# Frames

A frame $${e_{\mu}}$$ on $${U\subset M^{n}}$$ is defined to be a tensor field of bases for the tangent spaces at each point, i.e. $${n}$$ linearly independent smooth vector fields $${e_{\mu}}$$.

The concept of frame has a particularly large number of synonyms, including comoving frame, repère mobile, vielbein, $${n}$$-frame, and $${n}$$-bein (where $${n}$$ is the dimension). The dual frame, the $${1}$$-forms $${\beta^{\mu}}$$ corresponding to a frame $${e_{\mu}}$$, is also often simply called the frame.

When using particular coordinates $${x^{\mu}}$$, the frame $${e_{\mu}=\partial/\partial x^{\mu}}$$ is called the coordinate frame (AKA coordinate basis or associated basis); any other frame is then called a non-coordinate frame. A holonomic frame is a coordinate frame in some coordinates (though perhaps not the ones being used); this condition is equivalent to requiring that $${\left[e_{\mu},e_{\nu}\right]=0}$$, a result which is sometimes called Frobenius’ theorem. An anholonomic frame is then a frame that cannot be derived from any coordinate chart in its region of definition. Using a non-coordinate frame suited to a specific problem is sometimes called the method of moving frames.

 Δ Note that the distinction between holonomic and coordinate frames as defined here is often not made.

A frame cannot usually be globally defined on a manifold. A simple way to see this is by the example of the 2-sphere $${S^{2}}$$. Any drawing of coordinate functions on a globe will have singularities, such as the north and south poles when using latitude and longitude; these are points where the associated coordinate frame will either be undefined or will vanish. In general, there is no non-zero smooth vector field that can be defined on $${S^{n}}$$ for even $${n}$$ (this is sometimes called the hedgehog theorem, AKA hairy ball theorem, coconut theorem).

The above depicts the hedgehog theorem for $${S^{2}}$$, showing that any attempt to “comb the hair of a hedgehog” yields bald spots, in this case at the poles.

A manifold that can have a global frame defined on it is called parallelizable. Some facts regarding parallelizable manifolds include:

• All parallelizable manifolds are orientable (and therefore have a volume form), but as we saw with $${S^{2}}$$ the converse is not in general true
• Any orientable 3-manifold $${M^{3}}$$ is parallelizable $${\Rightarrow}$$ any 4-manifold $${M^{3}\times\mathbb{R}}$$ is parallelizable (important in the case of the spacetime manifold)
• Of the $${n}$$-spheres, only $${S^{1}}$$, $${S^{3}}$$, and $${S^{7}}$$ are parallelizable (this can be seen to be related to $${\mathbb{C}}$$, $${\mathbb{H}}$$, and $${\mathbb{O}}$$ being the only normed finite-dimensional real division algebras beyond $${\mathbb{R}}$$)
• The torus is the only closed orientable surface with a non-zero smooth vector field