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Analogies for Compact Two-point Homogeneous Spaces

In: Offbeat Integral Geometry on Symmetric Spaces

Author

Listed:
  • Valery V. Volchkov

    (Donetsk National University, Department of Mathematics)

  • Vitaly V. Volchkov

    (Donetsk National University, Department of Mathematics)

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.

Suggested Citation

  • Valery V. Volchkov & Vitaly V. Volchkov, 2013. "Analogies for Compact Two-point Homogeneous Spaces," Springer Books, in: Offbeat Integral Geometry on Symmetric Spaces, edition 127, chapter 0, pages 111-134, Springer.
    • Handle: RePEc:spr:sprchp:978-3-0348-0572-8_4
    DOI: 10.1007/978-3-0348-0572-8_4
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