Journal of Geometric Analysis

, Volume 20, Issue 3, pp 609–650 | Cite as

Hyperplane Envelopes and the Clairaut Equation

  • Robert J. FisherEmail author
  • H. Turner Laquer


The subject of envelopes has been part of differential geometry from the beginning. This paper brings a modern perspective to the classical problem of envelopes of families of affine hyperplanes. In the process, the classical results are generalized and unified.

A key step in this process is the use of “generalized immersions”. These have been described elsewhere but, briefly, every classical immersion defines a generalized immersion in a canonical way so that generalized immersions can be understood as ordinary immersions “with singularities.” Next, the concept of an envelope is given a modern definition, namely, an envelope is a generalized immersion solving the family that has a universal mapping property relative to all other full rank solutions.

The beauty of this approach becomes apparent in the “Envelope Theorem”. With one mild assumption, namely that the associated family of linear hyperplanes is immersed, it is proven that a family of affine hyperplanes always has an envelope, and that envelope is essentially unique.


Hyperplane envelope Generalized immersion Gauss map Affine Grassmann duality Clairaut equation 

Mathematics Subject Classification (2000)

53C42 53A07 53A15 35D05 


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  1. 1.
    Abraham, R., Marsden, J.E., Ratiu, T.: Manifolds, Tensor Analysis, and Applications, 2nd edn. Springer, New York (1988) zbMATHGoogle Scholar
  2. 2.
    Alekseevskij, D.V., Vinogradov, A.M., Lychagin, V.V.: Geometry I: Basic Ideas and Concepts of Differential Geometry. Encyclopaedia of Mathematical Sciences, vol. 28. Springer, New York (1991) Google Scholar
  3. 3.
    Arnold, V.I.: Singularities of Caustics and Wave Fronts. Kluwer Academic, Dordrecht (1990) Google Scholar
  4. 4.
    Arnold, V.I., Gusein-Zade, S.M., Varchenko, A.N.: Singularities of Differentiable Maps, vol. I. Birkhäuser, Boston (1985) zbMATHGoogle Scholar
  5. 5.
    Brunette, J.: The Clairaut equation: a study in the geometry of partial differential equations. Thesis, Idaho State University (1995) Google Scholar
  6. 6.
    Dieudonné, J.: Treatise on Analysis, vols. I, II, III, IV, V. Academic Press, New York (1969, 1970, 1972, 1977, 1974) Google Scholar
  7. 7.
    do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice-Hall, Englewood Cliffs (1976) zbMATHGoogle Scholar
  8. 8.
    Fisher, R.J., Laquer, H.T.: Generalized immersions and the rank of the second fundamental form. Pac. J. Math. 225(2), 243–272 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Goldschmidt, H.: Integrability criteria for systems of non-linear partial differential equations. J. Differ. Geom. 1, 269–307 (1967) zbMATHMathSciNetGoogle Scholar
  10. 10.
    Harris, J.: Algebraic Geometry: A First Course. Springer, New York (1992) zbMATHGoogle Scholar
  11. 11.
    Klingenberg, W.: A Course in Differential Geometry. Springer, New York (1978) zbMATHGoogle Scholar
  12. 12.
    Spivak, M.: A Comprehensive Introduction to Differential Geometry. Publish or Perish, Houston (1975) Google Scholar
  13. 13.
    Warner, F.W.: Foundations of Differentiable Manifolds and Lie Groups. Scott, Foresman and Company, Glenview (1971) zbMATHGoogle Scholar

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© Mathematica Josephina, Inc. 2010

Authors and Affiliations

  1. 1.Department of MathematicsIdaho State UniversityPocatelloUSA

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