Ma 157a, Introduction to Riemannian Geometry

Winter 2007

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. The course will conclude with a brief sketch of some more advanced topic, such as comparison geometry, hyperbolic structures on surfaces (Teichmüller space), or Perelman's proof of the Poincaré conjecture via Ricci flow. The course will be followed in Spring by Ma157b which will cover more advanced topics in Riemannian geometry.

Note: This is typically a quite small class, and contents will be tailored to suit the final audience. If you are interested in taking this course but can't make the time slot, please let me know; I will try to change the time to accommodate as many students as possible, subject to my own schedule and room availability.


A background in the basic topology of smooth manifolds (e.g. Ma 109c), as well as an understanding of the fundamental group from algebraic topology (e.g. Ma 109a or Ma 151a). More advanced knowledge of algebraic topology (e.g. Ma 151bc) is not needed.


Grades will be based on weekly homework assignments. In particular, there will be no exams in this course. The weekly homework will be due in class on Fridays, at the beginning of class. (Here "beginning of class" will be interpreted generously, say up to 10 minutes after I start, so no need to rush if your running a bit late.) Homework turned in after that, up through the beginning of class on Monday, will be graded but only count for half-credit. Beyond this, late homework will not be accepted, except in certain extreme circumstances, in which case you must ask me for an extension prior to the due date. However, your lowest homework grade will be dropped. If you won't be in class on Friday, turn your homework into my box outside the math department office at least 15 minutes prior to the start of class; late-but-not-too-late HW can be put in the same place.


I will not be following any particular text closely, and there is no required text for this course. Here are three sources that I'll be drawing on, all of which will be on reserve at Millikan library.

Initially, I'll start by covering some basic facts about smooth manifolds: vector fields, tensors, etc. (how much will depend on the classes' background). An additional source for this is:

Another great book on Riemannian geometry is

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).

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

Homework Assignments

HW #7: Due Friday, March 9. PDF.

HW #6: Due Friday, March 2. PDF.

No HW due Friday, Feb 23.

HW #5: Due Friday, Feb 16. PDF.

HW #4: Due Friday, Feb 9. PDF.

HW #3: Due Friday, Feb 2. PDF.

HW #2: Due Friday, January 19

Do any 5 of the following 9 problems from GHL.

2.11(a). 2.12(a, b). 2.25(b, c, d). 2.55(a,b). 2.57(b).

Notes: For 2.25(c), the second sentence should read: "Show that R2/G is diffeomorphic to the Klein bottle…". For problem 2.55(c), a clean answer requires the notion of the gradient grad f of a smooth function f on a Riemannian manifold (M, g). The gradient corresponds to the exterior derivative df under the isomorphim of TM to T*M induced by the metric g. That is, at a point p in M, for each v in TpM we have g( grad f, v) = df(v) = v(f).

HW#1: Due Friday January 12

Do any 4 of the exercises from the first chapter of Gallot-Hulin-Lafontaine, your choice. The exercises are spread throughout the text. Though they are often enumerated a), b), c) they are distinct problems and are counted as such toward your total of 4 problems.

This will not be a typical assignment; in future sets I will follow the usual path of assigning particular problems but give the diversity of the classes backgrounds this seems the best thing to do for this assignment.

Lecture notes:

Scans of lecture notes (PDF format): Part I (2.2Mb) and Part II (1.6Mb).

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