2010/11/25 by José Ricardo Arteaga B., B., José Ricardo Arteaga, M. Malakhaltsev +2
Mathematics · Medicine · Physics and Astronomy · #53B20 #53B21 #53C17 #Advanced Differential Geometry Research #Advanced Neuroimaging Techniques and Applications #Differential Geometry (math.DG) #FOS: Mathematics #Ophthalmology and Eye Disorders #math.DG #msc:53B20 #msc:53B21 #msc:53C17
paper · pdf · doi:10.48550/arxiv.1011.5683
19 pages, 4 figures
arxiv created 2010/11/25 · openalex publication_date 2010/11/25 · arxiv updated 2010/11/29 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28
In this paper we construct a functor from the category of two-dimensional Riemannian manifolds to the category of three-dimensional manifolds with generalized metric tensors. For each two-dimensional oriented Riemannian manifold (M,g) we construct a metric tensor g (in general, with singularities) on the total space SO(M,g) of the principal bundle of the positively oriented orthonormal frames on M. We call the metric g the Wagner lift of g. We study the relation between the isometry groups of (M,g) and (SO(M,g), g). We prove that the projections of the geodesics of (SO(M,g), g) onto M are the curves which satisfy the equation ∇(dγ)/(dt)(dγ)/(dt) = C K J (γ) - C2 K grad K, where K is the curvature of (M,g), J is the operator of the complex structure associated with g, and C is a constant. We find the properties of the solutions of this equation, in particular, for the case when (M,g) is a surface of revolution.