In his Disquisitiones generales circa superficies curvas (1827) [Gau], Gauss introduced the spherical image of an oriented surface in space. Each point P on the surface is mapped to that point Q on the unit sphere whose radius vector is parallel to the outward normal vector to the surface at P. The determinant of the derivative of the Gauss mapping at P is the Gaussian curvature K of the surface at P . If K(P) 0, a small simple closed curve about P on the surface is mapped to a simple closed curve about Q on the sphere. These two curves will have the same orientation if K(P) > 0 (P is elliptic), or opposite orientations if K(P) < 0 (P is hyperbolic). Gauss indicated that he intended to study the geometry of the spherical image near a point P with K(P) = 0 (a parabolic point), but he never published anything about his investigations.
Hilbert and Cohn-Vossen discussed several examples of Gauss mappings near parabolic points in their book Anschauliche Geometrie (1932) [HC, 529]. The spherical image of the bell surface has a "fold" along the image of the parabolic locus. The parabolic locus of the torus of revolution is two circles, each of which has a single point as spherical image. The monkey saddle has an isolated parabolic point, which is a ramification point of the Gauss map.
The Gauss map of the bell surface is stable: any sufficiently small perturbation of the map has the same topology. However, there is an arbitrarily small perturbation of the Gauss map of the monkey saddle which has a curve as its parabolic image, and which therefore has a different topological type. We shall see that such a perturbation can be realized as the Gauss map of a perturbed monkey saddle. This illustrates a theorem of Looijenga [Lo] and Bleeker and Wilson [BlW]: For an open dense set in the space of immersions of a surface in 3-space, the Gauss map is stable.
Whitney [Wh] showed that the only local singularities of a stable map of surfaces are folds and cusps. (The map (x, y) -> (x, y2) has a fold singularity at (x, y) = (0, 0), and (x,y) -> (x, y3 - xy) has a cusp singularity at (x, y) = (0, 0).) The singularities of the Gauss map of a surface are precisely the parabolic points. If the Gauss map is stable, almost all of the parabolic points are fold points of the Gauss map. In this paper we shall investigate the extrinsic geometry of a surface at a point where the Gauss map has a cusp. This not only puts classical geometric concepts in a new light, but also vividly illustrates the powerful techniques being developed in the theory of singularities of families of differentiable maps. We obtain a list of ten geometric characterizations of cusps of Gauss mappings, of which the following three are the most interesting.
Theorem. If the Gauss map N of the immersion X : M2 -> R3 is stable, then N has a cusp at P in M if and only if any one of the following statements is true:
1) The restriction of X to arbitrarily small neighborhoods of P has bitangent planes.
2) A ridge of X crosses the parabolic curve of X at P, and the principal curvature associated to the ridge is zero at P.
3) P is a parabolic point of X which is in the closure of the set of inflection points of asymptotic curves of X.
There are two types of Gaussian cusps: the elliptic cusp, at which the projection of the surface to its normal line has an extremum, and the hyperbolic cusp, at which this projection has a topological saddle. Each of the above three characterizations of cusps of the Gauss map reflects the difference between the two types. For an elliptic [hyperbolic] cusp, the nearby bitangent planes touch the surface at elliptic [hyperbolic] points. The line of curvature associated to the zero curvature direction at an elliptic [hyperbolic] cusp lies in the elliptic [hyperbolic] region. The configurations of asymptotic lines which occur near an elliptic cusp are topologically distinct from the configurations which occur near a hyperbolic cusp.
Characterization (1) occurs in the work of Zakalyukin [Z], Romero Fuster [Rom1], and Bruce [Bru1]. Characterization (3) occurs in the work of Arnold [A6], Kergosien and Thom [KeT] (see also [BaT]), Landis [Lan], and Platonova [Pl2]. Arnold has informed us that many of our results are known to the Russian school of singularity theory (cf. chapter 8 below). Much of their work is summarized in a forthcoming book [AGV]. A detailed investigation of cusps of Gauss mappings is also contained in the work of Y. L. Kergosien [Ke1] [Ke2] [Ke3].
We would like to express our debt to the pioneering work of Rene Thom on the geometry of singularities of differentiable maps [T1] [T2] [T4]. His description of focal surfaces using catastrophe theory [T4, p. 96], as developed by Porteous [Por1], was a starting point for our research.
Several other applications of singularity theory to differential geometry are discussed in Wall's survey "Geometric properties of generic differentiable manifolds" [Wa2] (cf. also [Bru2]). Many interesting ideas and examples in this vein occur in an unpublished manuscript of Michael Menn. An introduction to singularity theory is given by the survey articles of Callahan [C1], Arnold [A1], Wall [Wa1], and Levine [Le].
Our first chapter is about plane curves, where the interplay between the singularities of the Gauss map and singularities of families of functions takes on a very simple form. In the second chapter we introduce five examples which we shall use to illustrate our results. The singularities of the Gauss maps in these examples are analyzed using techniques of Bleeker and Wilson [BlW]. A description of the remarkable confluence of geometric properties which occurs at a Gaussian cusp appears in the following chapter. The proofs that these properties are equivalent for generic surfaces occur in chapters 4 through 7. We conclude with a short chapter summarizing current work on singularity theory and the extrinsic geometry of curves and surfaces.
The computer graphics illustrations in the original document [BaGM2] were created by Thomas Banchoff and Charles Strauss. Their film, "The Gauss map, a dynamic approach" includes our examples 2 and 3a (see [BaT]).
[Web graphics and mpeg movies by Dan Dreibelbis, using CenterStage and StageManager, external modules for Geomview. Images of mathematical equations were created by latex2html.]
We thank Rick Porter for a stimulating lecture on envelopes of plane curves and catastrophe theory which pointed out very fruitful directions for this research project.
Providence, Rhode Island 02912
Boston, Massachusetts 02115
University of Georgia
Athens, Georgia 30602
Supported in part by the National Science Foundation, Grants number
MC579-01310 and MC579-04905.