Canonical sphere bundles of the Grassmann manifold

  • Esteban Andruchow
  • Eduardo ChiumientoEmail author
  • Gabriel Larotonda
Original Paper


For a given Hilbert space \(\mathcal H\), consider the space of self-adjoint projections \(\mathcal P(\mathcal H)\). In this paper we study the differentiable structure of a canonical sphere bundle over \(\mathcal P(\mathcal H)\) given by
$$\begin{aligned} \mathcal R=\{\, (P,f)\in \mathcal P(\mathcal H)\times \mathcal H \, : \, Pf=f , \, \Vert f\Vert =1\, \}. \end{aligned}$$
We establish the smooth action on \(\mathcal R\) of the group of unitary operators of \(\mathcal H\), and it thereby turns out that the connected components of \(\mathcal R\) are homogeneous spaces. Then we study the metric structure of \(\mathcal R\) by endowing it first with the uniform quotient metric, which is a Finsler metric, and we establish minimality results for the geodesics. These are given by certain one-parameter groups of unitary operators, pushed into \(\mathcal R\) by the natural action of the unitary group. Then we study the restricted bundle \(\mathcal R_2^+\) given by considering only the projections in the restricted Grassmannian, locally modeled by Hilbert–Schmidt operators. Therefore we endow \(\mathcal R_2^+\) with a natural Riemannian metric that can be obtained by declaring that the action of the group is a Riemannian submersion. We study the Levi–Civita connection of this metric and establish a Hopf–Rinow theorem for \(\mathcal R_2^+\), again obtaining a characterization of the geodesics as the image of certain one-parameter groups with special speeds.


Sphere bundle Finsler metric Riemannian metric Geodesic Projection Flag manifold 

Mathematics Subject Classification (2010)

22E65 47B10 58B20 



  1. 1.
    Andruchow, E., Larotonda, G.: Hopf–Rinow theorem in the Sato Grassmannian. J. Funct. Anal. 255(7), 1692–1712 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Andruchow, E., Larotonda, G.: The rectifiable distance in the unitary Fredholm group. Studia Math. 196, 151–178 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Andruchow, E., Larotonda, G., Recht, L.: Finsler geometry and actions of the \(p\)-Schatten unitary groups. Trans. Am. Math. Soc. 362, 319–344 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Andruchow, E., Recht, L., Varela, A.: Metric geodesics of isometries in a Hilbert space and the extension problem. Proc. Am. Math. Soc. 135, 2527–2537 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Beltiţ\({{\breve{\rm a}}}\), D.: Smooth Homogeneous Structures in Operator Theory, Monographs and Surveys in Pure and Applied Mathematics 137. Chapman and Hall/CRC, Boca Raton (2006)Google Scholar
  6. 6.
    Beltiţ\({{\breve{\rm a}}}\), D., Ratiu, T., Tumpach, A.: The restricted Grassmannian, Banach Lie–Poisson spaces and coadjoint orbits. J. Funct. Anal. 247(1), 138–168 (2007)Google Scholar
  7. 7.
    Bottazzi, T., Varela, A.: Unitary subgroups and orbits of compact self-adjoint operators. Studia Math. 238, 155–176 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chiumiento, E.: Geometry of \(\mathfrak{I}\)-Stiefel manifolds. Proc. Am. Math. Soc. 138(1), 341–353 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Corach, G., Porta, H., Recht, L.: The geometry of spaces of projections in \(C^*\)-algebras. Adv. Math. 101(1), 59–77 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Davis, C., Kahan, W.M., Weinberger, H.F.: Norm-preserving dilations and their applications to optimal error bounds. SIAM J. Numer. Anal. 19, 445–469 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Durán, C.E., Mata-Lorenzo, L.E., Recht, L.: Metric geometry in homogeneous spaces of the unitary group of a \(C^*\)-algebra I. Minimal curves. Adv. Math. 184(2), 342–366 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gallot, S., Hulin, D., Lafontaine, J.: Riemannian Geometry. Universitext, 3rd edn. Springer, Berlin (2004)CrossRefzbMATHGoogle Scholar
  13. 13.
    Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. I. Reprint of the 1963 original. Wiley Classics Library. A Wiley-Interscience Publication. Wiley, New York (1996)Google Scholar
  14. 14.
    Koliha, J.J.: Range projections of idempotents in \(C^*\)-algebras. Demonstratio Math. 34(1), 91–103 (2001)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Lang, S.: Differential and Riemannian Manifolds. Graduate Texts in Mathematics, 160, 3rd edn. Springer, New York (1995)CrossRefGoogle Scholar
  16. 16.
    Kovarik, Z.V.: Manifolds of linear involutions. Linear Algebra Appl. 24, 271–287 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Krein, M.G.: The theory of self-adjoint extensions of semibounded Hermitian transformations and its applications. Mat. Sb. 20 (1947), 431–495; 21 (1947), 365–404 (in Russian) Google Scholar
  18. 18.
    Mata-Lorenzo, L.E., Recht, L.: Infinite-dimensional homogeneous reductive spaces. Acta Cient. Venezolana 43(2), 76–90 (1992)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Milnor, J.W., Stasheff, J.D.: Characteristic Classes. Princeton University Press, Princeton (1974)zbMATHGoogle Scholar
  20. 20.
    Pressley, A., Segal, G.: Loop Groups. Oxford Mathematical Monographs. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York (1986)zbMATHGoogle Scholar
  21. 21.
    Porta, H., Recht, L.: Minimality of geodesics in Grassmann manifolds. Proc. Am. Math. Soc. 100, 464–466 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Raeburn, I.: The relationship between a commutative Banach algebra and its maximal ideal space. J. Funct. Anal. 25(4), 366–390 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Riesz, F., Sz.-Nagy, B.: Functional Analysis. Ungar, New York (1955)zbMATHGoogle Scholar
  24. 24.
    Steenrod, N.E.: The classification of sphere bundles. Ann. Math. 45(2), 294–311 (1944)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Whitney, H.: On the theory of sphere-bundles. Proc. Natl. Acad. Sci. USA 26(2), 148–153 (1940)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Instituto Argentino de Matemática, ‘Alberto P. Calderón’CONICETBuenos AiresArgentina
  2. 2.Instituto de CienciasUniversidad Nacional de Gral. SarmientoLos PolvorinesArgentina
  3. 3.Departamento de MatemáticaFCE-UNLPLa PlataArgentina
  4. 4.Departamento de MatemáticaFCEyN-UBACiudad Autónoma de Buenos AiresArgentina

Personalised recommendations