Parallelizable sphere

Among positive-dimensional spheres, precisely

S^{1}

,

S^{3}

, and

S^{7}

are parallelizable. The circle and three-sphere inherit Lie-group frames from

U (1)

and

S U (2)

; the seven-sphere has a frame constructed from octonion multiplication.

Table of contents
- Quaternionic left-invariant frame on the three-sphere Parallelizable sphere

Quaternionic left-invariant frame on the three-sphere ( $q \cdot X_{i} = 0, X_{i} \cdot X_{j} = δ_{ij}$ )

For

q = (q_{0}, q_{1}, q_{2}, q_{3}) \in S^{3}

(- q_{1}, q_{0}, q_{3}, - q_{2})

,

(- q_{2}, - q_{3}, q_{0}, q_{1})

and

(- q_{3}, q_{2}, - q_{1}, q_{0})

are tangent and orthonormal. Under SU(2) as the three-sphere they are the left-invariant vector fields obtained by translating

- i

times the Pauli matrices. This explicitly realizes the parallelization of a Lie group by left translations.

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Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 313 / 1 / Solution

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