An Integrability Condition for Simple Lie Groups

In [6] H. E. Rauch pointed out that “the symmetric manifolds, far from being isolated phenomena of a special nature, derive their structure from certain parallelism and curvature properties which, when satisfied to a certain degree of approximation, delimit a general class of Riemannian manifolds with the same structure”. In addition, Rauch observed that these curvature properties can be viewed as the integrability condition of a certain set of partial differential equations. Rauch beautifully motivated the following comparison theorem and proved it in the case where the model symmetric space is of rank one, the general manifold simply connected, and equivalence is proved up to homeomorphism. The result envisioned by Rauch was finally proved in Min-Oo, Ruh [4], where the approximate integrability condition is formulated in terms of the curvature of an appropriate Cartan connection. The following theorem states the final result for small deviations from the standard geometry.