nLab
frame bundle

Definitions

Given a $k$-vector bundle $p:E\to M$ of finite rank $n$, its frame bundle (or bundle of frames in $E\to M$) is the bundle $FE\to M$ over the same base whose fiber over $x\in M$ is the set of all vector space bases of $E_x=p^{-1}(x)$. The frame bundle has a natural action of ${\mathrm{GL}}_n(k)$ given by an ordered change of basis which is free and transitive, i. e., the frame bundle is a principal ${\mathrm{GL}}_n(k)$-bundle.

The frame bundle of a manifold $M$ is the principal bundle $FTM\to M$ (also denoted $FM\to M$) of frames in the tangent bundle $TM$.

In the finite-dimensional case, the dual ${\mathrm{GL}}_n$-principal bundle $(FT{)}^{*}M$ is the coframe bundle of the manifold. This means that $F^{*}M=(FT{)}^{*}M$ is the associated bundle to $FTM{\times }_{{\mathrm{GL}}_n(k)}{\mathrm{GL}}_n(k)$ where the left action of ${\mathrm{GL}}_n(k)$ on ${\mathrm{GL}}_n(k)$ is given by right multiplication by inverses $g.h=h\cdot g^{-1}$. Also $FTM\cong (FT{)}^{*}M{\times }_{{\mathrm{GL}}_n(k)}{\mathrm{GL}}_n(k)$ using the same formula. Furthermore, the right action of ${\mathrm{GL}}_n(k)$ on this associated bundle is given by left multiplication by inverses on ${\mathrm{GL}}_n(k)$ factor.

Coframe bundle $F^{*}M$ has the following independent description. One looks at the set $𝒰(M)$ of tuples of the form $(p,(U,h))$ where $p\in U$ and $(U,h)$ is chart of the smooth structure on $M$, $U\subset M$, $h:U\to R^n$ (an atlas where $U$-s make a basis of topology suffices). ${\mathrm{GL}}_n(k)$ acts on the right on $𝒰(M)$ by

$$(p,(U,h))A:=(p,(U,A^{-1}h)).$$(p, (U, h)) A := (p, (U, A^{-1} h)).

Then $((p,(U,h))A)A\prime =(p,(U,h))(\mathrm{AA}\prime )$ holds. The total space $F^{*}M$ of the coframe bundle by the definition, as a set, consists of classes of equivalence of tuples in $𝒰(M)$ where $(p,(U,h))\sim (p\prime ,(U\prime ,h\prime ))$ iff $p=p\prime$ and the Jacobian matrix of the transition between charts at $h\prime (p)$ is the unit matrix: $J_{h\prime (p)}(h\circ (h\prime {)}^{-1})=I$. The left action of ${\mathrm{GL}}_n(k)$ is induced on the quotient. There is an obvious projection $\pi :[(p,(U,h)]\mapsto p$. To define the differential and principal bundle structure one charts $F^{*}M\to M$ with local trivializations from the neighborhoods of the form $U\times {\mathrm{GL}}_n(k)$, transfers the structure and checks that the transition functions are of the appropriate smoothness class and right ${\mathrm{GL}}_n(k)$-equivariant. The basic prescription is that to every chart $(U,h)$ one defines a map

$${\varphi }_h={\pi }^{-1}(U)\to U\times {\mathrm{GL}}_n(k),z\mapsto (\pi (z),J_{h(\pi (z))}(h\prime \circ h^{-1})),$$\phi_{h} = \pi^{-1}(U)\to U \times GL_n(k),\,\,\,\,\,\,z\mapsto (\pi(z), J_{h(\pi(z))}(h'\circ h^{-1})),

where $z=[(\pi (z),(U\prime ,h\prime ))]$ with $\pi (z)\in U\prime \cap U$. This does not depend on the choice of the chart $(U\prime ,h\prime )$ around $\pi (z)$. There is an equivariance

$$J_{h(\pi (zA))}(h\prime \circ h^{-1}))=A^{-1}{J}_{h(\pi (z))}(h\prime \circ h^{-1}))$$J_{h(\pi(z A))}(h'\circ h^{-1})) = A^{-1} J_{h(\pi(z))}(h'\circ h^{-1}))

and on intersection of $(U,h)$ and $(V,g)$

$$J_{h(\pi (z))}(h\prime \circ h^{-1}))=J_{g(\pi (z))}(h\prime \circ g^{-1})J_{h(\pi (z))}(g\circ h^{-1})$$J_{h(\pi(z))}(h'\circ h^{-1})) = J_{g(\pi(z))}(h'\circ g^{-1})J_{h(\pi(z))}(g\circ h^{-1})

Then ${\varphi }_h$ is onto and

$$({\varphi }_h\circ ({\varphi }_g{)}^{-1})(p,A)=(p,A{J}_{h(p)}(g\circ h^{-1})$$(\phi_h \circ (\phi_g)^{-1})(p,A) = (p, A J_{h(p)}(g\circ h^{-1})

what shows that the transition functions are smooth (where ${\mathrm{GL}}_n(k)$ has the standard differential structure).

References

• Wikipedia (English) frame bundle
• Shlomo Sternberg, Lectures on differential geometry, Prentice Hall 1964; Russian transl. Mir 1970