Principal curvatures and topology of codimension-one isometric immersions of complete manifolds into the spaces of constant holomorphic curvatures.

我们已经成功地对具有常全纯截面曲率的双曲空间中所有采用的完全实超曲面进行了分类，需要强调的是，主曲率的不变性并不是假设的，而是一个结果。研究的背景是距离函数的临界点理论。我们的思想应用于研究完备流形的割轨迹和共轭轨迹的Rauch猜想和径向曲率假设下的Alexandrov-Toponogov比较定理，从而得到了各种模型曲面下的新型拓扑球定理。这些结果将对研究内度量空间上的凸性有一定的参考价值。因此，将看到黎曼-芬斯勒几何的全球研究。

We have succeeded in classifying all of the curvature adopted complete real hyper surfaces in the hyperbolic spaces with constant holomorphic sectional curvature.It should be emphasized that the constancy of the principal curvatures is not assumed but a consequence. The back ground of the investigation is the critical point theory of distance functions. Our idea applies to the study of the Rauch conjecture on the cut loci and conjugate loci of complete manifolds and of the Alexandrov-Toponogov comparison theorem under the radial curvature assumption.As a consequence, we have obtained new types of topological sphere theorems under various model surfaces. These results will be useful for the study of convexity on inner metric spaces. Thus, the global study of Riemannian-Finsler Geometry will be seen.