【摘要】 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r₀>0,当r≤r₀,密度函数<2^1/2r,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r₀>0,当r≤r₀,密度函数小于某个比较函数,当r>r₀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.更多还原

【Abstract】 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 noncompact manifolds with sectional curvature≥0 or sectional curvature bounded below diffeomorphic to Rⁿ.Precisely,the main results are:For a noncompact Riemannian manifold with non negative sectional curvature,if the density function <2^1/2r for r≤r₀ with some constant r₀>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≤r₀ and the growth of the diameter is bounded above by some more weak comparison function for r>r₀ for some constant r₀> 0,then it is diffeomorphic to an Euclidean space.更多还原