Skip to main content
SpringerLink
Log in

Some Geometric Characterizations of the Hopf Fibrations of the Three-Sphere

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract.

The Fréchet manifold \({\cal E}/_{\!\sim}\) of all embeddings (up to orientation preserving reparametrizations) of the circle in S 3 has a canonical weak Riemannian metric. We use the characterization obtained by H. Gluck and F. Warner of the oriented great circle fibrations of S 3 to prove that among all such fibrations π:S 3→B, the manifold B consisting of the oriented fibers is totally geodesic in \({\cal E}/_{\sim }\), or has minimum volume or diameter with the induced metric, exactly when π is a Hopf fibration.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Besse A (1978) Manifolds All of Whose Geodesics are Closed. Berlin Heidelberg New York: Springer

  • E Binz (1980) ArticleTitleTwo natural metrics and their derivatives on a manifold of embeddings. Monatsh Math 89 275–288 Occurrence Handle10.1007/BF01659491 Occurrence Handle0428.58006 Occurrence Handle82h:58011

    Article  MATH  MathSciNet  Google Scholar 

  • H Gluck F Warner (1983) ArticleTitleGreat circle fibrations of the three-sphere. Duke Math J 50 107–132 Occurrence Handle84g:53056

    MathSciNet  Google Scholar 

  • G Kainz (1984) ArticleTitleA metric on the manifold of immersions and its Riemannian curvature. Monatsh Math 98 211–217 Occurrence Handle10.1007/BF01507749 Occurrence Handle0547.58016 Occurrence Handle86c:58020

    Article  MATH  MathSciNet  Google Scholar 

  • Kriegl A, Michor P (1997) The Convenient Setting for Global Analysis. Providence: Amer Math Soc

  • P Michor T Ratiu (1998) ArticleTitleOn the geometry of the Virasoro-Bott group. J Lie Theory 8 293–309 Occurrence Handle2000e:58005

    MathSciNet  Google Scholar 

  • M Salvai (2002) ArticleTitleAffine maximal torus fibrations of a compact Lie group. Int J Math 13 217–225 Occurrence Handle10.1142/S0129167X02001216 Occurrence Handle1058.53021 Occurrence Handle2003g:53031

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. FaMAF–CIEM, Córdoba, Argentina

    Marcos Salvai

Authors
  1. Marcos Salvai
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Partially supported by foncyt, Antorchas, ciem (conicet) and secyt (unc).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Salvai, M. Some Geometric Characterizations of the Hopf Fibrations of the Three-Sphere. Mh Math 147, 173–177 (2006). https://doi.org/10.1007/s00605-005-0327-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00605-005-0327-y

Search

Navigation