O ct 2 00 5 COMPLETE PROJECTIVE CONNECTIONS

ar X iv : m at h / 05 04 08 2 v 4 [ m at h . D G ] 1 5 A pr 2 00 6 COMPLETE PROJECTIVE CONNECTIONS

The first examples of complete projective connections are uncovered: on surfaces, normal projective connections whose geodesics are all closed and embedded are complete. On manifolds of any dimension, normal projective connections induced from complete affine connections with slowly decaying positive Ricci curvature are complete.

ar X iv : 0 81 0 . 23 57 v 1 [ m at h - ph ] 1 4 O ct 2 00 8 PROJECTIVE MODULE DESCRIPTION OF EMBEDDED NONCOMMUTATIVE SPACES

Noncommutative differential geometry over the Moyal algebra is developed following an algebraic approach. It is then applied to investigate embedded noncommutative spaces. We explicitly construct the projective modules corresponding to the tangent bundles of the noncommutative spaces, and recover from this algebraic formulation the metric, Levi-Civita connection and related curvature introduced...

ar X iv : m at h / 05 10 24 8 v 1 [ m at h . O A ] 1 2 O ct 2 00 5 THE CORONA FACTORIZATION PROPERTY

The corona factorization property is a property with connections to extension theory, K-theory and the structure of C *-algebras. This paper is a short survey of the subject, together with some new results and open questions.

ar X iv : m at h / 04 10 61 1 v 1 [ m at h . A G ] 2 8 O ct 2 00 4 ON RATIONAL CUSPIDAL PROJECTIVE PLANE CURVES

Complete Projective Connections

The first examples of complete projective connections are uncovered: normal projective connections on surfaces whose geodesics are all closed and embedded are complete, as are normal projective connections induced from complete affine connections with slowly decaying positive Ricci curvature. Moreover, I explain the relationship between projective and affine connections.