The Michigan Mathematical Journal

Random Manifolds Have No Totally Geodesic Submanifolds

Thomas Murphy and Frederick Wilhelm

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


For n4, we show that generic closed Riemannian n-manifolds have no nontrivial totally geodesic submanifolds, answering a question of Spivak. Although the result is widely believed to be true, we are not aware of any proof in the literature.

Article information

Michigan Math. J., Volume 68, Issue 2 (2019), 323-335.

Received: 25 May 2017
Revised: 17 January 2018
First available in Project Euclid: 12 April 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20]
Secondary: 53C40: Global submanifolds [See also 53B25] 53A99: None of the above, but in this section


Murphy, Thomas; Wilhelm, Frederick. Random Manifolds Have No Totally Geodesic Submanifolds. Michigan Math. J. 68 (2019), no. 2, 323--335. doi:10.1307/mmj/1555034652.

Export citation


  • [1] M. Berger, A panoramic view of Riemannian geometry, Springer Verlag, Berlin, 2007.
  • [2] R. Bryant,
  • [3] D. Ebin, On the space of Riemannian metrics, Bull. Amer. Math. Soc. 74 (1968), 1001–1003.
  • [4] L. Guijarro and F. Wilhelm, Restrictions on submanifolds via focal radius bounds, Math. Res. Lett. To appear.
  • [5] R. Hermann, Differential geometry and the calculus of variations, Elsevier Science, 2000.
  • [6] M. Hirsch, Differential topology, Grad. Texts in Math., Springer-Verlag, 1994.
  • [7] P. Petersen, Riemannian geometry, second edition, Grad. Texts in Math., 171, Springer Verlag, New York, 2006.
  • [8] J. Pitts, Existence and regularity of minimal surfaces on Riemannian manifolds, Math. Notes, 27, Princeton University Press, Princeton, 1981.
  • [9] R. Schoen and L. Simon, Regularity of stable minimal hypersurfaces, Comm. Pure Appl. Math. 34 (1981), 741–797.
  • [10] M. Spivak, Differential geometry, Vol. III, Publish or Perish, 1975.
  • [11] K. Tsukada, Totally geodesic submanifolds of Riemannian manifolds and curvature-invariant subspaces, Kodai Math. J. 19 (1996), no. 3, 395–437.