Cartan connections. (English) Zbl 0796.53037

Greene, Robert (ed.) et al., Differential geometry. Part 3: Riemannian geometry. Proceedings of a summer research institute, held at the University of California, Los Angeles, CA, USA, July 8-28, 1990. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 54, Part 3, 505-519 (1993).
The paper emphasizes the importance of the Cartan connections into a series of examples and applications, especially in the context of the comparison theorems for some classes of differentiable manifolds.
The author proposes a generalization of the concept of Cartan connection by considering the connection form with values not in a Lie algebra but in a vector space with skew product. Let \(P\) be a differentiable manifold of dimension \(m\) and let \([ ,\;]: \mathbb{R}^ m\times \mathbb{R}^ m\to \mathbb{R}^ m\) be a bilinear and skew symmetric map. The 1-form \(\omega: TP\to \mathbb{R}^ m\) is called a generalized Cartan connection if \(\omega(X)=0\) implies \(X= 0\); the form \(\Omega= d\omega+ [\omega,\omega]\) is said to be the generalized Cartan curvature of \(\omega\). If \(\Omega=0\) then the skew product used in the previous definitions defines a Lie algebra (the Jacobi identity is satisfied).
The author motivates the utility of this concept by discussing some recent results concerning the manifold parallelized by an \(\mathbb{R}^ m\)- values form.
For the entire collection see [Zbl 0773.00024].


53C20 Global Riemannian geometry, including pinching
53C30 Differential geometry of homogeneous manifolds