,\dots ,1)$ and determinant line equal to the Möbius line. It is also nonorientable and has no $SO(m)$-structure. This obstruction concerns the bundle's orientation, independently of the existence of a metric.

A real rank-$d$ vector bundle $p:E\to M$ is a continuous family of $d$-dimensional real vector spaces, locally trivial by homeomorphisms $p^{-1}(U)\cong U\times {\mathbb{R}}^d$ which are linear on every fiber. Transition matrices are continuous maps into ${\mathrm{GL}}_d(\mathbb{R})$. The Grassmannian ${\mathrm{Gr}}_k({\mathbb{R}}^n)$ is the space of $k$-dimensional linear subspaces; its topology can be defined by identifying each plane $W$ with its orthogonal projection matrix $P_W$. Local graph charts identify nearby planes with graphs of maps $W_0\to W_0^{\perp }$.

The tautological bundle is $E_{\mathrm{taut}}=\{(W,v):v\in W\}\to {\mathrm{Gr}}_k({\mathbb{R}}^n)$. In graph charts a basis of $W_0$ gives a continuously varying basis of each graph, proving local triviality. The universal quotient bundle has fiber $Q_W={\mathbb{R}}^n/W$. The map $[v]\mapsto (I-P_W)v$ identifies it continuously with the orthogonal-complement bundle $W^{\perp }$, whose graph charts similarly supply local frames. Thus $Q_{\mathrm{taut}}$ is a rank-$(n-k)$ quotient vector bundle.

For the given $E\to M$, at each $m$ choose a basis of $E_m$ and global sections of a vector bundle realizing its basis vectors, using the evaluation-surjectivity assumption. Those $d$ sections remain independent on a neighbourhood of $m$, since the corresponding determinant in a local trivialization is nonzero at $m$ and continuous. Compactness gives a finite cover by such neighbourhoods. Collect all the selected sections as $s_1,\dots ,s_N$; they span every fiber. They define a fiberwise surjective vector bundle morphismIts kernels have constant dimension $N-d$ and form the kernel bundle of a surjective vector bundle morphism. More concretely, in a local frame of $E$ the map has a continuous full-row-rank matrix $A(m)$ and its kernel projection isThis varies continuously and is unchanged by an invertible change of target frame. The Grassmannian as projection matrices therefore makesa continuous map. The fiber quotient map induced by $F_m$ is an isomorphism ${\mathbb{R}}^N/\ker F_m\to E_m$. The same local matrix formula gives continuously varying right inverses, so these fiber isomorphisms give a vector bundle isomorphism. This proves the classification of vector bundles by a universal quotient bundle:The displayed equality in the classification statement is understood as bundle isomorphism, not literal equality of total spaces.

Finally, the universal quotient bundle need not have a nowhere-zero section. Take ${\mathrm{Gr}}_1({\mathbb{R}}^2)={\mathbb{RP}}^1$. Its quotient identifies with $W^{\perp }$, a Möbius line bundle. Write $W_{\theta }=\mathrm{span}(\cos \theta ,\sin \theta )$, with $\theta$ modulo $\pi$. A section of the complement line has the form $a(\theta )(-\sin \theta ,\cos \theta )$, and being single-valued requires $a(\theta +\pi )=-a(\theta )$. If $a$ is nonzero anywhere, the intermediate value theorem forces a zero between that point and its translate by $\pi$; if it is nowhere nonzero the conclusion is immediate. Hence this rank-one quotient bundle has no nowhere-zero section.

Take the Möbius line bundle over $S^1$,A continuous section of a vector bundle corresponds to a continuous real-valued function $s:[0,1]\to \mathbb{R}$ with $s(1)=-s(0)$. If $s(0)=0$, it already vanishes. Otherwise its endpoint values have opposite signs, so the intermediate value theorem gives $s(t_0)=0$ for some $t_0\in (0,1)$. Thus this vector bundle has no nowhere-zero section. Every continuous section must vanish.

Under the standard smooth manifold hypotheses, the first answer is yes, although the isomorphism requires a choice. A partition of unity combines Euclidean inner products in manifold charts to produce a Riemannian metric $g$. Nonnegative local weights summing to one preserve positivity, and local finiteness preserves smoothness. The musical isomorphism isIt is smooth and fiberwise invertible, with inverse having local matrix $(g_{ij}{)}^{-1}$. It is therefore a vector bundle isomorphism. There is no preferred such map before a metric or equivalent additional structure is chosen.

The second answer is no. On $S^1=\mathbb{R}/2\pi \mathbb{Z}$, compare the trivial vector bundle $S^1\times \mathbb{R}$ with the Möbius line bundleBoth have rank one. A continuous section of a vector bundle of $L$ is represented on $[0,2\pi ]$ by a continuous function $s$ satisfying $s(2\pi )=-s(0)$. If it were nowhere zero, the intermediate value theorem would prevent its endpoint signs from being opposite. Thus $L$ has no nowhere-zero section. The trivial bundle has the section $1$, and a bundle isomorphism would carry that section to a nowhere-zero section of $L$, a contradiction. Hence

Over ${\mathrm{Gr}}_1({\mathbb{R}}^2)={\mathbb{RP}}^1$, the universal quotient is the orthogonal-complement line bundle, a Möbius line bundle. Writing a section as $a(\theta )(-\sin \theta ,\cos \theta )$ forces $a(\theta +\pi )=-a(\theta )$. Continuity then forces a zero, so no nowhere-zero section exists.