Differential Geometry from a Singularity Theory Viewpoint

The following sections are included:

• Preface
• Contents

The study of curves and surfaces in the Euclidean space is a fascinating and important subject in differential geometry. We highlight in this chapter how singularity theory can be used not only to recover classical results on curves and surfaces in a simpler and more elegant way but also how it reveals the rich and deep underlying concepts involved…

In this chapter, we consider some aspects of the extrinsic geometry of a submanifold M of dimension n of the Euclidean space ℝ^n+r, with r ≥ 1. When r = 1, M is a hypersurface and has locally a well defined unit normal vector field which can be extended to the whole hypersurface if it is orientable. This normal vector field defines a map, called the Gauss map, from M to the unit hypersphere in ℝⁿ⁺¹. A great deal of the extrinsic geometry of the hypersurface M can be derived from the Gauss map and its derivative map, the shape operator. The shape operator is, at each point of M, a self adjoint operator from the tangent space of M at p to itself. Its eigenvalues are called the principal curvatures and when they all are equal at a given point p, the point p is called an umbilic point. A hypersurface is totally umbilic if all its points are umbilic points. Totally umbilic hypersurfaces form our models of hypersurfaces and the extrinsic geometry of a general hypersurface in ℝⁿ⁺¹ is studied in subsequent chapters by looking at its contact with these model hypersurfaces…

The birth of singularity theory can be traced back to the pioneering work of Whitney [Whitney (1955)] where he showed that maps from the plane to the plane have, in general, only folds and cusps singularities. Mather introduced in his seminal papers [Mather (1968, 1969a,b,c, 1970)] groups that act on the set of map-germs and set the foundations for the study of finite determinacy of map-germs, a concept linked to the existence of versal deformations and versal unfoldings. A versal deformation, is in some sense, one that contains all possible deformations of the initial map-germ. This has a wide range of applications in mathematics and other fields of science (see for example [Koenderink (1990); Poston and Stewart (1996); Thom (1983)])…

Following Felix Klein's approach, a geometry of submanifolds in an ambient space X depends on the description of their properties which are invariant under the action of a transformation group of X ([Klein (1974)]). Each geometry has its class of model submanifolds which is invariant under the action of the transformation group of X. These submanifolds are homogeneous spaces (they look locally the same at all points). We can then associate geometrical invariants to a submanifold M by comparing it with the models in such a way that an invariant at a given point p in M is defined as that of the model that better approximates M at p. For instance, the curvature of a plane curve at a point p is the curvature of the osculating circle at p. The osculating circle has the highest possible contact with the curve at p among all circles passing through the point p. The model submanifolds do not need to be unique at each point. This is illustrated by the focal hyperspheres at a point p on a hypersurface in ℝⁿ, n ≥ 3. There are n focal hyperspheres at p and this leads to the hypersurface having n-principal curvatures, where each principal curvature is the curvature of an osculating hypersphere. The osculating hyperspheres at a point p are the hyperspheres with highest contact with a hypersurface at p. We investigate in this chapter the concept of contact between submanifolds as a singularity theory tool for the study of differential geometry of submanifold of ℝⁿ (or of any manifold)…

A caustic in Euclidean space is the envelope of light rays reflected on a concave surface. It is the set of points in the space which are mostly illuminated by these rays. Caustics can be visible (the reader may try to catch one on a surface of a cup of coffee or tea)…

In the previous chapters, we considered the general framework of Lagrangian and Legendrian singularities as well as that of contact between submanifolds in ℝⁿ. We shall apply here the results from those chapters to study the extrinsic geometry of surfaces embedded in the Euclidean 3-space ℝ³…

We consider in this chapter the extrinsic differential geometry of a surface M immersed in ℝ⁴. The study of the second order geometry of M is considered in depth in [Little (1969)]. The work of Little inspired much of recent work on the subject…

With the aim of illustrating how the singularity techniques can be applied to analyse the extrinsic geometry of surfaces in higher codimensions we discuss in this chapter the geometrical properties associated to the contacts of surfaces with hyperplanes and hyperspheres in ℝ⁵. In this, as well as in higher codimensional cases, the curvature ellipse at each point of the surface determines a proper subspace of the normal space and both, its dimension and relative positions with respect to the considered point are relevant in the description of the second order geometry of the surface at this point. Moreover, differently from the case of surfaces in ℝ⁴, the directions leading to degenerate singularities of height functions at each point are not finite. These directions determine a cone in the normal space at the considered point. Attending to the behaviour of this cone we can distinguish among different types of points. The corresponding properties are described in §8.1. The generic singularities of the height functions on surfaces in ℝ⁵ are analysed in §8.2. Analogously to the case of surfaces in ℝ⁴, this setting leads naturally to the introduction of binormal and asymptotic directions at each point of the surface. Since the number of normal directions leading to degenerate singularities of height functions at each point is not finite, we use the concept of binormal direction for those leading to singularities of height functions on the surface with codimension higher than one, which only occur in a finite number of normal directions at every point. The corresponding contact directions are the asymptotic directions at the point. Alternative characterisations of asymptotic directions in terms of normal sections and of the geometrical behaviour of orthogonal projections of the surface into 3- and 4-spaces are also given. We see in Proposition 8.9 that the number of asymptotic directions at each point is at least one and at most five. The possible local configurations, described in Proposition 8.10, were determined in [Romero Fuster, Ruas and Tari (2008)]…

The Lorentz-Minkowski space, also called Minkowski space-time, provides a mathematical setting in which Einstein's theory of special relativity is most conveniently formulated. In this setting, the three ordinary dimensions of space are combined with a single dimension of time to form a four-dimensional manifold for representing space-time. The geometry on this space is defined by the Poincaré's group or the group of Lorentz transformations. The geometrical properties of submanifolds of Minkowski space can be studied, in a similar way to those of Euclidean space, through the analysis of their contact with model submanifolds invariant by the Lorentz transformations group. The situation is richer in this case than in the Euclidean one due to the different possibilities for the type of the submanifolds and of the models. The submanifolds can be spacelike, timelike or lightlike and the models can be taken to be the lightlike hyperplanes or hyperspheres…

In the previous chapters we used singularity theory techniques to study a submanifold M of Euclidean space and of Minkowski space-time locally at a given point on M. In this chapter, we illustrate how those techniques can be used to obtain global results on M (which we shall assume to be closed, i.e., compact without boundary). The approaches we adopt here are the following…

The following sections are included:

• Bibliography
• Index