Closed Lie Subgroup Research Articles

Stability of closed Lie subgroups in compact Lie groups

Stability of closed Lie subgroups in compact Lie groups

Classifications of bundle connection pairs by parallel translation and lassos

Let M be a connected manifold and G be a closed Lie subgroup of GL( V)—the general linear group on a finite dimensional vector space V. Denote by ( Ω m ) the space of H 1-loops in M starting at a fixed point ( m). Let M be the set of P ϵ C ∞(Ω m, G) , modulo conjugation by an element of G, such that P( στ) = P( σ) P( τ) for σ, τ ϵ Ω m (στ is the concatenation of the loops σ and τ), P( σ′) = P( σ) if σ′ is reparameterization of σ, and the differential of P satisfies a “locality” condition. It is shown that the bundle connection pairs ( E, ▽) (up to equivalence), with structure group G and fiber V, are in one to one correspondence with M —a similar result has been announced by Kobayashi. The correspondence is induced by the parallel translation operators of connections. Furthermore, if the manifold ( M) is simply connected, then the space of bundle connection pairs can be classified by a collection of Lie algebra valued 1-forms on the manifold Ω m (called integrated lassos). These 1-forms are related to the differentials of elements of M . This last result generalizes Weil's characterization of U(1)—line bundle connection pairs by the curvature 2-form. It is also a generalization of Gross' results to base manifolds ( M) other than R n .

Homologies and elations in compact, connected projective planes

In a compact, connected topological projective plane, let Ω be a closed Lie subgroup of the group of all axial collineations with a fixed axis A. We compare the set З\\ A consisting of the centres of all non-identical homologies in Ω to orbits of the group Ω [A] of all elations contained in Ω and of its connected component θ = (Ω [A]) 1 . It is shown that З\\ A is the union of at most countably many θ-orbits; moreover, З\\ A turns out to be a single θ-orbit whenever the connected component of Ω contains non-identical homologies. This result is analogous to a well-known theorem of André for finite planes. It has numerous consequences for the structure of collineation groups of compact, connected projective planes.