On continuous families of geometric Seifert conemanifold structures

In this paper, dedicated to Prof. Lou Kauffman, we determine the Thurston’s geometry possesed by any Seifert fibered conemanifold structure in a Seifert manifold with orbit space (Formula presented.) and no more than three exceptional fibers, whose singular set, composed by fibers, has at most three components which can include exceptional or general fibers (the total number of exceptional and singular fibers is less than or equal to three). We also give the method to obtain the holonomy of that structure. We apply these results to three families of Seifert manifolds, namely, spherical, Nil manifolds and manifolds obtained by Dehn surgery on a torus knot (Formula presented.). As a consequence we generalize to all torus knots the results obtained in [Geometric conemanifolds structures on (Formula presented.), the result of (Formula presented.) surgery in the left-handed trefoil knot (Formula presented.), J. Knot Theory Ramifications 24(12) (2015), Article ID: 1550057, 38pp., doi: 10.1142/S0218216515500571] for the case of the left handle trefoil knot. We associate a plot to each torus knot for the different geometries, in the spirit of Thurston.

Homotheties and topology of tangent sphere bundles

We prove a Theorem on homotheties between two given tangent sphere bundles SrM of a Riemannian manifold (M,g) of dim ≥ 3, assuming different variable radius functions r and weighted Sasaki metrics induced by the conformal class of g. New examples are shown of manifolds with constant positive or with constant negative scalar curvature which are not Einstein. Recalling results on the associated almost complex structure I^G and symplectic structure ω^G on the manifold TM , generalizing the well-known structure of Sasaki by admitting weights and connections with torsion, we compute the Chern and the Stiefel-Whitney characteristic classes of the manifolds TM and SrM.

Riemannian questions with a fundamental differential system

Introduzimos o leitor ao estudo de um sistema diferencial exterior fundamental, descoberto anteriormente pelo autor, que se pode sempre associar a qualquer dada variedade riemanniana M de dimensão n+1. Depois de recordarmos a geometria do fibrado de esferas tangente SM--->M com a métrica de Sasaki, apresentamos o sistema de formas diferencias de grau n que complementa a conhecida estrutura de contacto de SM. A partir daí vemos como o sistema diferencial se aplica ao estudo de problemas métricos em hipersuperfícies de M, bem como a outros que são próprios de SM, e as diversas questões que se podem colocar neste novo contexto.