Contents

28 2 Differential geometry without a metricframe: a bundle with this property is called a principal bundle. Framebundles play an essential role in the theory underlying Cartan’s methodfor testing equivalence of metrics (see Chapter 9).Given a point in p ∈ F (M) and any tangent vector v at π(p) ∈M,tangent to a curve γ(t) say, we can use the connection to define theparallelly-transported frame at neighbouring points of γ(t), i.e. to definea corresponding lifted curve in F (M). The set of all such lifted curvesdefines an n-dimensional plane at p, called horizontal, and the tangentto the lifted curve at p defines a vector in T (F M)), the horizontal lift ofv. The distribution of horizontal subspaces in fact completely defines theconnection. The basis corresponding to p can be lifted to give a uniquelydefined basis of horizontal vectors at p. One-forms on M can be lifted tothe horizontals in F (M) byπ ∗ ;atapointp, they have a uniquely-definedbasis given by lifting the basis of 1-forms dual to the basis defined by p.Now consider a general curve through p ∈ F (M) with tangent vectorV , and a frame {e a }. The change of frame along the horizontal part ofV , defined by π ∗ (V ), is given by the usual connection, while the changedue to the vertical part of V , i.e. the part tangent to the fibre, is given by(2.5) with L a b = ̂Γ b acV c , where ̂Γ b ac depends on the parametrization ofGL(n, R). The quantities ̂Γ b ac can be added to Γ b ac to define a connectionΓ a b on the bundle; the formulae for covariant derivatives given in §2.9, inparticular (2.76), can then all be extended to F (M). Moreover, given theconnection (and the structure group) the 1-form fields ω a and Γ a b area uniquely defined basis on F (M). Similar remarks apply to the variousrestricted (co)frame bundles.From this connection, one can define a curvature in F (M). Direct calculationshows (see e.g. Araujo et al. (1992)) that the non-zero componentsof the curvature of F (M) atp are just given by those of the usualcurvature at π(p) in the frame {e a } which p represents, and (2.85) stillapplies although Γ a b and Θ a b now refer to the connection and curvatureon F (M). Part of the reason is that since the action of the structuregroup on a fibre maps the horizontal subspaces to one another in a uniqueway, transport in the vertical direction has no holonomy and correspondinglycomponents of the curvature in the vertical direction are zero. Oneshould note that the curvature components are invariantly-defined scalarson F (M) since they are known for a given point in M and frame.The exterior derivative of the components of curvature on F (M) obeysdR abcd = R abcd;e ω e +R ebcd Γ e a+R aecd Γ e b+R abed Γ e c+R abce Γ e d , (2.87)where R abcd;e is evaluated, in the tetrad given by p, atπ(p) ∈M, andsimilar equations hold for higher derivatives.