On the geometry and trigonometry of homogeneous 3-manifolds with 4-dimensional isometry group

• DOI: 10.1007/s00209-005-0883-9
• Publication Date: 2006-07-24T08:44:03Z
• Authors: Sven Engel
• Keywords: 0101 mathematics; 01 natural sciences

In this article we study the geometry of the family of simply connected homogeneous 3-manifolds (M, g K,τ ) given as a principal bundle over a 2-manifold of constant curvature such that the curvature form is constant. We give explicit results for the conjugate radius, normal Jacobi fields and the cut locus on (M, g K,τ ). Moreover, we determine the trigonometry on (M, g K,τ ) by a complete set of trigonometric laws.