Location
Dates
Presentation
The Theory of Submanifolds emerged as a natural generalization of the classical study of curves and surfaces of three-dimensional Euclidean space through the methods of differential calculus. In the last century, this theory has become a vast sub-area of Differential Geometry with numerous correlated lines of research and a variety of techniques that allow obtaining many deep research results, both local and global. A significant part of these techniques originated in Geometric Analysis, whose scope includes the use of geometric methods in the study of partial differential equations (PDE’s) and, conversely, the application of the theory of PDE’s to Differential Geometry. In the last decades, an important school of geometers from Brazil, Spain, Colombia and Mexico has been working intensively on the theory of submanifolds (surfaces, hypersurfaces, isometric immersions) and on Geometric Analysis, initiated by pioneers such as Manfredo do Carmo, Greg Galloway, Santiago López de Medrano and Antonio Martínez Naveira.
In this research school, the students will have contact with geometers, some of them oriented by the pioneers cited above. The introductory courses of the first week will teach the techniques of classical geometry of curves and surfaces in Euclidean and/or Lorentzian spaces, covering intrinsic (Gaussian curvature) and extrinsic (mean curvature) invariants from a global point of view.
Scientific program is available on the local website of the School: https://naturales.ues.edu.sv/ecimpasv2024/
Official language of the school: Spanish
Administrative and scientific coordinators
Scientific program
Course 1: "introductory course - The aim of this course is to present some basic notions on the geometry of curves and surfaces in the Euclidean space. We will start with a brief treatment of regular curves in R3, focusing on the study of its local properties that appear naturally while studying surfaces. Specifically, we will introduce the essential concepts of parametrized curves, arc length, curvature, torsion and the Frenet trihedron. We will start the study of surfaces developing the concept of a regular surface in R3 and some fundamental notions as the tangent plane and the first fundamental form of a surface. Later, we will introduce one of the most important concepts for the study of the extrinsic geometry of surfaces: the Gauss map. The Gauss map will allow us to define the second fundamental form, which will be essential in order to introduce two of the most common ways to measure the curvature of a surface, the mean and the Gaussian curvature. We will show that the Gaussian curvature is, in fact, an intrinsic concept, since it is invariant by local isometries, whereas the mean curvature is extrinsic. Some important families of regular surfaces such as ruled and minimal surfaces will be introduced. Along the course many examples of well-known curves and surfaces will be presented. This course represents the basis for an introduction to the “Geometric Analysis on Surfaces”, which will be studied in a different course. ", Sandra Carolina GARCÍA MARTÍNEZ (Universidad Nacional de Colombia, El Salvador)
Course 2: "introductory course - The goal of this minicourse is to give motivation and geometric insight into Lorentzian Geometry. First, we give some preliminaries on Indefinite Linear Algebra on real vector spaces, discussing several topics on causal character, isometry groups, and canonical forms for self-adjoint linear operators. Next, we take a quick look at curves in three-dimensional Lorentz-Minkowski space. We finish with the geometric study of non-degenerate surfaces, also in the three-dimensional case.", Oscar PALMAS (Universidad Nacional Autónoma de México, El Salvador)
Course 3: "introductory course - The extension of the analysis of differential operators in domains of Euclidean space to the context of smooth manifolds is a modern and relevant topic that involves geometric and analytical concepts, and has received a lot of attention in recent literature. In this minicourse, we will establish the basic tools of geometric analysis as the first contact for researchers who are interested in working on subjects in this area. We will cover the following subjects: A review of Riemannian geometry: smooth manifolds, Riemannian metric, Levi-Civita connection, and curvatures; Differential operators on Riemannian manifolds: gradient, divergent, Laplacian, and Hessian; Tensors and the differential operators: contracted Bianchi's second identity, Bochner's formula; Isometric immersions.", Walcy SANTOS (Universidade Federal de Rio de Janeiro , El Salvador)
Course 4: "advanced course - The development of Lorentzian geometry has been closely related to the theory of Relativity. In this minicourse we explore some active areas of research in Lorentzian geometry, spanning from an abstract mathematical setting to novel applications in General Relativity. Topics will include causal theory, geometry of null hypersurfaces, and Lorentzian length spaces.", Didier SOLIS (Universidad Autónoma de Yucatán, El Salvador)
Course 6: "advanced course - In this minicourse we will compute a lower bound for the scalar curvature of a gradient Einstein soliton under a certain assumption on its potential function. We will establish an asymptotic behavior of the potential function on a noncompact gradient shrinking Einstein soliton. As a result, we will obtain the finiteness of its fundamental group and its canonical weighted volume. We will also prove the validity of the weak maximum principle at infinity for a drifted Laplacian. Additionally, we will prove some geometrical and analytical results for constructing gradient Einstein solitons that are realized as warped metrics, and we will give a few explicit examples.", José GOMES (Universidad Federal de Sao Carlos, El Salvador)
Website of the school
How to participate
For registration and application to a CIMPA financial support, follow the instructions given here. https://www.cimpa.info/fr/node/40
Deadline for registration and application: April 5, 2024