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

-sheeted cover inside the unordered Stiefel bundle. It is a trivial cover because the frame labels are global. Selecting subsets of a direct-sum framing decomposes the cover as the disjoint union of fiber products of the two summands' subset covers.

Complex Stiefel manifold ( $V_{k} (C^{n}) = U (n) / U (n - k)$ )

The space of ordered orthonormal complex

k

-frames in

C^{n}

, for

0 \leq k \leq n

2 nk - k^{2}

. Completing a frame to a unitary basis gives the displayed homogeneous-space description;

V_{0}

is a point and

V_{n} (C^{n})

U (n)

V_{k}

is exterior with smallest positive degree

2 n - 2 k + 1

, the complement bundle has Euler degree

2 (n - k)

, in which base cohomology vanishes. Its Euler class is consequently zero. The Gysin ring splitting for an odd-dimensional sphere bundle adds one exterior generator of degree

2 (n - k) - 1

. The initial base

V_{0}

is a point. This gives the full integral cohomology ring and in particular its torsion-freeness. At

k = n

\prod_{j = 0}^{n - 1} (1 + t^{2 j + 1})

n - k

inside the trivial

C^{n}

bundle. The orthogonal projector

I - \sum_{i} v_{i} v_{i}^{*}

is smooth of constant rank, giving local bundle frames. Its underlying real bundle has canonical orientation and rank

2 (n - k)

. Its sphere bundle is exactly

V_{k + 1} (C^{n})

, by appending the unit vector in the complement to the frame.

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The Stiefel manifold, denoted as $ V_k(\mathbb{R}^n) $, is a mathematical object that describes the space of orthonormal k-frames in an n-dimensional Euclidean space $\mathbb{R}^n$. More specifically, it consists of all matrices $ A \in \mathbb{R}^{n \times k} $ whose columns are orthonormal vectors in $\mathbb{R}^n$.