Journal of the Mathematical Society of Japan

Spaces of nonnegatively curved surfaces

Taras BANAKH and Igor BELEGRADEK

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We determine the homeomorphism type of the space of smooth complete nonnegatively curved metrics on $S^2$, $RP^2$, and $\mathbb{C}$ equipped with the topology of $C^\gamma$ uniform convergence on compact sets, when $\gamma$ is infinite or is not an integer. If $\gamma=\infty$, the space of metrics is homeomorphic to the separable Hilbert space. If $\gamma$ is finite and not an integer, the space of metrics is homeomorphic to the countable power of the linear span of the Hilbert cube. We also prove similar results for some other spaces of metrics including the space of complete smooth Riemannian metrics on an arbitrary manifold.

Article information

Source
J. Math. Soc. Japan, Volume 70, Number 2 (2018), 733-756.

Dates
First available in Project Euclid: 18 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1524038672

Digital Object Identifier
doi:10.2969/jmsj/07027344

Subjects
Primary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60]
Secondary: 57N20: Topology of infinite-dimensional manifolds [See also 58Bxx]

Keywords
nonnegative curvature space of metrics infinite dimensional topology absorbing

Citation

BANAKH, Taras; BELEGRADEK, Igor. Spaces of nonnegatively curved surfaces. J. Math. Soc. Japan 70 (2018), no. 2, 733--756. doi:10.2969/jmsj/07027344. https://projecteuclid.org/euclid.jmsj/1524038672


Export citation

References

  • M. T. Anderson and J. Cheeger, $C^\alpha$-compactness for manifolds with Ricci curvature and injectivity radius bounded below, J. Differential Geom., 35 (1992), 265–281.
  • T. Banakh, Operator images homeomorphic to $\Sigma^\omega$, Dissertationes Math. (Rozprawy Mat.), 387 (2000), 53–81.
  • T. Banakh and R. Cauty, Interplay between strongly universal spaces and pairs, Dissertationes Math. (Rozprawy Mat.), 386 (2000), 38.
  • T. Banakh, T. Dobrowolski and A. Plichko, The topological and Borel classification of operator images, Dissertationes Math. (Rozprawy Mat.), 387 (2000), 37–52.
  • C. Blanc and F. Fiala, Le type d'une surface et sa courbure totale, Comment. Math. Helv., 14 (1941–42), 230–233.
  • I. Belegradek and J. Hu, Connectedness properties of the space of complete nonnegatively curved planes, Math. Ann., 362 (2015), 1273–1286.
  • I. Belegradek and J. Hu, Erratum to: Connectedness properties of the space of complete nonnegatively curved planes, Math. Ann., 364 (2016), 711–712.
  • B. Bojarski, P. Hajlasz and P. Strzelecki, Sard's theorem for mappings in Hölder and Sobolev spaces, Manuscripta Math., 118 (2005), 383–397.
  • M. Bestvina and J. Mogilski, Characterizing certain incomplete infinite-dimensional absolute retracts, Michigan Math. J., 33 (1986), 291–313.
  • C. Bessaga and A. Pelczyński, Selected topics in infinite-dimensional topology, Monografie Matematyczne, Tom 58, [Mathematical Monographs, 58], PWN–-Polish Scientific Publishers, Warsaw, 1975.
  • T. Banakh, T. Radul and M. Zarichnyi, Absorbing sets in infinite-dimensional manifolds, Mathematical Studies Monograph Series, 1, VNTL Publishers, L'viv, 1996.
  • R. Cauty, T. Dobrowolski and W. Marciszewski, A contribution to the topological classification of the spaces $C_p(X)$, Fund. Math., 142 (1993), 269–301.
  • T. Dobrowolski and J. Mogilski, Problems on topological classification of incomplete metric spaces, Open problems in topology, North-Holland, Amsterdam, 1990, 409–429.
  • S. Donaldson, Riemann surfaces, Oxford Graduate Texts in Mathematics, 22, Oxford University Press, Oxford, 2011.
  • T. Dobrowolski and H. Toruńczyk, Separable complete ANRs admitting a group structure are Hilbert manifolds, Topology Appl., 12 (1981), 229–235.
  • C. J. Earle and A. Schatz, Teichmüller theory for surfaces with boundary, J. Differential Geometry, 4 (1970), 169–185.
  • H. D. Fegan and R. S. Millman, Quadrants of Riemannian metrics, Michigan Math. J., 25 (1978), 3–7.
  • D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001, reprint of the 1998 edition.
  • M. W. Hirsch, Differential topology, Graduate Texts in Mathematics, 33, Springer-Verlag, New York, 1994, Corrected reprint of the 1976 original.
  • J. Hu, Complete nonnegatively curved spheres and planes, Ph.D. thesis, Georgia Tech, 2015.
  • K. Kuratowski, Topology, I, New edition, revised and augmented, Translated from the French by J. Jaworowski, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe, Warsaw, 1966.
  • J. L. Kazdan and F. W. Warner, Curvature functions for compact $2$-manifolds, Ann. of Math. (2), 99 (1974), 14–47.
  • E. L. Lima, On the local triviality of the restriction map for embeddings, Comment. Math. Helv., 38 (1964), 163–164.
  • J. Mogilski, Characterizing the topology of infinite-dimensional $\sigma$-compact manifolds, Proc. Amer. Math. Soc., 92 (1984), 111–118.
  • R. S. Palais, Foundations of global non-linear analysis, W. A. Benjamin, Inc., New York-Amsterdam, 1968.
  • P. Petersen, Convergence theorems in Riemannian geometry, Comparison geometry (Berkeley, CA, 1993–94), Math. Sci. Res. Inst. Publ., 30, Cambridge Univ. Press, Cambridge, 1997, 167–202.
  • S. Smale, Diffeomorphisms of the 2-sphere, Proc. Amer. Math. Soc., 10 (1959), 621–626.
  • F. Trèves, Topological vector spaces, distributions and kernels, Academic Press, New York, 1967.
  • T. Yagasaki, Homotopy types of diffeomorphism groups of noncompact 2-manifolds.v3.