Parallelizable manifold

In mathematics, a differentiable manifold ${\displaystyle M}$ of dimension n is called parallelizable

$${\displaystyle \{V_{1},\ldots ,V_{n}\}}$$

on the manifold, such that at every point ${\displaystyle p}$ of ${\displaystyle M}$ the

$${\displaystyle \{V_{1}(p),\ldots ,V_{n}(p)\}}$$

at ${\displaystyle p}$. Equivalently, the

has a global section on ${\displaystyle M.}$

A particular choice of such a basis of vector fields on ${\displaystyle M}$ is called a parallelization (or an absolute parallelism) of ${\displaystyle M}$.

Examples

• An example with ${\displaystyle n=1}$ is the circle: we can take V₁ to be the unit tangent vector field, say pointing in the anti-clockwise direction. The torus of dimension ${\displaystyle n}$ is also parallelizable, as can be seen by expressing it as a cartesian product of circles. For example, take ${\displaystyle n=2,}$ and construct a torus from a square of graph paper with opposite edges glued together, to get an idea of the two tangent directions at each point. More generally, every Lie group G is parallelizable, since a basis for the tangent space at the identity element can be moved around by the action of the translation group of G on G (every translation is a diffeomorphism and therefore these translations induce linear isomorphisms between tangent spaces of points in G).
• A classical problem was to determine which of the
  normed division algebras of the real numbers, complex numbers, quaternions, and octonions, which allows one to construct a parallelism for each. Proving that other spheres are not parallelizable is more difficult, and requires algebraic topology
  .
• The product of parallelizable manifolds is parallelizable.
• Every orientable closed three-dimensional manifold is parallelizable.^[3]

Remarks

• Any parallelizable manifold is orientable.
• The term
  rigged manifold) is most usually applied to an embedded manifold with a given trivialisation of the normal bundle, and also for an abstract (that is, non-embedded) manifold with a given stable trivialisation of the tangent bundle
  .
• A related notion is the concept of a π-manifold.[4] A smooth manifold ${\displaystyle M}$ is called a π-manifold if, when embedded in a high dimensional euclidean space, its normal bundle is trivial. In particular, every parallelizable manifold is a π-manifold.

See also

• Chart (topology)
• Differentiable manifold
• Frame bundle
• Kervaire invariant
• Orthonormal frame bundle
• Principal bundle
• Connection (mathematics)
• G-structure

Notes