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Stiefel manifold

In mathematics, the Stiefel manifold ${\\displaystyle V_{k}(\\mathbb {R} ^{n})}$ is the set of all orthonormal k-frames in ${\\displaystyle \\mathbb {R} ^{n}.}$ That is, it is the set of ordered orthonormal k-tuples of vectors in ${\\displaystyle \\mathbb {R} ^{n}.}$ It is named after Swiss mathematician Eduard Stiefel. Likewise one can define the complex Stiefel manifold ${\\displaystyle V_{k}(\\mathbb {C} ^{n})}$ of orthonormal k-frames in ${\\displaystyle \\mathbb {C} ^{n}}$ and the quaternionic Stiefel manifold ${\\displaystyle V_{k}(\\mathbb {H} ^{n})}$ of orthonormal k-frames in ${\\displaystyle \\mathbb {H} ^{n}}$ . More generally, the construction applies to any real, complex, or quaternionic inner product space.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Stiefel_manifold)