Hipersuperfícies conformemente euclidianas com curvatura média ou escalar constante
In this work we study conformally flat hypersurfaces f: M3 ^ Q4(c) with three distinct principal curvatures in a space form with constant sectional curvature c, under the assumption that either its mean curvature H or its scalar curvature S is constant. In case H is constant, first we extend to any...
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| Format: | doctoral thesis |
| Status: | Published version |
| Publication Date: | 2016 |
| Country: | Brasil |
| Institution: | Universidade Federal de São Carlos (UFSCAR) |
| Repository: | Repositório Institucional da UFSCAR |
| Language: | Portuguese |
| OAI Identifier: | oai:repositorio.ufscar.br:20.500.14289/8794 |
| Online Access: | https://repositorio.ufscar.br/handle/20.500.14289/8794 |
| Access Level: | Open access |
| Keyword: | Hipersuperfícies conformemente euclidianas Hipersuperfícies mínimas Curvaturas principais distintas Conformally flat hypersurfaces Minimal hypersurfaces Distinct principal curvatures Constant mean CIENCIAS EXATAS E DA TERRA::MATEMATICA CIENCIAS EXATAS E DA TERRA::MATEMATICA::GEOMETRIA E TOPOLOGIA |
| Summary: | In this work we study conformally flat hypersurfaces f: M3 ^ Q4(c) with three distinct principal curvatures in a space form with constant sectional curvature c, under the assumption that either its mean curvature H or its scalar curvature S is constant. In case H is constant, first we extend to any c G R a theorem due to Defever when c = 0 and show that there is no such hypersurface if H = 0. Our main results are for the minimal case H = 0. If c = 0, we prove that f (M3) is an open subset of a generalized cone over a Clifford torus in an umbilical hypersurface Q4(c) C Q4(c), c > 0, with c > c if c > 0. For c = 0, we show that, besides the cone over the Clifford torus in S3 C R4, there exists precisely a one-parameter family of (congruence classes of) minimal isometric immersions f: M3 ^ R4 with three distinct principal curvatures of simply-connected conformally flat Riemannian manifolds. Assuming S to be constant, we only study the case c = 0. We prove that f (M3) is an open subset of a cylinder over a surface of nonzero constant Gauss curvature in R3. |
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