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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?

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Note that the sectional curvature corresponding to two tangent vectors at a point depends only on the plane they span, and the curvature tensor is determined by the sectional curvatures. So local frames don't really distinguish Riemannian curvature among curvature tensors associated to other connections. Perhaps a more "global" approach to Riemannian geometry in this sense would involve the geometry of 2-Grassmannian bundles. – Paul Siegel Mar 29 '11 at 10:56
Thanks Paul. It seems intriguing that idea of considering 2-Grassmannian bundles. – Feri Mar 30 '11 at 2:41
up vote 5 down vote accepted

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.

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Thanks Deane! That's close to what I also thought, though I still hope to see a comparison theorem in riemannian geometry proved with exterior calculus and exploiting the structure of some Lie group :) – Feri Mar 30 '11 at 2:40

I would say you do the same: Even if the notations are different, you have the same notions (like Levi-Civita connection, Riemannian curvature,..). The equation you obtain in these two different approaches look different (for example for an geodesic) but they have the same geometric meaning. Therefore, you get the same proofs, even if they look different. In the end, you have two different languages, but you can translate (easily).

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Sigurdur Helgason in his book "Differential Geometry,Lie Groups and Symmetric Spaces" proves various results like geodesic is locally length minimizing,compact group of isometries acting on a manifold of nonpositive curvature have a fixed point and many more without using any variational techniques involving jacobi fields but rather cartan structural equations.

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