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frame bundle

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Differential geometry

differential geometry

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Frame and coframe bundles

Definition

Traditional

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

Formalization of frame bundles in differential cohesion is discussed there in the section Differential cohesion – Frame bundles

Properties

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.

References

Revised on December 22, 2014 13:41:09 by Urs Schreiber (127.0.0.1)