In the original text, you define a Riemannian metric to be a nonvanishing symmetric 2-form. This is not right. The symmetric 2-form must be positive definite and not just nonvanishing.

]]>If one were to look at Lorentzian metrics rather than Riemannian, then even though there is symmetry, there is still a topological obstruction, as Aaron and others have already pointed out.

Things which are positive, such as Riemannian metrics, positive measures, and norms, are not as manifestly geometric as, say, differential forms, because they are now also encoding some analysis (magnitudes, lengths, bounds, inequalities, etc.) as well as geometry.

]]>In terms of reducing the structure group of , the splitting of corresponds to a reduction of the structure group to a structure group, which is only possible if we can find a global section of . You can see the space of p/q splittings floating around in the Grassmannian fibers of this bundle.

]]>That is, what obstruction (if any) is there to the existence of a pseudo-Riemannian metric of a given signature?

There are obstructions (as Aaron pointed out). The existence of such a form is equivalent to the -bundle associated to the tangent bundle having sections. In the case where , this bundle is the projectivization of the tangent bundle (hence the theorem Aaron mentioned). In general, it’s just going to get nastier. You’re going to get some relations on characteristic classes (coming from the kernel of the pullback map on cohomology ), but I’m not sure if those will be sufficient.

]]>One can obviously identify with the set of inner products on . Let be your favorite inner product, and consider the map on the set of inner products times sending . This is the identity at 1, and the map to a point at 0, and an inner product at every point (since the sum of inner products is an inner product, which is also what makes the proof using partitions of unity go).

]]>It’s a more general theorem that all reductive algebraic groups over the reals contract onto their maximal compact subgroups. The magic words are Iwasawa decomposition, which shows that every such group is the product (topologically) of the maximal compact, and a unipotent subgroup, which is necessarily contractible.

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