Tounderstend differential geometry,i will begin byinvestigating the following propertiesofthecontinuous mappingf. In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a riemannian metric. This concise guide to the differential geometry of curves and surfaces can be recommended to. A topological space is compact if every open cover i. D m is a coordinate patch in m, then the composite mapping fx. The curves and surfaces treated in differential geometry are defined by functions which can be differentiated a certain number of times. The definition of geodesic curvature, and the proof that it is intrinsic. The discipline owes its name to its use of ideas and techniques from differential calculus, though the modern subject often uses algebraic and purely geometric techniques instead. It talks about the differential geometry of curves and surfaces in real 3space. Self adjointness of the shape operator, riemann curvature tensor of surfaces, gauss and codazzi mainardi equations, and theorema egregium revisited. Illustrations by pressley, elementary differential geometry.
Im here with a simple question and a somewhat harder one. Minimal surfaces are beautiful geometric objects with interesting. Of particular interest are minimal surfaces in manifolds of constant curvature, such as the euclidean space \\mathbbr 3\, the hyperbolic space \\mathbbh 3\, and the sphere \s3\. The concepts are similar, but the means of calculation are different. Modern differential geometry of curves and surfaces with mathematica, third edition by alfred gray, elsa abbena, simon salamon.
So on a minimal surface all geodesics have zero torsion, a result extremely easy to establish. Lecture notes on minimal surfaces emma carberry, kai fung, david glasser, michael nagle, nizam ordulu. Triply periodic minimal surfaces australian national university. Apr 14, 2006 this first course in differential geometry presents the fundamentals of the metric differential geometry of curves and surfaces in a euclidean space of three dimensions. Before we can talk about the deformation we need a definition. Gaussian and mean curvature, minimal surfaces, and gauss.
The book mainly focus on geometric aspects of methods borrowed from linear algebra. Properties of families of curves and surfaces are also studied see, for example, congruence. Dec, 2019 a beginners course on differential geometry. Free differential geometry books download ebooks online. Since grays death, authors abbena and salamon have. These notes are an attempt to summarize some of the key mathematical aspects of differential geometry,as they apply in particular to the geometry of surfaces in r3. Modern differential geometry of curves and surfaces with. Finally, understanding such compactness properties can in turn be used to understand properties of the minimal surfaces themselves. Novel link between discrete differential geometry discrete affine minimal surfaces and cagd smooth patchworks from bezier surfaces of degree 1, 1 a geometric approach to discrete affine minimal surfaces, based on smooth patchworks.
Patches and surfaces differential geometry physics forums. A flat surface is a regular surface and special class of minimal surface on which gaussian curvature vanishes everywhere. If you like to play with maple i recommend the book by oprea for the handson experience with curves and surfaces on the computer. In the parameterization of the surface patch as a graph over the square.
Differential geometry of surfaces and minimal surfaces. In mathematics, the weierstrassenneper parameterization of minimal surfaces is a classical piece of differential geometry. Alfred enneper and karl weierstrass studied minimal surfaces as far back as 1863. Harmonicity, minimal surfaces, toric bezier patches. We present a systematic and sometimes novel development of classical differential differential, going back to. Jul 16, 2015 i give another two examples here on surfaces of revolution and an application to the sphere where we see a different patch in contrast to the monge patch seen in part 2 of lecture 12. Although basic definitions, notations, and analytic descriptions. Math4030 differential geometry 201516 cuhk mathematics. We see, from example 2, that theorem removes much of the differential geometry behind minimal surfaces, and instead deals with the analysis of the holomorphic function by taking a deeper look at the conditions in the previous theorem, we can actually find a general formulation for. The first nine chapters focus on the theory, treating the basic properties of curves and surfaces, the mapping of surfaces, and the absolute geometry of a surface. Minimal surfaces are surfaces for which the mean curvature is zero every where.
I am using the textbook elementary differential geometry by oneill which i cant read for the life of me. Differential geometry of curves and surfaces by manfredo do carmo see also. In mathematics, the weierstrassenneper parameterization of minimal surfaces is a classical piece of differential geometry alfred enneper and karl weierstrass studied minimal surfaces as far back as 1863. The concept of minimal isothermal patch, previously introduced in section 16. Triply periodic minimal surfaces welcome to the epinet. We present a systematic and sometimes novel development of classical differential differential, going back to euler, monge, dupin, gauss and many others. Novel link between discrete differential geometry discrete affine. Lecture notes on minimal surfaces mit opencourseware. John roes book 7 is a pleasant exposition of geometry with a di.
California state university, san bernardino csusb scholarworks theses digitization project john m. Polymerforschung, ackermannweg 10, 55128 mainz, germany these notes are an attempt to summarize some of the key mathe. Minimal surfaces are among the most important objects studied in differential geometry. In differential geometry the properties of curves and surfaces are usually studied on a small scale, i. Interactive 3d geometry and visualization geodesic surveyor compute geodesics on polyhedral surfaces model viewer view and manipulate polyhedral models caustics in differential geometry by oliver knill and michael teodorescu, an hcrp project that includes. If you want a book on manifolds, then this isnt what youre looking for though it does say something about manifolds at the end. In mathematics, a minimal surface is a surface that locally minimizes its area. Sold by itemspopularsonlineaindemand and ships from amazon fulfillment. Written by an outstanding teacher and mathematician, it explains the material in the most effective way, using vector notation and technique. Feb 23, 2010 im completely confused with patches, which were introduced to us very briefly we were just given pictures in class. A minimal surface described by the weierstrassenneper data or has an associated family of minimal surfaces given by, respectively, or.
Some fundamentals of the theory of surfaces, some important parameterizations of surfaces, variation of a surface, vesicles, geodesics, parallel transport and. Presenting theory while using mathematica in a complementary way, modern differential geometry of curves and surfaces with mathematica, the third edition of alfred grays famous textbook, covers how to define and compute standard geometric functions using mathematica for constructing new curves and surfaces from existing ones. This is a classic result in differential geometry and is worth mentioning in these posts on minimal surfaces. A geometric approach to discrete affine minimal surfaces, based on smooth patchworks. The traditional intro is differential geometry of curves and surfaces by do carmo, but to be honest i find it hard to justify reading past the first 3 chapters in your first pass do it when you get to riemannian geometry, which is presumably a long way ahead. This means they are equally convex and concave at all points and their form is therefore saddlelike, or hyperbolic. Do carmo only talks about manifolds embedded in r n, and this is somewhat the pinnacle of the traditional calc sequence.
A point p on a regular surface is classified based on the sign of as given in the following table gray 1993, p. Classical minimal surfaces in euclidean space by examples. Given that the parametric form of a surface patch is known, this chapter deals with determining the differential properties of the patch to facilitate composite fitting. A first course in curves and surfaces preliminary version summer, 2016 theodore shifrin university of georgia dedicated to the memory of shiingshen chern, my adviser and friend c 2016 theodore shifrin no portion of this work may be reproduced in any form without written permission of the author, other than.
Additionally, the minimal case provides a model for more general situations where the mean curvature is only controlled in some sense. Elementary differential geometry curves and surfaces the purpose of this course note is the study of curves and surfaces, and those are in general, curved. These notes are intended as a gentle introduction to the di. Geometry of curves and surfaces weiyi zhang mathematics institute, university of warwick september 18, 2014. From the point of view of local geometry, a minimal surface is equivalently described as one that is equally bent in all directions so as to have zero average curvature, e. Modeling smooth surfaces from bilinear patches, motivated by applications in architecture. Definition of curves, examples, reparametrizations, length, cauchys integral formula, curves of constant width.
Elementary differential geometry curves and surfaces. Pfau library 1997 differential geometry of surfaces and minimal surfaces. Homework equations for a mapping to be a patch, it must be onetoone injective and regular. Euler lagrange equation which is a second order partial differential equation pde. The fact that they are equivalent serves to demonstrate how minimal surface theory lies at the crossroads of several mathematical disciplines, especially differential geometry, calculus of variations, potential theory, complex analysis and mathematical physics local least area definition. Unbordered minimal surfaces have the property that each point is the center of a small patch that behaves like a soapfilm relative to its boundary contour. Barrett oneill, in elementary differential geometry second edition, 2006. Pdf soap films, differential geometry, and minimal surfaces. Novel link between discrete differential geometry discrete affine minimal surfaces and cagd smooth patchworks from bezier surfaces of degree 1, 1. Moreover, if it is algebraic then is there an algorithm to derive its defining polynomial. I give another two examples here on surfaces of revolution and an application to the sphere where we see a different patch in contrast to the monge patch seen in part 2 of lecture 12. The final chapter considers the applications of the theory to certain important classes of surfaces. Modern differential geometry of curves and surfaces with mathematica. But one can state definitively that you cannot cover a surface with nonvanishing gaussian curvature.
What book a good introduction to differential geometry. Surfaces math 473 introduction to differential geometry. Minimal surfaces are defined within the language of differential geometry as surfaces of zero mean curvature. Differential geometry of curves the differential geometry of curves and surfaces is fundamental in computer aided geometric design cagd. R3 is minimal if and only if it can be locally expressed as the graph of a solution of. Evidently, fxd is contained in m, so the definition of surface in r 3 is satisfied. Differential geometry spring 2010 this course will present an introduction to differential geometry of curves and surfaces in 3space. For gaussbolyailobachevsky space, the gaussian curvature is. The terminology of complex derivatives in section 22. We will only consider minimal surfaces in threemanifolds, this is both for. Topics to be covered include first and second fundamental forms, geodesics, gaussbonnet theorem, and minimal surfaces.
Minimal surfaces can be defined in several equivalent ways in r 3. Basics of euclidean geometry, cauchyschwarz inequality. The case of minimal surfaces in \\mathbbr 3\ is a classical subject. Browse other questions tagged differential geometry minimal surfaces or ask your own. Differential geometry, branch of mathematics that studies the geometry of curves, surfaces, and manifolds the higherdimensional analogs of surfaces.
914 515 170 340 588 1477 451 747 1360 1032 652 867 695 859 360 1363 337 634 1609 1211 565 137 192 865 94 1498 151 9 1324