Many books on differential geometry develop the geometry in the setting of principal bundles or moving frames. But when it comes the time to do riemannian geometry they leave all that nice machinery and just talk about the Riemann tensor, sectional curvature, Jacobi's equation and the first and second variations of engergy. 


So, isn't there a way to tackle the classic problems in global riemannian geometry (for example the theorems that one can see in do Carmo's book) with Cartan's methods using moving frames? 



How do I see sectional curvature in the principal bundle (or in Cartan's) approach to riemannian geometry?
Moving frames and differential forms are primarily useful for exact formal pointwise computations involving local differential invariants of a geometric structure (such as a Riemannian metric) and proving theorems that follow from such computations. An example might be the uniqueness of Riemannian metrics with constant sectional curvature. An impressive amount of differential geometry can be studied in this way, as shown in the work of Elie Cartan, S. S. Chern (see, for example, his papers generalizing Gauss-Bonnet and constructing Chern classes), and Robert Bryant.


The approach is less useful when working globally or semiglobally and when studying geometric inequalities rather than exact identities. The power of studying Jacobi fields along geodesics lies in the comparison theorems that originate in Sturm-Liouville theory. Here, formal computations are needed only to reduce the original geometric setup to a self-adjoint linear second order ODE, and the computations are best done with respect to a properly chosen orthonormal frame of vector fields (parallel along geodesics). The Cartan differential form approach is less convenient here.


The same is true when considering variational formulas for various energy integrals that are useful in differential geometry. It is possible to do the computation using differential forms and the formula for the Lie derivative, but for most of us find it more natural to work with vector fields and/or local co-ordinates when doing these computations.


The upshot, as I have already said elsewhere, is that I find it quite handy to be able to do computations using any of the different approaches and choose the one that feels most comfortable at any given moment.


