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The unit tangent bundle of a Riemannian manifold $M$ is the <sphere bundle> of its <tangent bundle>. Its points are pairs $(x,v)$ with $x\in M$, $v\in T_xM$, and $\lVert v\rVert=1$.

Unit tangent bundle

Stiefel manifold

Cohomology of the unit tangent bundle of an even-dimensional sphere

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 114 / 5 / Solution

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The unit tangent bundle is a fundamental concept in differential geometry and is used in the study of manifolds, particularly in the context of differential geometry and geodesic flows. Given a smooth manifold \$ M \$, the unit tangent bundle, denoted as \$ U(TM) \$, consists of all unit tangent vectors at every point in \$ M \$.


Unit tangent bundle ( $STM$ )

Stiefel manifold ( $V_{k} (R^{n})$ )

Cohomology of the unit tangent bundle of an even-dimensional sphere ( $H^{*} (ST S^{2 n}; Z)$ )

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Unit tangent bundle (STM)

Stiefel manifold (Vk​(Rn))

Cohomology of the unit tangent bundle of an even-dimensional sphere (H∗(STS2n;Z))