frame bundle


Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Frame and coframe bundles



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

The frame bundle of a manifold MM is the principal bundle FTMMF T M \to M (also denoted FMMF M \to M) of frames in the tangent bundle TMT M.

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

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

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

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

ϕ h=π 1(U)U×GL n(k),z(π(z),J h(π(z))(hh 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=[(π(z),(U,h))]z = [(\pi(z), (U',h'))] with π(z)UU\pi(z)\in U'\cap U. This does not depend on the choice of the chart (U,h)(U',h') around π(z)\pi(z). There is an equivariance

J h(π(zA))(hh 1))=A 1J h(π(z))(hh 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)(U,h) and (V,g)(V,g)

J h(π(z))(hh 1))=J g(π(z))(hg 1)J h(π(z))(gh 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 ϕ h\phi_h is onto and

(ϕ h(ϕ g) 1)(p,A)=(p,AJ h(p)(gh 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 GL n(k)GL_n(k) has the standard differential structure).

In differential cohesion

In a context of differential cohesion, then the frame bundle (or higher order frame bundle) of a VV-manifold is the principal bundle (principal infinity-bundle) to which the infinitesimal disk bundle is the canonically associated bundle (associated infinity-bundle)

See at differential cohesion – Frame bundles.


The canonical differential 1-form

The frame bundle Fr(X)Fr(X) carries a canonical differential 1-form with values in n\mathbb{R}^n.

αΩ 1(Fr(X), n) \alpha \in \Omega^1(Fr(X), \mathbb{R}^n)

This is defined as follows. Let pFr(X)p \in Fr(X) be a point in the frame bundle π:Fr(X)X\pi \colon Fr(X)\to X over some point xXx \in X, hence a linear isomorphism p:T x np \colon T_x \simeq \mathbb{R}^n. For vT pFr(X)v \in T_p Fr(X) a tangent vector to the frame bundle, its projection π *vT xX\pi_\ast v \in T_x X is a tangent vector to XX. Then the value of α\alpha on vv is the image of this π *(v)\pi_\ast(v) under the isomorphism pp

α(v)p(π *(v)). \alpha(v) \coloneqq p(\pi_\ast(v)) \,.

(Sternberg 64, section VII, (2.2))

Relation to GG-structures

A choice sub-bundle of a frame bundle which is a GG-principal bundle for GGL(n)G\hookrightarrow GL(n) defines a G-structure. See there for more.


Revised on May 13, 2015 12:46:23 by Urs Schreiber (