Let Mn+p+q2 be a(n+p+q)-dimensional Riemannian manifold,Mn+p1(c1) be a(n+p)-dimensional submanifold with constant curvature c1 in Mn+p+q2, and Mn be a compact pseudo-umbilical submanifold with parallel mean curvature vector in Mn+p1(c1).
Furthermore it can infer that (Φ,M~2) is smooth 2-dimension immersion from M~2 into R~3 and embedding sub-manifold compact in R~3 when M~2 is 2-dimension convex manifold smooth and compact of R~3 and rank_xΦ=2 for any x∈M~2.
Compactness propertiesCn andC Pn for locally compact groupsG are introduced generalizing the finiteness propertiesFn andF Pn for discrete groups.
The propertyC1 resp.C2 is equivalent withG having a compact set of generators, resp.
In the preceding paper [AT] compactness propertiesCn andCPn for locally compact groups were introduced.
In this paper we prove that the homogeneous spaceG/K has a structure of a globally symmetric space for every choice ofG andK, especially forG being compact.
The cohomology algebra of the classifying space of a compact Lie group admits the structure of ann-Hopf algebra, wheren is the order of the Weyl group; the homology with dual structure is also ann-Hopf algebra.