CAJ | 학술논문

借助于临界点理论和亏函数的估计,得到了非负截曲率以及截曲率有下界的完备非紧流形微分同胚于欧氏空间的一些新的条件.并证明了下面的结果:完备非紧非负截曲率Riemann流形上,若对某个常数r_0＞0,当r≤r_0,密度函数＜√2r,则该流形微分同胚于欧氏空间;完备非紧截曲率有下界的Riemann流形上,若对某个常数r_0＞0,当r≤r_0,密度函数小于某个比较函数,当r＞r_0时,直径增长小于另一无关的比较函数,则该流形微分同胚于欧氏空间.
차조우림계점이론화우함수적고계,득도료비부절곡솔이급절곡솔유하계적완비비긴류형미분동배우구씨공간적일사신적조건.병증명료하면적결과:완비비긴비부절곡솔Riemann류형상,약대모개상수r_0＞0,당r≤r_0,밀도함수＜√2r,칙해류형미분동배우구씨공간;완비비긴절곡솔유하계적Riemann류형상,약대모개상수r_0＞0,당r≤r_0,밀도함수소우모개비교함수,당r＞r_0시,직경증장소우령일무관적비교함수,칙해류형미분동배우구씨공간.
In this paper, by virtue of the critical point theory, and using the estimate of excess function, the author obtains certain new conditions to make a complete noncom-pact manifolds with sectional curvature ≥0 or sectional curvature bounded below diffeo-morphic to R~n. Precisely, the main results are: For a noncompact Riemannian manifold with non-negative sectional curvature, if the density function ≤√2r for r≤r_0 with some constant r_0＞0, then it is diffeomorphic to an Euclidean space; For a noncompact Riemannian manifold with sectional curvature bounded below, if the density function is bounded above by some comparison function for r≤r_0 and the growth of the diameter is bounded above by some more weak comparison function for r＞r_0 for some constant 0, then it is diffeomorphic to an Euclidean space.