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Geometry and topology of maximal antipodal sets and related topics

Bang-Yen Chen

Abstract

Maximal antipodal sets of Riemannian manifolds were introduced by the author and T. Nagano in [Un invariant géométrique riemannien, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 5, 389--391]. Since then maximal antipodal sets have been studied by many mathematicians and they shown that maximal antipodal sets are related to several important areas in mathematics. The main purpose of this paper is thus to present a comprehensive survey on geometry and topology of maximal antipodal sets and also on their applications to several related topics.

Geometry and topology of maximal antipodal sets and related topics

Abstract

Maximal antipodal sets of Riemannian manifolds were introduced by the author and T. Nagano in [Un invariant géométrique riemannien, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 5, 389--391]. Since then maximal antipodal sets have been studied by many mathematicians and they shown that maximal antipodal sets are related to several important areas in mathematics. The main purpose of this paper is thus to present a comprehensive survey on geometry and topology of maximal antipodal sets and also on their applications to several related topics.
Paper Structure (43 sections, 56 theorems, 26 equations)

This paper contains 43 sections, 56 theorems, 26 equations.

Table of Contents

  1. Maximal antipodal sets and two-numbers
  2. $(M_{+},M_{-})$-theory for compact symmetric spaces
  3. E. Cartan's classification of irreducible compact symmetric spaces
  4. $(M_{+},M_{-})$-theory
  5. Descriptions of great antipodal sets
  6. Great antipodal sets of classical Lie groups
  7. Great antipodal sets of exceptional Lie groups and exceptional symmetric spaces
  8. Great antipodal sets of symmetric spaces of compact type
  9. Expansion of antipodal sets and homogeneous antipodal sets
  10. Links between two-numbers and topology
  11. Two-numbers and Euler numbers
  12. Two-numbers and covering maps
  13. A link between two-number and projective rank in algebraic geometry
  14. Holomorphic two-number of compact hermitian symmetric space
  15. Symmetric $R$-spaces
  16. ...and 28 more sections

Key Result

Proposition 2.1

Let $M=G/K$ be a compact symmetric space. Then, for each antipodal point $p$ of $o\in M$, the isotropy subgroup $K$ at $o$ acts transitively on the polar $M_+(p)$. Further, we have $K(p)=M_+(p)$ and $K(p)$ is connected. Hence, $M_+(p)=K/K_p$, where $K_p=\{k\in K: k(p)=p\}$.

Theorems & Definitions (69)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Remark 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Remark 2.8
  • Remark 3.1
  • Theorem 4.1
  • ...and 59 more