Geometry & Topology

Directed immersions of closed manifolds

Mohammad Ghomi

Full-text: Open access


Given any finite subset X of the sphere Sn, n2, which includes no pairs of antipodal points, we explicitly construct smoothly immersed closed orientable hypersurfaces in Euclidean space Rn+1 whose Gauss map misses X. In particular, this answers a question of M Gromov.

Article information

Geom. Topol., Volume 15, Number 2 (2011), 699-705.

Received: 25 October 2010
Accepted: 13 March 2011
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53A07: Higher-dimensional and -codimensional surfaces in Euclidean n-space 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]
Secondary: 57R42: Immersions 58K15: Topological properties of mappings

Gauss map spherical image directed immersion convex integration h-principle closed hypersurface parallelizable manifold


Ghomi, Mohammad. Directed immersions of closed manifolds. Geom. Topol. 15 (2011), no. 2, 699--705. doi:10.2140/gt.2011.15.699.

Export citation


  • Y Eliashberg, N Mishachev, Introduction to the $h$–principle, Graduate Studies in Math. 48, Amer. Math. Soc. (2002)
  • M Ghomi, Gauss map, topology, and convexity of hypersurfaces with nonvanishing curvature, Topology 41 (2002) 107–117
  • M Ghomi, Shadows and convexity of surfaces, Ann. of Math. $(2)$ 155 (2002) 281–293
  • M Ghomi, Tangent bundle embeddings of manifolds in Euclidean space, Comment. Math. Helv. 81 (2006) 259–270
  • M Ghomi, M Kossowski, $h$–principles for hypersurfaces with prescribed principal curvatures and directions, Trans. Amer. Math. Soc. 358 (2006) 4379–4393
  • M Ghomi, S Tabachnikov, Totally skew embeddings of manifolds, Math. Z. 258 (2008) 499–512
  • M Gromov, Partial differential relations, Ergebnisse der Math. und ihrer Grenzgebiete (3) 9, Springer, Berlin (1986)
  • M Gromov, Spaces and questions, from: “Visions in mathematics: GAFA 2000 (Tel Aviv, 1999)”, (N Alon, J Bourgain, A Connes, M Gromov, V D Milman, editors), Geom. Funct. Anal., Special Volume, Part I (2000) 118–161
  • P Hartman, L Nirenberg, On spherical image maps whose Jacobians do not change sign, Amer. J. Math. 81 (1959) 901–920
  • M W Hirsch, On imbedding differentiable manifolds in euclidean space, Ann. of Math. $(2)$ 73 (1961) 566–571
  • J Milnor, On the immersion of $n$–manifolds in $(n{+}1)$-space, Comment. Math. Helv. 30 (1956) 275–284
  • U Pinkall, Regular homotopy classes of immersed surfaces, Topology 24 (1985) 421–434
  • C Rourke, B Sanderson, The compression theorem. II. Directed embeddings, Geom. Topol. 5 (2001) 431–440
  • D Spring, Directed embeddings of closed manifolds, Commun. Contemp. Math. 7 (2005) 707–725
  • D Spring, The golden age of immersion theory in topology: 1959–1973. A mathematical survey from a historical perspective, Bull. Amer. Math. Soc. $($N.S.$)$ 42 (2005) 163–180
  • H Wu, The spherical images of convex hypersurfaces, J. Differential Geometry 9 (1974) 279–290