During these years, Riemannian Geometry has undergone many dramatic – velopments. Here is not Sylvestre Gallot, Dominique Hulin, Jacques Lafontaine . During these years, Riemannian Geometry has undergone many dramatic – velopments. Here is not By Sylvestre Gallot, Dominique Hulin, Jacques Lafontaine. Greene, Robert E. Review: S. Gallot, D. Hulin and J. Lafontaine, Riemannian geometry. Bull. Amer. Math. Soc. (N.S.) 21 (), no. 1,

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Sloan Office Phone: Or just drop by — even if I can’t talk to you then, we’ll set up a time to meet. 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 hulinn curvature Euclidean, spherical, and hyperbolic geometryGrassmanians, Lie groups, and symmetric spaces.

Then the course will cover topological and geometric consequences of curvature such as the Cartan-Hadamard theorem.

Greene : Review: S. Gallot, D. Hulin and J. Lafontaine, Riemannian geometry

The course will be followed in Spring by Mab which will cover more advanced topics in Riemannian geometry. 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 lafontxine my own schedule and room availability.

Prerequisites A background in the basic topology laafontaine smooth manifolds e.

Ma cas well as an understanding of the fundamental group from algebraic topology e. Ma a or Ma a. More advanced knowledge of algebraic topology e. Ma goemetry is not needed.


Grading 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 glalot Friday, turn your homework into my lzfontaine 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.

Texts 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: An additional source for this is: Boothby, An introduction to differentiable manifolds and Riemannian geometryAcademic Press.

Another great book on Riemannian geometry is. This is not a textbook which carefully covers foundations of the field, but an page attempt to survey all of modern Riemannian geometry.

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.

Riemannian Geometry by Sylvestre Gallot, Dominique Hulin, Jacques Lafontaine

The following book is a nice elementary account of this. Due Friday, March 9.

Due Friday, March 2. No HW due Friday, Feb Due Friday, Feb Due Friday, Feb 9. Due Friday, Feb 2. The exercises are spread throughout the text. Though they are often enumerated abc 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.