On Affine Differential Geometry of Surfaces in R^3

On Affine Differential Geometry of Surfaces in R3. A study from the viewpoint of qualitative Ordinary Differential Equations (ODEs) and Singularity Theory. 1. Introduction The geometry of surfaces is a classical subject in mathematics that has a long tradition of interesting results and whose range of applicability comprehends a wide spectrum of disciplines including theoretical and applied physics, material engineering and design, computer vision, and mathematical biology, among others. This broad range of applications stem from the simple fact that surfaces are ubiquitous in everyday life because we live immersed in a three-dimensional space. Many techniques have been developed to study different aspects of the geometry of surfaces, including algebraic, analytic and topological tools. For example, according to Felix Klein’s “Erlangen Program”, the use of geometric transformation groups provides a fundamental method for the classification of different geometries. The study of properties of geometric objects that are invariant under a given transformation group G is called the geometry subordinated to G. In particular, Euclidean geometry can be understood in this way when the group G is the group of the so-called Euclidean motions, i.e., translations and rotations in R3. Similarly, affine geometry is associated to the group of affine transformations, where in addition to the set of rotations, the whole collection of all invertible linear transformations is admitted as a transformation group. 2. State of the art To provide a rigorous context for the statement of our research problem, we start with a brief review of the required notions of classical differential geometry. Consider a smooth germ of an immersion α:M→R3 of a surface M into Euclidean 3-space. In Euclidean differential geometry the fundamental forms of α at a point p∈M are defined as the symmetric bilinear forms on the tangent space TpM given as follows (see [11]): • The first fundamental form: Iα(p; w1, w2) =〈Dα(p; w1), Dα(p; w2)〉. • The second fundamental form: IIα(p; w1, w2) = −〈DNα (p; w1), Dα(p; w2)〉. Here 〈 · , · 〉 denotes the Euclidean inner product on R3, D denotes the total derivative of the quantity to its right, the tangent vectors w1, w2 ∈ TpM , and Nα is the unit normal associated to the immersion: Nα = (αu ∧ αv)/|αu ∧ αv|, where the pair (u, v) : U⊂M → R2 denotes a chart on the surface M , “∧” stands for the vector (or wedge) product in R3, and αu := ∂α/∂u , αv := ∂α/∂v. A vector w ∈ TpM for which the normal curvature kn(p; w) = IIα(p; w, w)/Iα(p; w, w), vanishes, is called an asymptotic direction of α at p. A regular curve c:[a, b] → M, whose tangent line is an asymptotic direction is called an asymptotic line of α. Through every point p of the hyperbolic region Hα of the immersion α; characterized by the condition that the Gaussian curvature Kα = det DNα is negative, pass two transverse asymptotic lines of α, tangent to the two asymptotic directions through p. When it is non-empty, the region Hα is bounded by the set (generically a regular curve) Pα of parabolic points of α, on which Kα vanishes. On Pα, the pair of asymptotic directions degenerate into a single one. The parabolic points will be regarded here as the singularities of the asymptotic net. Along Pα , the asymptotic directions are transversal to the parabolic set except at isolated points called the cusp of Gauss points. The asymptotic lines near to the cusp of Gauss points have been well studied, see for example [12, 16] where it is shown that, generically, the asymptotic lines near to the cusp of Gauss points behave as in Figure 1: Figure 1. Asymptotic lines near Gauss cuspidal points: folded saddle (left), folded focus (center), and folded node (right). The closed asymptotic lines and extended closed asymptotic lines were also studied in [12]. Asymptotic lines, together with geodesics and principal curvature lines are studied in classical differential geometry by many authors. For principal curvature lines, which are smooth curves such that the tangent at each point is an eigenvector of DNα, there is a classification of local topological models near the umbilic points (see Figure 2). Figure 2. (Darbouxian umbilics). Generic and stable topological models of principal curvature lines near an umbilic point of index 1/2 (left and center) and index −1/2 (right). Closed principal curvature lines were also studied, first by C. Gutierrez and J. Sotomayor in [14]. For more details on these results see [13]. It is natural to think about these kind of questions concerning special geometric properties on surfaces but in a more general context where the group of transformations is enlarged to include a broader spectrum of interesting situations. In fact, this is an important and necessary step to obtain new perspectives that will be useful in applications and the first natural setting to consider (after the Euclidean one) is precisely given by affine differential geometry that will be the subject of our research proposal. Affine differential geometry. The affine differential geometry of surfaces is the study of properties of surfaces in three-dimensional space that are invariant under the group of unimodular affine transformations ASL(R3). A survey about the origins of this subject of research can be found in [1]. Affine differential geometry has been studied by many authors and it is a subject that has regained considerable interest in current research, see for example [2, 3, 4, 5, 6, 8, 9, 10, 17, 18]. Analogously to the Euclidean case, for a smooth germ of an immersion α:M → R3 of a surface M in 3-space, affine fundamental forms of α at point p are also defined as particular symmetric bilinear forms on the tangent space TpM as we now explain (see [8, 18]). The affine first fundamental form or Berwald–Blaschke metric is given at a point p∈M by (Ia)α (p; w1, w2) = |Kα|−1/4 IIα(p; w1, w2). Furthermore, the co-normal vector ν to M at p is given by ν(p) = |Kα(p)|-1/4 Nα(p). There is a single transversal field ξ defined on to M , such that {αu,αv,ξ} is a frame on the surface. The vector field ξ is called affine normal field and locally it is uniquely determined up to a direction sign. The affine normal vector satisfies Dξ⊂TM . Thus, for p and w ∈ TpM , Dξ(w) = Bij w, where the Bij is the affine shape operator. The affine third fundamental form is defined as (IIIa)α = 〈 Dν, Dξ 〉, where 〈 · , · 〉 is the Euclidean inner product on R3. Affine curvature lines and affine asymptotic lines can be defined analogously to the Euclidean case. In [7], the conditions under which the affine curvature lines and Euclidean curvature lines are the same are considered. In [5], the authors studied the affine curvature lines near to affine umbilic points and near others singularities of the principal fields of affine principal directions. In particular, it is shown that near singularities, the principal affine field of affine directions has 17 generic different topological models, which is very different from the Euclidean case which only has three (Figure 2) generic models. In [6], the study of closed affine curvature lines has been also undertaken but, apart from affine curvature lines, the study of Blaschke asymptotic lines has not been considered yet in the literature. Statement of the research problem. Recently, a definition of the affine normal curvature on M at a point p was given in [10] as knα(p; w) = (IIIa)α(p; w) / (Ia)α(p; w) , where w ∈ TpM is not an asymptotic direction. Analogous to Euclidean geometry, we say that w ∈ TpM is an affine asymptotic direction or Blaschke asymptotic direction of the immersion α at p if knα(p; w) vanishes. A regular (differentiable) curve c:I → M whose tangent line is a Blaschke asymptotic direction is called a Blaschke asymptotic line of M . Since the local behavior of the Blaschke asymptotic lines near singularities is still unknown, further research is needed to understand and describe its properties. This is precisely the aim of our research proposal, and it will open the possibility to employ new tools from affine geometry for different applications, that we mention briefly below. To summarize, our research proposal is to provide a detailed study of the local behavior of the Blaschke asymptotic lines of affine surfaces in three-dimensional space. This behaviour is currently unknown and to complete this stuy several aspects need to be considered as listed in the specific objectives of this document. Justification. This direction of research can be considered as part of the qualitative theory of binary differential equations of principal and asymptotic lines developed by C. Gutierrez, J. Sotomayor, R. Garcia in [14, 15, 12] and others. The techniques involved belong essentially to the fields of Dynamical Systems, Differential Geometry and Singularity Theory. Historically, this line of research goes back to the works of Monge, Dupin, Darboux, Andronov, Peixoto. Although Differential Geometry is a very fruitful area of Mathematics that has a long history of more than a century of worldwide developments, in Colombia it has been gaining traction as a research field only in the last two decades. Whithin this field, the particular area of Affine Differential Geometry is one of the most underrepresented in the research interests of Colombian geometers and one of the purposes of this project is to firmly establish a line of research in this direction, taking advantage of the fact that the postdoctoral candidate is one of a very few that has had the oportunity to be part of and collaborate with well-known research groups in affine geometry. Indeed, several Brazilian universities host some of the leading experts in the topic proposed in our research statement and the close contact of the postoctoral candidate with them constitute and excellent oportunity to achieve our aim. For this reason we intend to offer a seminar in Affine Differential Geometry where students can participate and learn the basic material needed to initiate their thesis graduation work in this area. We expect this to have a very positive impact in the development of mathematics in general, and geometry in particular, in Colombia. Concerning the impact of this research proposal for applications, it is worth noting that classical invariants under rigid motions (Euclidean case) are heavily used in many aspects of computer graphics and geometric modeling. The affine case, being much more general, allows to extend the tools and techniques of classical Gaussian geometry to a broader set of situations and applications. For example, in [2, 3] the authors have introduced affine differential geometry tools into the world of computer graphics. It is to be expected that applications of affine differential geometry will become very relevant in areas such as scientific imaging (specially medical imaging reconstruction techniques), tomography, as well as computer vision and related fields. References. [1] A. F. Agnew, A. Bobe, W. G. Boskoff, and B. D. Suceava. Tzitzeica and the origins of affine differential geometry. Historia Mathematica, 36:161–170, 2009. [2] M. Andrade. Cálculo de estruturas afins e aplicação às Isossuperfı́cies. PhD thesis, Pontifı́cia Universidade Católica do Rio de Janeiro, 2011. [3] M. Andrade and T. Lewiner. Cálculo e estimação de invariantes geométricos: Uma introdução às geometrias Euclidiana e Afim. 28 Colóquio Brasileiro de Matemática, 2011. [4] M. Barajas. Dinâmica das linhas de curvatura de superfı́cies no espaço afim. Master’s thesis, Instituto de Matemática e Estatı́stica - Universidade Federal de Goiás (IME-UFG), 2013. [5] M. Barajas, M. Craizer, and R. Garcia. Affine curvature lines of surfaces in 3-space. Preprint, 2018. [6] M. Barajas and R. Garcia. On closed affine curvature lines. (in preparation), 2018. [7] S. Buchin. On the theory of lines of curvature of the surface. Tohoku Math. Journal, 30:457–467, 1929. [8] S. Buchin. Affine differential geometry. Routledge, 1983. [9] E. Calabi. Hypersurfaces with maximal affinely invariant area. American Journal of Mathematics, 104:91–126, 1982. [10] D. Davis. Affine Differential Geometry and Singularity Theory. PhD thesis, University of Liverpool, 2008. [11] M. P. do Carmo. Differential geometry of curves and surfaces. Prentice-Hall lnc., Englewood Cliffs, New Jersey, 1976. [12] R. Garcia, C. Gutierrez, and J. Sotomayor. Structural stability of asymptotic lines on surfaces immersed in R3. Bulletin des sciences mathematiques, pages 599–622, 1999. [13] R. Garcia and J. Sotomayor. Differential Equations of Classical Geometry, a Qualitative Theory. 27 Coloquio Brasileiro de Matematica, Rio de Janeiro, 2009. [14] C. Gutierrez and J. Sotomayor. Structural stable configurations of lines of principal curvature. Asterisque, pages 98–99, 1982. [15] C. Gutierrez and J. Sotomayor. Lines of curvature and umbilic points on surfaces, Brazilian 18 Math. Coll. reprinted as: Structurally configurations of lines of curvature and umbilic points on surfaces. Monografias del IMCA, Lima, Peru, 1998. IMPA, 1991. [16] S. Izumiya, C. Romero, M. A. S. Ruas, and F. Tari. Differential geometry from a singularity theory viewpoint. World Scientific Publishing Company, 2016. [17] J. Loftin. Survey on affine spheres. In Handbook of geometric analysis, Adv. Lect. Math., 13:161–191, 2010. [18] K. Nomizu and T. Sasaki. Affine differential geometry. Cambridge University Press, 1994.