A parallelizable manifold is a smooth -manifold whose tangent bundle is trivial, equivalently one admitting smooth vector fields that form a basis of every tangent space. Every Lie group is parallelizable because a basis at the identity extends to a global frame by left translation.
Among positive-dimensional spheres, precisely , , and are parallelizable. The circle and three-sphere inherit Lie-group frames from and ; the seven-sphere has a frame constructed from octonion multiplication.
For , the vectors , and are tangent and orthonormal. Under SU(2) as the three-sphere they are the left-invariant vector fields obtained by translating times the Pauli matrices. This explicitly realizes the parallelization of a Lie group by left translations.
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A **parallelizable manifold** is a differentiable manifold that has a global frame of vector fields. This means there exists a set of smooth vector fields that span the tangent space at every point of the manifold, and these vector fields can be chosen to vary smoothly. In more formal terms, a manifold \( M \) is said to be parallelizable if there exists a smooth bundle of vector fields \( \{V_1, V_2, ...