RIEMANNIAN GEOMETRY- TOPICS COVERED, SPRING 2000
1. The tangent bundle of a manifold; vector
fields. Connections, covariant derivative
along curves and maps. The geodesic flow. Exponential map.
2. Torsion-free connections. Partitions of unity,
existence of metrics and Levi-Civita
connection. Normal neighborhoods. Geodesics minimize locally. Gauss's lemma. Ex:
geodesics of the hyperbolic plane.
3. Riemannian manifolds as metric spaces. Hopf-Rinow theorem; complete manifolds.
4. Curvature tensor- definition, basic identities.
Sectional, Ricci and scalar curvature.
Einstein manifolds. Schur's lemma, Einstein implies constant scalar. Curvature tensor in
constant sectional curvature
5. Riemannian submanifolds- induced connection, second
fundamental form. The Gauss and
Codazzi curvature equations. Codimension-one case: "shape operator". Examples: sectional
curvatures of n-sphere and hyperbolic n-space (hyperboloid model). Totally geodesic submanifolds.
6. The isometry group of the hyperbolic plane.
Hyperbolic distance formulas. Elliptic,
hyperbolic, parabolic isometries. Conformal group of n-sphere= isometry group of hyperbolic
(n+1)-space. Iwasawa decompositions.
7. Jacobi equation and Jacobi fields. Constant curvature
examples; relationship with the
differential of exp. Taylor expansions of norm-squared of Jacobi fields. Conjugate points.
The Cartan- Hadamard theorem.
8. Cartan's theorem on curvature and local isometry. Simply-connected
constant sectional curvature. Non- simply- connected manifolds: covering metric,
properly discontinuous action of the fundamental group.
9. Energy minimizing and length-minimizing curves. First
and second variations of energy.
The index form on vector fields along a curve. Null eigenvectors= Jacobi fields vanishing at
endpoints. Geodesics do not minimize past first conjugate point.
10. Theorems of Bonnet-Myers and Synge-Weinstein.
Compact manifolds of positive curvature
(even-dimensional implies simply-connected).
11. The first cut point along a geodesic. Conjugate
locus and cut locus. Injectivity radius; lower
bound in positive curvature. Distance between conjugate points under given curvature bounds.
Compactness of the cut locus.
12. Rauch comparison theorem. Index lemma. Applications:
lengths of curves from curvature
estimates, geodesic triangles on Hadamard manifolds (law of cosines, angle inequalities).
Existence of closed geodesics in free homotopy classes (Cartan's theorem). Preissman's
theorem (fund. group in negative curvature).
13. Volume of geodesic balls/spheres in terms of Jacobi
fields on Hadamard manifolds.
Volume comparison theorem for geodesic balls; exponential volume growth in negative
curvature. Growth of balls in univ. cover vs. growth of fundamental group (no proofs) .
Volume growth for complete manifolds of non-negative Ricci (no proofs).
14. (optional class) Symmetric spaces; definition, symmetric
spaces are homogeneous,
basic examples. Locally symmetric spaces (= parallel curvature tensor, Cartan's theorem).
Lie groups with bi-invariant metrics: one-parameter subgroups are geodesics through the
identity. Levi-Civita connection and Lie bracket of left-invariant vector fields. The
curvature tensor and sectional curvatures.
15. (Homework sets)) I). Recovering connection from parallel
transport; complete manifolds and
lengths of divergent curves. II) Killing fields; Riemannian products III) geodesics on Hadamard
manifols are length-minimizing. Non-existence of lines on complete manifolds of K>0.