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A G-structure on a real rank-$m$ vector bundle is a principal $G$ reduction of its frame bundle. Equivalently it has local frames whose transition matrices take values in $G$, with the reduction consisting of their right $G$-orbits. An orthogonal structure on a real vector bundle is equivalent to a fiber metric; a special orthogonal reduction is equivalent to that metric together with an orientation of a vector bundle.

An orientation of a smooth manifold is a continuous choice of orientation of a vector space in every tangent space, equivalently an orientation of a vector bundle on its tangent bundle. It can be specified by an atlas with positive Jacobian determinants on all overlaps or by a nowhere-vanishing top-degree differential form. On a connected orientable manifold there are two choices, interchanged by reversing every tangent-space orientation.

// Target: differential-geometry.bigb

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.

An orientation of a vector bundle of real rank $d$ over $R$ is a coherent choice of generator of $H^d(E_b,E_b\setminus \{0\};R)$ in every fibre. For a complex bundle of rank $m$, a complex basis gives the real basis $(v_1,i{v}_1,\dots ,v_m,i{v}_m)$. A complex change of basis has positive real determinant $|{\det }_{\mathbb{C}}A{|}^2$, so these local choices agree and give the canonical orientation of a complex vector bundle over $\mathbb{Z}$, hence over every commutative ring $R$.

Let $D(E)$ and $S(E)$ be the disk and sphere bundles of an $R$-oriented rank-$d$ bundle over compact $B$. A Thom class $u_E\in H^d(D(E),S(E);R)$ restricts to the chosen generator on every fibre. The Thom isomorphism theorem states thatThe Euler class of a vector bundle is $e(E)=s^{\ast }{u}_E\in H^d(B;R)$ for the zero section $s$. Substituting the Thom isomorphism into the long exact sequence of the pair $(D(E),S(E))$ gives the Gysin sequence of a sphere bundle

Apply this to the Hopf fibration and induct on $n$. If $x=c_1(\mathcal{O}(1))$ has degree two, the result is the cohomology ring of complex projective space

The Künneth theorem over a principal ideal domain gives a natural short exact sequence with tensor-product and Tor terms; the sequence splits, though not naturally. Here all groups are free, so the Tor term vanishes andIf a graded-ring automorphism sends $x$ to $ax+by$, then $(ax+by{)}^{n+1}=0$. The coefficients of the nonzero mixed monomials force $ab=0$; the same applies to the image of $y$. Invertibility then forces a signed permutation matrix on the basis $x,y$. Conversely, complex conjugation on either factor changes the sign of its degree-two generator, and swapping the factors exchanges $x$ and $y$. Thus the realizable group isthe group of all signed $2$ by $2$ permutation matrices, as described by the cohomology automorphisms of a product of two complex projective spaces.