Sphere bundle - Imported Knowlpedia Content

Definition

Let $E\to M$ be a positive-rank real vector bundle with bundle metric $g$ . Its unit sphere bundle is

S(E)=\{v\in E:g(v,v)=1\},

with projection obtained by restricting $E\to M$ . If $E$ has rank $r$ , then $S(E)\to M$ is a smooth fiber bundle with typical fiber $S^{r-1}$ . In a local orthonormal trivialization it is identified with $U\times S^{r-1}$ . Thus the metric selects a smooth unit sphere in each vector-space fiber, while the bundle's transition data describes how those spheres are glued.

Basic properties

The closed unit disk bundle

D(E)=\{v\in E:g(v,v)\leq 1\}

is a bundle with fiber the closed disk $D^r$ , and its fiberwise boundary is $S(E)$ . The antipodal map $v\mapsto-v$ defines a free involution on every sphere bundle.

Different bundle metrics on a fixed vector bundle yield isomorphic sphere bundles. One may construct a fiberwise positive automorphism carrying one metric to the other, so the isomorphism type depends on $E$ , not on the particular metric used to draw its unit spheres.

Examples

For the trivial bundle $M\times\mathbb R^r$ , the sphere bundle is $M\times S^{r-1}$ .

For a Riemannian manifold $M$ , the sphere bundle of the tangent bundle is the unit tangent bundle. Its points are unit tangent vectors.

If $L$ is a real line bundle, then $S(L)\to M$ has fiber $S^0=\{-1,1\}$ and is a two-sheeted covering. For the Möbius line bundle this covering is connected and nontrivial.

A Hermitian complex bundle of complex rank $n$ , viewed with its underlying real metric, has unit sphere fibers $S^{2n-1}$ .

Conventions and scope

Some authors use “sphere bundle” for any fiber bundle whose fiber is a sphere, whether or not it arises from a vector bundle. Others use it for the fiberwise one-point compactification of $E$ , whose fiber is $S^r$ , rather than for $S(E)$ , whose fiber is $S^{r-1}$ .

References

D. Husemoller, Fibre Bundles, 3rd ed., Springer, 1994. DOI record. Relevant: chapter 3, sphere and disk bundles associated to vector bundles.