MATHEMATICS 6170 Introduction to Riemannian Geometry Fall semester 2003

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Riemannian Geometry: A Modern Introduction, by Isaac Chavel, ISBN 0-521-48578-9

Riemannian Geometry

by M.P. Do Carmo, ISBN 3-7643-3490-8 MWF 10:45-11:35 NS 202

instructor:

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Prof. Nick Korevaar

LCB 204 581-7318 korevaar@math.utah.edu M 2-2:50 p.m., T 10:15-11:15 a.m., 2-2:50 p.m. W 2-2:30 p.m., F 8:30-9:30 a.m., or arranged

6170 home page: www.math.utah.edu/ ∼korevaar/6170fall03

prerequisites: The ideal prerequisite for Math 6170 is a course on differentiable manifolds, like Math 6510 in our catalog. Alternate prerequisites for motivated students are the equivalent of both Math 4530 - Euclidean Curves and Surfaces, and Math 5520 - Introduction to Algebraic and Geometric Topology. The zeroth chapter of Do Carmo is a 35 page review of differentiable manifold essentials.

course outline: Riemannian Geometry is the geometry of curved spaces - precisely the study of differentiable manifolds which also have a Riemannian metric - an inner product for tangent vectors. This infinitesimal inner product yields infinitesimal elements of arclenth, area and volume and you can integrate these to compute curve lengths, surface areas, volumes. The metric also yields a natural way to differentiate vector fields along curves, called the Riemannian connection. Riemannian manifolds have a natural metric space structure: define the distance between points to be the length of the shortest curve (geodesic) connecting them. So, basically, you can do Calculus and analysis on Riemannian Manifolds.

The local geometry of Riemannian manifolds is controlled by the Riemann curvature tensor - for example it affects volume growth of balls, whether geodesics bend towards or away from each other, the interior angles of geodesic triangles, and it measures by how much second derivatives fail to commute. You’ve probably seen the effects of this tensor in two dimensions, where it is equivalent to Gauss curvature, and where its effects are well-exhibited in the constant curvature spaces - the sphere, Euclidean space, and hyperbolic space.

The first part of Math 6170 will expand on the local geometry ideas sketched in the preceding two paragraphs, chapters 1-4 of Chavel. In the second part of the course we will study global results - topological, geometric and analytic properties of manifolds which arise from pointwise curvature conditions. These will mostly be drawn from the later chapters of Chavel and Do Carmo. For example, Preissman’s Theorem says that the only non-trivial abelian subgroup of the fundamental group of a negatively curved compact manifold is the integers Z. Another example is the sphere theorem: If the Riemannian sectional curvature of a compact manifold is bounded strictly between 1/4 and 1, then the manifold is topologically an n-sphere. This theorem is sharp.

grading: Homework will be assigned regularly. You will provide written solutions, and we may have a weekly problem session as well at which these can be discussed. You will also be responsible for a final project - the presentation to the class of an interesting paper or result from Riemannian geometry. If you already know your area of specialization we may be able to find a paper in that area which uses a Riemannian setting. Otherwise, there are a number of global theorems which are accessible.