Abstract Let M^2 be an umbilic-free surface in the unit sphere S^3. Four basic invariants of M^2 under the Moebius transformation group of S^3 are Moebius metric g, Blaschke tensor A, Moebius second fundamental form B...Abstract Let M^2 be an umbilic-free surface in the unit sphere S^3. Four basic invariants of M^2 under the Moebius transformation group of S^3 are Moebius metric g, Blaschke tensor A, Moebius second fundamental form B and Moebius form φ. We call the Blaschke tensor is isotropic if there exists a smooth function λ such that A = λg. In this paper, We classify all surfaces with isotropic Blaschke tensor in S^3.展开更多
Let x : M^n-1→ R^n be an umbilical free hypersurface with non-zero principal curvatures. Two basic invariants of M under the Laguerre transformation group of Rn are Laguerre form C and Laguerre tensor L. In this pap...Let x : M^n-1→ R^n be an umbilical free hypersurface with non-zero principal curvatures. Two basic invariants of M under the Laguerre transformation group of Rn are Laguerre form C and Laguerre tensor L. In this paper, we prove the following theorem: Let M be an (n-1)-dimensional (n 〉 3) complete hypersurface with vanishing Laguerre form and with constant Laguerre scalar curvature R in R^n, denote the trace-free Laguerre tensor by L = L -1/n-1tr(L)· Id. If supM ||L||=0,then M is Laguerre equivalent to a Laguerre isotropic hypersurface; and if supM ||L^-||≡√(n-1)(n-2)R/ (n-1)(n-2)(n-3) , M isLaguerre equivalent to the hypersurface x^- : H^1× S^n-2 → R^n.展开更多
文摘Abstract Let M^2 be an umbilic-free surface in the unit sphere S^3. Four basic invariants of M^2 under the Moebius transformation group of S^3 are Moebius metric g, Blaschke tensor A, Moebius second fundamental form B and Moebius form φ. We call the Blaschke tensor is isotropic if there exists a smooth function λ such that A = λg. In this paper, We classify all surfaces with isotropic Blaschke tensor in S^3.
基金supported by Scienticfic Research Fund of Yunnan Provincial Education Department(Grant No.2014Y445)supported by Scienticfic Research Fund of Yunnan Provincial Education Department(Grant No.2015Y101)Yunnan Applied Basic Research Young Project
文摘Let x : M^n-1→ R^n be an umbilical free hypersurface with non-zero principal curvatures. Two basic invariants of M under the Laguerre transformation group of Rn are Laguerre form C and Laguerre tensor L. In this paper, we prove the following theorem: Let M be an (n-1)-dimensional (n 〉 3) complete hypersurface with vanishing Laguerre form and with constant Laguerre scalar curvature R in R^n, denote the trace-free Laguerre tensor by L = L -1/n-1tr(L)· Id. If supM ||L||=0,then M is Laguerre equivalent to a Laguerre isotropic hypersurface; and if supM ||L^-||≡√(n-1)(n-2)R/ (n-1)(n-2)(n-3) , M isLaguerre equivalent to the hypersurface x^- : H^1× S^n-2 → R^n.