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New gap results on the 4†dimensional sphere

Author

Listed:
  • Ezequiel Barbosa
  • Allan Freitas
  • Antonio Rosivaldo Gonçalves

Abstract

A result showed by M. Gursky in ensures that any metric g on the 4†dimensional sphere S4 satisfying Ricg=3g and injg(S4)≥π34 is isometric to the round metric. In this note, we prove that there exists a universal number i0 such that any metric g on the 4†dimensional sphere S4 satisfying Ricg=3g and injg(S4)≥π34−i0 is isometric to the round metric. Moreover, there exists a universal ε0>0 such that any metric g on the 4†dimensional sphere S4 with nonnegative sectional curvature, Ricg=3g and 89π2−ε0≤Volg(S4) is isometric to the round metric. This last result slightly improves a rigidity theorem also proved in .

Suggested Citation

  • Ezequiel Barbosa & Allan Freitas & Antonio Rosivaldo Gonçalves, 2017. "New gap results on the 4†dimensional sphere," Mathematische Nachrichten, Wiley Blackwell, vol. 290(17-18), pages 2755-2758, December.
    • Handle: RePEc:bla:mathna:v:290:y:2017:i:17-18:p:2755-2758
    DOI: 10.1002/mana.201600219
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