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  "body": "\n## Example\n\nLet $1 \\le k \\le n$.\nThe real Stiefel manifold $V_k(\\R^n)$ is the set of orthonormal $k$-frames in $\\R^n$:\n$$V_k(\\R^n) = \\set{(v_1, \\ldots, v_k) \\in (\\R^n)^k : \\ip{v_i}{v_j} = \\delta_{ij} \\text{ for all } i, j},$$\nwith the [[subspace topology|0WS9P12FQR]] of $(\\R^n)^k$.\nIt is a compact [[smooth manifold|F45VQ47GPV]] of dimension $nk - k(k+1)/2$.\n\nCheck: $V_k(\\R^n)$ is closed and bounded in $(\\R^n)^k$, hence compact.\nIt is the level set of the map $(v_1, \\ldots, v_k) \\mapsto (\\ip{v_i}{v_j})_{i \\le j}$ to the space of symmetric $k \\times k$ matrices at the identity matrix, and this value is regular.\nThe space of symmetric $k \\times k$ matrices has dimension $k(k+1)/2$, which gives the dimension.\n\n## Remarks\n\nSpecial cases: $V_1(\\R^n) = S^{n-1}$, and $V_n(\\R^n)$ is the orthogonal group $O(n)$.\nFor $V_2(\\R^3)$ the dimension is $6 - 3 = 3$, and indeed $(v_1, v_2) \\mapsto (v_1, v_2, v_1 \\times v_2)$ identifies it with the rotation group $SO(3)$.\n\nStiefel manifolds give standard [[fiber bundles|CX4M8K61AB]].\nThe map $V_k(\\R^n) \\to S^{n-1}$, $(v_1, \\ldots, v_k) \\mapsto v_1$, is a fiber bundle with fiber $V_{k-1}(\\R^{n-1})$.\nThe map $V_k(\\R^n) \\to G_k(\\R^n)$ that sends a frame to the subspace it spans is a [[principal $O(k)$-bundle|8E37E74XVP]] over the Grassmannian.\nUsing these bundles one shows that $V_k(\\R^n)$ is $(n - k - 1)$-connected; as $n \\to \\infty$, the bundle $V_k(\\R^\\infty) \\to G_k(\\R^\\infty)$ becomes the [[universal bundle|1HDH716Z24]] for $O(k)$.\n\n## Explanations\n\n### L1 human\n\nA point of the Stiefel manifold $V_k(\\R^n)$ is a list of $k$ unit vectors in $\\R^n$ that are all perpendicular to each other.\nFor one vector, this is just a point of the unit sphere.\nFor two perpendicular unit vectors in space, the third vector of a right-handed frame is fixed by the first two.\nSo these pairs correspond to the rotations of space.\nThe question whether the sphere $S^{n-1}$ has $k$ independent tangent vector fields is the question whether the bundle $V_{k+1}(\\R^n) \\to S^{n-1}$ has a section.\n",
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