Analogies for Compact Two-point Homogeneous Spaces

Abstract

The operators $\mathfrak{U}_\delta$ which we studied in Chapter 3 have analogues in the compact case. In this chapter we study their properties for compact two-point homogeneous spaces of dimension > 1. These are the Riemannian manifolds M with the property that for any two pairs of points $(p_1,\,p_2)\,{\rm and}\,(q_1,q_2)\,{\rm satisfying}\, d((p_1,\,p_2)\,=\,(q_1,q_2)\,{\rm where}\, d{\rm \,is\, the \,distance \,on}\, M$, there exists an isometry mapping $(p_1\,to\,p_2)\,{\rm and}\,(p_1\,to\,p_2).$ By virtue of Wang’s classification (see Helgason [H5, Chapter 1, § 4]) these are also the compact symmetric spaces of rank one. Unlike the non-compact case, the treatment in this chapter is based on the realizations of the spaces under consideration. Accordingly, the use of Lie theory is minimal.