Tuesday 12-14, Endenicher Allee 60, SR 1.008
Wednesday 16-18, Endenicher Allee 60, SR 1.008
Final Exam dates:
February 5-6
March 26-27
Week 1 (Oct 9-10):
Curvature for surfaces in R^3,
examples of abstract surfaces (flat torus and hyperbolic surface),
completeness (exponential map, Hopf-Rinow),
curvature (covariant derivatives, geodesics, sectional curvature),
covering spaces ((non)-examples, characterization as local diffeomorphisms with path lifting property),
proof of Cartan-Hadamard theorem.
Week 2 (Oct 16-17):
Convexity of the distance function on Hadamard manifolds,
comparison theorems (Toponogov triangle comparison, and volume comparison),
strict convexity of balls,
closest point projections (euclidean and hyperbolic examples, contraction property),
finding flat rectangles using closest point projections,
parallel geodesics bound flat strips.
Week 3 (Oct 23-24):
Centers of compact sets in Hadamard manifolds,
torsionfreeness of fundamental groups,
translation axes,
minsets,
isometric splitting for union of parallels (differential geometry vs metric space arguments),
flat torus theorem,
discussion of HW1.
Week 4 (Oct 30-31):
Heisenberg manifold doesn't have nonpositively curved metric,
nilpotent groups,
Isometries of H^2 and fixed points on the closed unit disk,
Geodesic ray compactification (description and identification with a closed n-ball),
elliptic-hyperbolic-parabolic classification of isometries of Hadamard manifold,
special groups of isometries (preserving points, lines, and points at infinity),
H^2 x H^2 example.
Week 5 (Nov 6-7):
Horofunction compactification (description, identification with geodesic ray compactification),
Horoballs/horospheres (examples),
Busemann functions (different characterizations),
Parabolic isometries preserve a horosphere,
infimum displacement via orbits (subadditive function lemma, convexity trick),
orbits of hyperbolics,
orbits of parabolics with positive inf displacement (sublinear tracking, two limit points at infinity).
Week 6 (Oct 13-14):
Construction of closed hyperbolic manifolds (Hyperboloid model, quadratic forms),
compactness criteria,
dealing with torsion,
Guest lecture by Tam Nguyen Phan: "The almost negatively curved 3-sphere" (if you missed it, she is giving the same talk at 3pm on Wednesday 21st in 2.008.)
Week 7 (Oct 20-21):
Hilbert modular surfaces (units, cusps, flat tori),
Branched covers (examples and constructions),
Gromov-Thurston manifolds. (link).
Week 8 (Oct 27-28):
Almost nilpotent groups of isometries,
Bieberbach Theorem,
Margulis Lemma,
Application: Lower bound on volume of -1<K<0 manifolds.
Week 9 (Dec 4):
Guest lecture by Tam Nguyen Phan: "Nash's C^1 isometric embedding theorem''. Here is Nash's paper and also Gromov's recent take on it.
Week 10 (Dec 11-12):
More Margulis lemma,
Bounding the number of generators for the fundamental group,
Gromov finiteness theorem (statement and examples),
proof of Gromov finiteness theorem (small tubes, modifying the injectivity radius function).
Week 11 (Dec 18-19):
Linear bounds on Betti numbers in terms of volume,
packing arguments, doubling trick,
nerves, convex covering lemma (look in 4G.3 of Hatcher if you want to see the algebraic topology version).
The symmetric space for SL(n,R) (description via positive definite symmetric, determinant one matrices and as SL(n,R)/SO(n)) following notes of Rick Schwartz,
totally geodesic flats in SL(n,R)/SO(n).
Week 12 (Jan 8-9):
Statement of rank rigidity and motivation,
the angle metric and the l-metric on the ideal boundary (basic properties and comparison),
construction and uniqueness of geodesics in the angle metric,
the angle metric has “curvature =<1”.
References: Chapter 4 of BGS or II.4 of Ballmann’s lectures. Also see p. 285-289 of Bridson-Haefliger.
Week 13 (Jan 15-16):
Centers for subsets of the sphere,
centers for subsets of ideal boundary with the angle metric,
fix sets of parabolic isometries are star-shaped in the angle metric, have canonical centers at which horospheres are preserved by the parabolic.
The unit tangent bundle, geodesic flow, (non)-wandering vectors (III.1 of Ballmann's lectures).
Visibility manifolds (definition, K<-epsilon<0 Hadamard manifolds are visibility manifolds).
Week 14 (Jan 22-23):
Transitivity properties of the geodesic flow, Eberlein's theorem (for finite volume visibility manifolds, geodesic flow is topologically transitive),
Jacobi equation, variation through geodesics, parallel Jacobi fields, rank of a vector/manifold,
Integrating infinitesimal flats to actual flats.
For examples of interesting (not products or symmetric spaces) higher rank manifolds in non-negative curvature, see this paper by Spatzier and Strake.
See these lectures (I,II,III) by Keith Burns for the the tangent bundle to the tangent bundle, the Liouville measure, geodesic flow (I:16:20-47), its differential via Jacobi fields (II: 18:40-34) and more (e.g. ergodicity).
Week 15 (Jan 29-30):
These are optional and will not be graded, but doing them (or at least thinking about them) should help you follow the class.
Homework 1
Homework 2
Homework 3
Homework 4
Homework 5
Homework 6
Homework 7
All surfaces, except for the sphere and projective plane, have complete nonpositively curved metrics. In higher dimensions, the nonpositively curved manifolds form a diverse yet tightly constrained class of spaces, including such things as hyperbolic manifolds and moduli spaces of flat, volume one tori. These spaces can be studied from geometric, analytic, algebraic, dynamical and arithmetic points of view. There are many different nonpositively curved manifolds (locally symmetric spaces, as well as manifolds constructed via branch cover and hyperbolization methods) yet, at the same time, the nonpositive curvature puts strong constraints on their topology.
In the first half of the course, we will introduce the basic tools for studying nonpositively curved manifolds (and more general nonpositively curved spaces). These include triangle comparison, volume comparison, closest point projections to convex sets and the Cartan-Hadamard theorem. All these follow from convexity of distance functions in nonpositively curved geometry. Next, we will introduce the ideal boundary of a simply connected nonpositively curved manifold (more concisely called a Hadamard manifold), describe it in several ways using geodesic rays and also Busemann functions, define the Tits metric on the ideal boundary and derive its basic properties. Then, we describe how the classification of isometries of the hyperbolic plane into three types (hyperbolic, parabolic, and elliptic isometries) generalizes to Hadamard manifolds. All results described so far are consequences of nonpositive curvature (an upper curvature bound). In the presence of a lower curvature bound (K>-1) one also knows that isometries that move a given point by a small amount are well understood (more precisely, they generate a virtually nilpotent group). This is known as the Margulis lemma, and it is key to understanding the ''thin'' part in the thick-thin decomposition of a nonpositively curved manifold.
In the second half of the course, we will cover some results about the structure of closed and finite volume nonpositively curved manifolds. This will probably include some subset of the following: Gromov-Schroeder finiteness theorems, Betti number bounds in terms of volume, rigidity results for nonpositively curved manifolds.
During the course we will also describe various constructions of nonpositively curved manifolds, including locally symmetric (arithmetic) manifolds, Gromov-Thurston branched covers, and maybe manifolds obtained via hyperbolization and reflection group techniques.
We will primarily follow the book
Manifolds of nonpositive curvature by Ballmann, Gromov, and Schroeder.
A good reference for arithmetic constructions of nonpositively curved manifolds (and many other things) is
Introduction to Arithmetic groups by Witte-Morris available here.
A down-to-earth description of the symmetric space for SL(n,R) can be found in
The symmetric space for SL(n,R) by Rick Schwartz, available here.
A good reference that covers rank rigidity (and also nonpositively curved spaces more general than manifolds) is
Lectures on spaces of nonpositive curvature by Ballmann available here.
For an illuminating discussion of Berger's holonomy theorem (and an introduction to many, many other things about Riemannian geometry) see chapter 13 (especially p. 643-645) of Berger's
A panoramic view of Riemannian Geometry by Berger.
Differential geometry: Smooth manifolds, tangent spaces, vectors fields.
The following bits of Riemannian geometry and topology are good to have as background for the class.
Riemannian geometry: Riemannian metric, curvature, geodesics, completeness. Chapters 8-10 of Morse Theory by Milnor are a good place to read up on this. Comparison Theorems in Riemannian Geometry by Cheeger and Ebin is also a good source (containing much more than we need).
Topology: fundamental group and covering spaces (e.g. Chapter 1 of Algebraic Topology by Hatcher available here).