Ma 157a, Introduction to Riemannian Geometry

Winter 2004

Course Description

Ma 157a is an introductory course in Riemannian geometry. The course will begin with an overview of Riemannian manifolds including such basics as geodesics, curvature, and the exponential map. As examples, the course will emphasize things like spaces of constant curvature (Euclidean, spherical, and hyperbolic geometry), Grassmanians, Lie groups, and symmetric spaces. Then the course will cover topological and geometric consequences of curvature such as the Cartan-Hadamard theorem. Possible concluding topics include sketches of comparison geometry, Cheeger-Gromov compactness, or entropy characterizations of hyperbolic metrics (Mostow rigidity a la Besson-Courtois-Gallot). The course will be followed in Spring by Ma157b which will cover more advanced topics in Riemannian geometry.

Note: This year, the curriculum of Ma151 and Ma157 have been changed. In particular, the material here used to be covered in Ma151c. Thus those who took Ma151c last year will probably not be interested in this course, though they might well be interested in Ma157b.


A background in the basic topology of smooth manifolds, e.g. Ma151a. The course will be designed so that it can be taken by those currently enrolled in Ma151b.


There will be weekly homework assignments that account for 60% of the course grade. The remaining 40% of the course grade will come from the final project. This will be a 6-10 page paper written on a topic in Riemannian geometry. The topic will be chosen in consolation with me in the 5th or 6th week of class. For ideas, you can consult Berger's Panorama listed below. The final project will be due on the last day of class, Wednesday March 10.


The main text for this course will be

Gallot, Hulin, and Lafontaine, Riemannian Geometry, Springer-Verlag, Universitext, Second Edition, 1990 or later

Unfortunately, this book is out of print. I will get something set up where you can buy a photocopy of the text (not sure what this will cost as Springer hasn't gotten back to me about the copyright clearance fee). Alternatively, you could by a used copy online somewhere (e.g. here). With regard to this, the Second Edition was reprinted several times, and so appears with different publishing dates (1990, 1993, 1994, and 1996 have been observed in the wild). If the published date is 1987, that's the first edition, which you probably want to avoid. There will also be a copy of this text on reserve on the first floor of Millikan.

Initially, I'll start by covering some basic facts about smooth manifolds: vector fields, tensors, etc. (how much will depend on the classes' background). As the main text covers these topics in a fairly condensed fashion, you may also want to consult the following excellent source, which is on reserve on the first floor of Millikan:

Boothby, An introduction to differentiable manifolds and Riemannian geometry, Academic Press.

Another great book on Riemannian geometry is

Berger, A Panoramic View of Riemannian Geometry, Spinger, 2003.

This is not a textbook which carefully covers foundations of the field, but an 800 page attempt to survey all of modern Riemannian geometry. It is a great place to see what Riemannian geometry is all about, and also to get further intuition about basic concepts (there are several hundred figures and innumerable examples). I suggest you look through this book to find the topic for your paper, and then use the bibliography to get more detailed sources.

A copy of this book will also be on reserve in Millikan; unlike the others, I've set it up with a 3 day loan period to give you sufficient time to look through it. Also, Berge's book only costs $70 which is quite reasonable considering its bulk.

There is another point of view one can take on Riemannian geometry which deemphasizes the role of differentiability and focuses on more intrinsically metric-space notions. In particular, it is possible to talk about a general path metric space with curvature bounded above or below. This point of view is based on comparing geodesic triangles in your metric space with triangles in model geometries like the Euclidean plane and the round 2-sphere. This is called Comparison Geometry, and I sometimes find this point of view more appealing and geometric than the traditional one. The following book is a nice elementary account of this

Burago, Burago, and Ivanov. A Course in Metric Geometry. AMS Graduate Studies in Mathematics, Vol 33, 2001.

Finally, some other standard texts are

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