Parallelizable manifold

In mathematics, a differentiable manifold

M

{\displaystyle M}

of dimension n is called parallelizable if there exist smooth vector fields

{

V

1

,

…

,

V

n

}

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

on the manifold, such that at every point

p

{\displaystyle p}

of

M

{\displaystyle M}

the tangent vectors

{

V

1

(

p

)

,

…

,

V

n

(

p

)

}

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

provide a basis of the tangent space at

p

{\displaystyle p}

. Equivalently, the tangent bundle is a trivial bundle, so that the associated principal bundle of linear frames has a global section on

M

.

{\displaystyle M.}

A particular choice of such a basis of vector fields on

M

{\displaystyle M}

is called a parallelization (or an absolute parallelism) of

M

{\displaystyle M}

.

Examples

An example with

n

=

1

{\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

n

{\displaystyle n}

is also parallelizable, as can be seen by expressing it as a cartesian product of circles. For example, take

n

=

2

,

{\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 spheres Sn are parallelizable. The zero-dimensional case S0 is trivially parallelizable. The case S1 is the circle, which is parallelizable as has already been explained.