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The Stiefel manifold can be equipped with a uniform measure, i.e. a Borel measure that is invariant under the action of the groups noted above. For example, V 1 ( R 2 ) {\displaystyle V_{1}(\mathbb {R} ^{2})} which is isomorphic to the unit circle in the Euclidean plane, has as its uniform measure the natural uniform measure ( arc length ) on ...
Any two knots in 3-space are equivalent by regular homotopy, though not by isotopy. This curve has total curvature 6π, and turning number 3.. The Whitney–Graustein theorem classifies the regular homotopy classes of a circle into the plane; two immersions are regularly homotopic if and only if they have the same turning number – equivalently, total curvature; equivalently, if and only if ...
The set of k-frames (particularly the set of orthonormal k-frames) in a given vector space X is known as the Stiefel manifold, and denoted V k (X). A k-frame defines a parallelotope (a generalized parallelepiped); the volume can be computed via the Gram determinant.
In the special case in which E is the tangent bundle TM over the base manifold M, if a spin structure exists then one says that M is a spin manifold. Equivalently M is spin if the SO( n ) principal bundle of orthonormal bases of the tangent fibers of M is a Z 2 quotient of a principal spin bundle.
The Stiefel–Whitney numbers of the tangent bundle of a smooth manifold are called the Stiefel–Whitney numbers of the manifold. They are known to be cobordism invariants. It was proven by Lev Pontryagin that if B is a smooth compact ( n +1)–dimensional manifold with boundary equal to M , then the Stiefel-Whitney numbers of M are all zero ...
Eduard Stiefel (right) together with Otto Volk [] in Oberwolfach, 1978. Eduard L. Stiefel (21 April 1909 – 25 November 1978) was a Swiss mathematician.Together with Cornelius Lanczos and Magnus Hestenes, he invented the conjugate gradient method, and gave what is now understood to be a partial construction of the Stiefel–Whitney classes of a real vector bundle, thus co-founding the study ...
A repair kit or service kit is a set of items used to repair a device, commonly comprising both tools and spare parts. Many kits are designed for vehicles, such as cars, boats, airplanes, motorbikes, and bicycles, and may be kept with the vehicle in order to make on-the-spot repairs.
Stephen Smale (born July 15, 1930) is an American mathematician, known for his research in topology, dynamical systems and mathematical economics.He was awarded the Fields Medal in 1966 [2] and spent more than three decades on the mathematics faculty of the University of California, Berkeley (1960–1961 and 1964–1995), where he currently is Professor Emeritus, with research interests in ...
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