**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
manifolds of

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.