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Existence of the H ~ 2 × R and S ~ 2 × R complete the regular Gaussian Surfaces

Author: WangDongMei
Tutor: YuZuHuan
School: Capital Normal University
Course: Basic mathematics
Keywords: Gauss curvature extrinsic curvature height function
CLC: O186.12
Type: Master's thesis
Year: 2009
Downloads: 23
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Abstract


In this paper,we mainly study the existence of complete surfaces with constant Gauss curvature in H2×R and S2×R.In section 1,we introduce some knowledge about this subject.In section 2,we prove that in H2×R there exist complete revolution surfaces with constant Gauss curvature≥-1;in S2×R there exist complete revolution surfaces with constant Gauss curvature＞1 or = 0.In section 3,we prove that there is no complete surface with constant Gauss curvature＜-1 in H2×R and S2×R.In section 4,we mainly study the existence of complete surfaces with constant Gauss curvature∈（0,1） or[-1,0) in S2×R.we prove that in S2×R there is no complete surface with constant Gauss curvature∈（0,1）;in the case of Gauss curvature∈[-1,0),there is no complete surface with constant Gauss curvature which Gauss curvature and height function’s module satisfy some relationship;there is no compact surface with constant Gauss curvature∈[-1,0);there is no complete surface with constant Gauss curvature＜(2-51/2)/4.

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CLC: > Mathematical sciences and chemical > Mathematics > Geometry, topology > Differential geometry,integral geometry > Differential Geometry > Riemannian geometry
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