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Its fibre over a point of quaternionic projective space is the quaternionic line represented by that point. As a real vector bundle it has rank four, and its unit sphere bundle is $S^{4n+3}\to {\mathbb{HP}}^n$. Right multiplication by a chosen imaginary unit equips it with complex rank two. Under the complex inclusion into quaternionic projective space, it restricts to $L\oplus \overline{L}$, where $L$ is the complex tautological bundle. Thus its Euler class restricts to $-c_1(L^{\ast }{)}^2$.

Over Real projective space ${\mathbb{RP}}^n$, the real tautological line bundle isOn the standard chart $U_i=\{[x]:x_i\ne 0\}$, let $s_i([x])=x/x_i$. A local trivialization sends $([x],t)$ to $([x],t{s}_i([x]))$, with inverse $([x],v)\mapsto ([x],v_i)$. These maps prove that this is a real line bundle. Its unit sphere bundle is the antipodal map quotient $S^n\to {\mathbb{RP}}^n$. Its mod-two Euler class of a real line bundle is the degree-one generator of the mod-two cohomology ring of real projective space.

For a real vector bundle over a compact base, its Thom space is the quotient of its disk bundle by its sphere bundle, with the collapsed sphere bundle as basepoint. Equivalently it is the one-point compactification of the total space. An oriented bundle's Thom class and Thom isomorphism identify the reduced cohomology of this space with shifted cohomology of the base.

For an oriented rank-four real vector bundle $E\to S^4$, its sphere bundle has fibre $S^3$. Let $m=⟨e(E),[S^4]⟩$ be its Euler class evaluated on the orientation class. The Gysin sequence gives integral homology $\mathbb{Z}$ in degrees $0,7$, $\mathbb{Z}/m\mathbb{Z}$ in degree $3$, and a copy of $\mathbb{Z}$ in degree $4$ exactly when $m=0$, with all other groups zero. Here $\mathbb{Z}/0\mathbb{Z}=\mathbb{Z}$. A transition function $v\mapsto q^{k}v{q}^l$ on the unit quaternions has Euler number $k+l$, up to an overall orientation sign: evaluating it at a fixed unit vector gives the power map $q\mapsto q^{k+l}$, whose mapping degree is $k+l$. The same homology calculation follows by applying the Mayer–Vietoris sequence to the two trivializations over the hemispheres.

A tubular neighborhood of a smooth embedded submanifold is a neighborhood identified with a neighborhood of the zero section in its normal bundle. Radial deformation of punctured fibers gives the sphere bundle, and excision identifies the associated pair with a normal disk-bundle pair.

The unit tangent bundle of a Riemannian manifold $M$ is the sphere bundle of its tangent bundle. Its points are pairs $(x,v)$ with $x\in M$, $v\in T_{x}M$, and $\Vert v\Vert =1$.

For a real rank-$r$ vector bundle $E\to B$, the Thom isomorphism theorem with ${\mathbb{F}}_2$ coefficients identifies the relative cohomology sequence of its disk bundle and sphere bundle withThe multiplier is the top Stiefel–Whitney class, also called the mod-two Euler class. No orientation of $E$ is required. For the real tautological line bundle, the sphere bundle is the antipodal cover of Real projective space; this sequence proves that all powers of the degree-one generator through the dimension of the base are nonzero.