Abstract:

Let G be a non-compact Lie group, and ξ its Lie algebra. The structures of Lie group G and its Lie algebra ξ are carefuly studied, and the metric and Cartan decomposition of G and ξ are given. From the relation between Lie group and symmetric space, and important necesary and sufficient condition for existence of focal points of a submanifold in a non-compact symmetric space is obtained. At the same time, a computational method for the multiplicity of a focal point is also obtained.

Key words: Ｎｏｎｃｏｍｐａｃｔｓｙｍｍｅｔｒｉｃｓｐａｃｅ, Ｓｕｂｍａｎｉｆｏｌｄｓ, Ｆｏｃａｌｐｏｉｎｔ, Ｓｈａｐｅｏｐｅｒａｔｏｒ, Ａｂｅｌｉａｎｎｏｒｍａｌｂｕｎｄｌｅ