"Parallelizable" redirects here. For the computer science usage, see parallel algorithm.
In mathematics, a differentiable manifold of dimension n is called parallelizable[1] if there exist smooth vector fields
on the manifold, such that at every point of the tangent vectors
provide a basis of the tangent space at . Equivalently, the tangent bundle is a trivial bundle,[2] so that the associated principal bundle of linear frames has a global section on
A particular choice of such a basis of vector fields on is called a parallelization (or an absolute parallelism) of .
^Bishop, Richard L.; Goldberg, Samuel I. (1968), Tensor Analysis on Manifolds, New York: Macmillan, p. 160
^Milnor, John W.; Stasheff, James D. (1974), Characteristic Classes, Annals of Mathematics Studies, vol. 76, Princeton University Press, p. 15, ISBN 0-691-08122-0
and 26 Related for: Parallelizable manifold information
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 parallelizablemanifolds. The structures of b P n + 1 {\displaystyle bP_{n+1}} and the quotient...
parallelizablemanifold, including any compact Lie group, has Euler characteristic 0. The Euler characteristic of any closed odd-dimensional manifold...
boundary of the manifold into which it is embedded. Orientation of a vector bundle Parallelizable – A smooth manifold is parallelizable if it admits a...
Benjamin, New York-Amsterdam x+203 pp.MR0258020 Bott–Duffin inverse Parallelizablemanifold Thom's and Bott's proofs of the Lefschetz hyperplane theorem Atiyah...
vector bundle TM. Hence, the four-dimensional spacetime manifold M must be a parallelizablemanifold. The tetrad field was introduced to allow the distant...
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow...
sphere is nontrivial—i.e., S 2 n {\displaystyle S^{2n}} is not a parallelizablemanifold, and cannot admit a Lie group structure. For odd spheres, S2n−1...
the cyclic subgroup represented by homotopy spheres that bound a parallelizablemanifold, πS n is the nth stable homotopy group of spheres, and J is the...
By definition, a manifold M {\displaystyle M} is parallelizable if and only if the tangent bundle is trivial. By definition, a manifold M {\displaystyle...
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 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...
embedding (or immersion). Let ( M , g ) {\displaystyle (M,g)} be a Riemannian manifold, and S ⊂ M {\displaystyle S\subset M} a Riemannian submanifold. Define...
acts transitively on the Lie group Every Lie group is parallelizable, and hence an orientable manifold (there is a bundle isomorphism between its tangent...
{\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 GL^{+}(4,\mathbb {R} )} . A world manifold X {\displaystyle X} is said to be parallelizable if the tangent bundle T X {\displaystyle TX} and...
{\displaystyle bP_{n+1}} is the cyclic subgroup of n-spheres that bound a parallelizablemanifold of dimension n + 1 {\displaystyle n+1} , π n S {\displaystyle \pi...
classify manifolds in higher dimension (they are not a complete set of invariants): for instance, orientable 3-manifolds are parallelizable (Steenrod's...
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...
118–121. Zbl 0070.30401. Forster, Otto (1967). "Some remarks on parallelizable Stein manifolds". Bulletin of the American Mathematical Society. 73 (5): 712–716...
connection on T X {\displaystyle TX} ) is well defined only on a parallelizablemanifold X {\displaystyle X} . In field theory, one meets a problem of physical...
diffeomorphism classes of exotic (4n − 1)-spheres which bound parallelizablemanifolds involves Bernoulli numbers. Let ESn be the number of such exotic...
topological space and the action is continuous, G is a Lie group, X is a smooth manifold and the action is smooth, G is an algebraic group, X is an algebraic variety...
at each point in the manifold. This is possible globally in a continuous fashion if and only if the manifold is parallelizable. As before, frames can...
One of the basic structure theorems about Milnor fibers is they are parallelizable manifoldspg 75. Milnor fibers are special because they have the homotopy...
after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric. Killing fields are the infinitesimal...
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