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  "body": "\n## Definition\n\nLet $k \\ge 0$ be an integer.\nA *sphere bundle* with fiber $S^k$ is a [[fiber bundle|CX4M8K61AB]] $\\pi \\colon E \\to B$ whose fiber is the [[$k$-sphere|T62369RFH6]] $S^k$.\n\n## Remarks\n\nMany authors also require the structure group to be the orthogonal group $O(k+1)$ acting on $S^k$, so that the transition functions are rotations and reflections.\nThe main example: let $V \\to B$ be a real vector bundle of rank $k + 1$ with a continuous inner product.\nThen the unit sphere bundle $S(V) = \\set{v \\in V : \\abs{v} = 1}$ is a sphere bundle with fiber $S^k$ and structure group $O(k+1)$.\nA sphere bundle with fiber $S^1$ is called a circle bundle.\n\nExample: the unit tangent bundle of $S^2$, the set of pairs $(x, v)$ with $x, v \\in S^2$ and $v \\perp x$, is a circle bundle over $S^2$.\nIts total space is homeomorphic to $\\R P^3$.\n\nNon-example: the Möbius band over the circle is a fiber bundle, but its fiber is a closed interval, not a sphere, so it is not a sphere bundle.\nIts boundary, however, is a sphere bundle with fiber $S^0$: it is the connected double cover of the circle.\n",
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