Skip to main content
Log in

The classification and curvature of biquotients of the form \(Sp(3)/\!\!/Sp(1)^2\)

  • Published:
Annals of Global Analysis and Geometry Aims and scope Submit manuscript

Abstract

We show there are precisely \(15\) inhomogeneous biquotients of the form \(Sp(3)/\!\!/Sp(1)^2\) and show that at least \(8\) of them admit metrics of quasi-positive curvature.

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

Access this article

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

  1. Atiyah, M.F., Hirzebruch, F.: Riemann–Roch theorems for differentiable manifolds. Bull. Am. Math. Soc. 65, 276–281 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  2. Berger, M.: Les variétés Riemanniennes homogénes normales simplement connexes á courbure strictement positive. Ann. Scuola Norm. Sup. Pisa 15, 179–246 (1961)

    MATH  MathSciNet  Google Scholar 

  3. Borel, A., Hirzebruch, F.: Characteristic classes and homogeneous spaces i. Am. J. Math. 80, 458–538 (1958)

    Article  MathSciNet  Google Scholar 

  4. Cheeger, J.: Some example of manifolds of nonnegative curvature. J. Differ. Geo. 8, 623–625 (1973)

    MATH  MathSciNet  Google Scholar 

  5. DeVito, J.: The classification of simply connected biquotients of dimension 7 or less and 3 new examples of almost positively curved manifolds. University of Pennsylvania, Thesis (2011)

  6. Eschenburg, J.: Freie isometrische aktionen auf kompakten Lie-gruppen mit positiv gekr\(\ddot{\text{ u }}\)mmten orbitr\(\ddot{\text{ a }}\)umen. Schriften der Math. Universit\(\ddot{\text{ a }}\)t M\(\ddot{\text{ u }}\)nster 32 (1984)

  7. Eschenburg, J.: Cohomology of biquotients. Manuscripta Math. 75, 151–166 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  8. Eschenburg, J., Kerin, M.: Almost positive curvature on the gromoll-meyer 7-sphere. Proc. Am. Math. Soc. 136, 3263–3270 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  9. Fulton, W., Harris, J.: Representation Theory. A First Course. Springer, Berlin (2004)

    Google Scholar 

  10. Gromoll, D., Meyer, W.: An exotic sphere with nonnegative sectional curvature. Ann. Math. 100, 401–406 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  11. Kerin, M.: Some new examples with almost positive curvature. Geom. Topol. 15, 217–260 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  12. Kerin, M.: On the curvature of biquotients. Math. Ann. 352, 155–178 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  13. Kerr, M., Tapp, K.: A note on quasi-positive curvature conditions. Differ. Geom. Appl. 34, 63–79 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  14. Mal’cev, : On semisimple subgroups of Lie groups. Am. Math. Soc. Transl. 1, 172–273 (1950)

    Google Scholar 

  15. O’Neill, B.: The fundamental equations of a submersion. Michigan Math. J. 13, 459–469 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  16. Petersen, P., Wilhelm, F.: Examples of Riemannian manifolds with positive curvature almost everywhere. Geom. Topol. 3, 331–367 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  17. Singhof, W.: On the topology of double coset manifolds. Math. Ann. 297, 133–146 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  18. Tapp, K.: Quasi-positive curvature on homogeneous bundles. J. Differ. Geom. 65, 273–287 (2003)

    MATH  MathSciNet  Google Scholar 

  19. Wilhelm, F.: An exotic sphere with positive curvature almost everywhere. J. Geom. Anal. 11, 519–560 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  20. Wilking, B.: Manifolds with positive sectional curvature almost everywhere. Invent. Math. 148, 117–141 (2002)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jason DeVito.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

DeVito, J., DeYeso, R., Ruddy, M. et al. The classification and curvature of biquotients of the form \(Sp(3)/\!\!/Sp(1)^2\) . Ann Glob Anal Geom 46, 389–407 (2014). https://doi.org/10.1007/s10455-014-9430-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10455-014-9430-4

Keywords

Mathematics Subject Classification (2010)

Navigation