A piece of a Delaunay surface
Geometry
110.645, MTW 11, Maryland 202
Dr Mark Haskins
E-mail address: mhaskin@math.jhu.edu
Telephone: 410-516-4047
Department: Mathematics
Office: Krieger 312
Office Hours: M 9.30-10.30, T&W 3-4
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A piece of a Delaunay surface
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Riemannian Geometry 110.645, MTW 11, Maryland 202 Dr Mark Haskins E-mail address: mhaskin@math.jhu.edu Telephone: 410-516-4047 Department: Mathematics Office: Krieger 312 Office Hours: M 9.30-10.30, T&W 3-4 |
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HW 1: due Weds Feb 7. Postscript, PDF or DVI version. HW 2: due Mon Feb 19. Postscript, PDF or DVI version. |
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Your grade for this class will have two components each worth 50%. I. Homework: there will be homework assignments due roughly each week. I encourage people to talk about the homework; however, everyone must turn in their own assignment. Homework assignments will be available on this webpage. II. Project: there will be a project due roughly at the end of the semester. The project will involve both writing a paper and giving a talk on a subject related to the material of the course. I will give some advice about possible subjects, but you will ultimately choose the subject. You are not expected to undertake original research for the paper. There is some possibility of being able to do a group project. Further details will be announced. |
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The main text for the course is "Riemannian Geometry" by Gallot, Hulin and Lafontaine (Second Edition) published by Springer. Unfortunately this book is currently out of stock at the publishers with no immediate plans for a reprinting. Photocopies of the first 30 pages will be handed out on the the first class day. Copies of the complete book should be available from Printing Services in the basement of Garland Hall sometime during the first week of class. Another useful text is the lecture notes of Karsten Grove, "Riemannian Geometry: A Metric Entrance". The course will probably start off following Grove's presentation. I will order copies of these from the University of Aarhus during the first week of class for those who want a copy. There are many other useful books for Riemannian geometry and for background information on smooth manifolds and differential topology. For more information on smooth manifolds try the books by M.W. Hirsch, Guillemin & Pollack and F.W. Warner. For the classical differential geometry of curves and surfaces in 3-space a good source is "Differential Geometry of Curves and Surfaces" by Manfredo do Carmo. To help visualize curves and surfaces Mathematica can be quite helpful. The book "Modern Differential Geometry of Curves and Surfaces with Mathematica" by Alfred Gray is a very useful guide to exploring differential geometry via Mathematica. Other texts you might find helpful are: Do Carmo, "Riemannian Geometry", Chavel, "Riemannian Geometry: A Modern Introduction" and Morgan, "Riemannian Geometry". |
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This course is intended as an introduction at the graduate level to the venerable subject of Riemannian geometry. Since this is already a mature subject we will only scratch its surface. The goal rather is to equip you with the basic tools and provide you with some sense of direction so that you can go on to make your own exploration of this beautiful subject. Some of the basic notions in Riemannian geometry include: connections, covariant derivatives, parallel transport, geodesics and curvature. We will also need to say something about the standard modern setting for global Riemannian geometry which is to say -- smooth manifolds -- and what kinds of structures are instrinsic to them (smooth maps, tangent bundle, cotangent bundle, differential forms among these). Following that one finds a rich interaction between the topology of a smooth manifold (a global property) and the kinds of Riemannian metrics they admit (a local property) -- the simplest examples being the theorems of Myers and Cartan. Also central to geometry this century has been the relation between analysis on manifolds (for example properties of the Laplace operators) and their topology and geometry. We will see some basic examples of this kind of interaction. The precise structure of the course will, however, be influenced by the background and interests of the class. In particular, how much discussion of smooth manifolds occurs in class will depend on the need for it. |
| Last updated 28 January 2001 Mark Haskins |