Equivalent conditions for triviality of a principal bundle

Let $\pi:P\to M$ be a principal G-bundle with structure group $G$.

Theorem (TFAE: principal bundle triviality)

The following are equivalent:

1. (Bundle isomorphism with the product)
   $P$ is trivial, i.e. there exists a principal bundle isomorphism

   $$\Phi:P \stackrel{\cong}{\longrightarrow} M\times G$$

   commuting with the projections to $M$ and intertwining the right $G$-actions (see trivial principal bundle ).

2. (Existence of a global section)
   $P$ admits a smooth global section $s:M\to P$ with $\pi\circ s=\mathrm{id}_M$ (see section for the general notion of section, and compare the principal-bundle criterion global section implies triviality together with its converse trivial bundles admit sections ).

3. (Transition functions can be made trivial)
   There exists a bundle atlas for $P$ whose transition functions are all the identity element of $G$ on overlaps. Equivalently, the transition cocycle is equivalent (cohomologous) to the trivial cocycle (see transition functions and cocycle condition ).

4. (A global equivariant trivialization map)
   There exists a smooth map $f:P\to G$ such that

   $$f(pg)=g^{-1}f(p)\qquad\text{for all }p\in P,\ g\in G.$$

   (This is the condition that $f$ is equivariant for the right action on $P$ and the right-translation action on $G$; compare equivariant maps .)

The equivalence (1) $\Leftrightarrow$ (2) is the most frequently used in practice: a global section picks a preferred point in each fiber, and translating that point by $G$ produces the explicit trivialization.

Examples

1. Hopf bundle is not trivial.
   The Hopf fibration $S^3\to S^2$ is a nontrivial principal $U(1)$-bundle (see Hopf fibration ). By the theorem, it admits no global section; this is a standard instance of a nontrivial principal bundle with no global section .

2. Triviality from a global gauge choice on a trivial bundle.
   On $P=M\times G$, the map $s(x)=(x,e)$ is a global section, so the theorem recovers triviality immediately. In this case, choosing a section is exactly choosing a global “gauge,” and it identifies principal connections with $\mathfrak g$-valued $1$-forms on $M$ (compare flat connection on a trivial bundle and pure gauge connections ).

3. Principal bundles over the circle for connected groups.
   If $G$ is connected, every principal $G$-bundle over $S^1$ is trivial, hence admits a global section. This aligns with the clutching description in principal bundles over S1 via clutching , where all clutching data become equivalent to the identity when $G$ has a single connected component.