完备流形上射线的密度函数与拓扑

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.

On the density function of rays and topology of complete manifolds

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.