In mathematics, a differentiable manifold M {\displaystyle M} of dimension n is called parallelizable if there exist smooth vector fields { V 1 , … , V...

1 {\displaystyle bP_{n+1}} represented by n-spheres that bound parallelizable manifolds. The structures of b P n + 1 {\displaystyle bP_{n+1}} and the quotient...

In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow...

By definition, a manifold M {\displaystyle M} is parallelizable if and only if the tangent bundle is trivial. By definition, a manifold M {\displaystyle...

manifold is called parallelizable whenever it admits a parallelization. Every Lie group is a parallelizable manifold. The product of parallelizable manifolds...

parallelizable manifold, including any compact Lie group, has Euler characteristic 0. The Euler characteristic of any closed odd-dimensional manifold...

Benjamin, New York-Amsterdam x+203 pp.MR 0258020 Bott–Duffin inverse Parallelizable manifold Thom's and Bott's proofs of the Lefschetz hyperplane theorem Atiyah...

{\displaystyle X} is trivial. In particular, every Hilbert manifold is parallelizable. Every smooth Hilbert manifold can be smoothly embedded onto an open subset of...

{\displaystyle TU\cong U\times {\mathbb {R} ^{n}}} . Since not every manifold is parallelizable, a vielbein can generally only be chosen locally (i.e. only on...

vector bundle TM. Hence, the four-dimensional spacetime manifold M must be a parallelizable manifold. The tetrad field was introduced to allow the distant...

the cyclic subgroup represented by homotopy spheres that bound a parallelizable manifold, πS n is the nth stable homotopy group of spheres, and J is the...

{\displaystyle bP_{n+1}} is the cyclic subgroup of n-spheres that bound a parallelizable manifold of dimension n + 1 {\displaystyle n+1} , π n S {\displaystyle \pi...

pairs of pants along their boundary components. Parallelizable – A smooth manifold is parallelizable if it admits a smooth global frame. This is equivalent...

is non-trivial by the hairy ball theorem. In general, a manifold is said to be parallelizable if, and only if, its tangent bundle is trivial. Vector bundles...

acts transitively on the Lie group Every Lie group is parallelizable, and hence an orientable manifold (there is a bundle isomorphism between its tangent...

classify manifolds in higher dimension (they are not a complete set of invariants): for instance, orientable 3-manifolds are parallelizable (Steenrod's...

sphere is nontrivial—i.e., S 2 n {\displaystyle S^{2n}} is not a parallelizable manifold, and cannot admit a Lie group structure. For odd spheres, S2n−1...

diffeomorphism classes of exotic (4n − 1)-spheres which bound parallelizable manifolds involves Bernoulli numbers. Let ESn be the number of such exotic...

representation of the operator Δ M {\displaystyle \Delta _{M}} if the manifold is not parallelizable, i.e. if the tangent bundle is not trivial, there is no canonical...

{\displaystyle GL^{+}(4,\mathbb {R} )} . A world manifold X {\displaystyle X} is said to be parallelizable if the tangent bundle T X {\displaystyle TX} and...

mathematics, an immersion is a differentiable function between differentiable manifolds whose differential pushforward is everywhere injective. Explicitly, f :...

does give S7 one important property: parallelizability. It turns out that the only spheres that are parallelizable are S1, S3, and S7. By using a matrix...

embedding (or immersion). Let ( M , g ) {\displaystyle (M,g)} be a Riemannian manifold, and S ⊂ M {\displaystyle S\subset M} a Riemannian submanifold. Define...

anomaly. This imposes that Σ {\displaystyle \Sigma } must be a parallelizable 2d manifold, which is also a strong restriction: for example, if Σ {\displaystyle...

exist; but not when M is a 2-sphere. A manifold that does have a global moving frame is called parallelizable. Note for example how the unit directions...

its frame bundle admits a global section. In this case, the manifold is called parallelizable. If P {\displaystyle P} is a principal G {\displaystyle G}...

connection on T X {\displaystyle TX} ) is well defined only on a parallelizable manifold X {\displaystyle X} . In field theory, one meets a problem of physical...

{\displaystyle p} is a frame at x {\displaystyle x} . It follows that a manifold is parallelizable if and only if the frame bundle of M {\displaystyle M} admits...

One of the basic structure theorems about Milnor fibers is they are parallelizable manifoldspg 75. Milnor fibers are special because they have the homotopy...

118–121. Zbl 0070.30401. Forster, Otto (1967). "Some remarks on parallelizable Stein manifolds". Bulletin of the American Mathematical Society. 73 (5): 712–716...