MATH 538-001 44849 Riemannian Geometry II,

Time TTh 1230-1345, Location SARAR-107.


Instructor: Dimiter Vassilev     Office :  SMLC, Office 326  Email:  Phone Number: 505 277 2136


Office Hours: Tuesday & Thursday 2:00pm-3:00pm. You are also welcome to stop by anytime you have a question.


Description: Continuation of MATH 537 with emphasis on adding more structures. Comparison theorems in Riemannian geometry, Riemannian submersions, Bochner theorems with relation to topology of manifolds, Riemannian Foliations, Complex and Kaehler geometry, Sasakian and contact geometry. Prerequisite: 537.

HOMEWORK: Homework and course related material will be posted on UNMLearn. The general rule is that you should do the homework problems in order to maximize what you learn in the class.

ATTENDANCE: Attendance at UNM is mandatory, see policy.

Textbooks: The following books will be useful references for a large part of the covered material.

Schedule and homework per week

  1. Sep. 15: Review- exponential map, geodesics, Gauss' lemma and metric balls. HW
  2. Jan. 22: no class on Tuesday. Properties of small balls. Uniform convex neigborhood. The Hopf-Rinow theorem. HW
  3. Jan 29: Finish the proof of the Hopf-Rinow's theorem and some corollaries for characterization and properties of minimal geodesics. Isometries and local isometries - properties and rigidity. Covering maps and local isometries. HW
  4. Feb 5: Criteria for a local isometry or exp to be a Riemannian covering. Riemannian covers and group actions. HW
  5. Feb 12: Orbit spaces. Isometry vs. distance preserving (Myers-Steenrod' theorem). HW
  6. Feb 19: Existence of closed (peridoic) geodesics. Riemannian submersions - warped products; horizontal and vertical sub-bundles, horizontal lifts . HW
  7. Feb 26: Geodesics and Riemannian submersions; covariant derivatives of horizontal lifts along horizontal lifts. The fundamental tensors of a Riemannian submersion. Minimal and totally geodesic sub-manifolds. HW
  8. Mar 5: (Ehrsmann) completeness for a submersion, horizontal lifts of curves in the base, properties of Riemannian submersions with complete total space - equidistance of fibers, locally trivial fibration (Hermann's theorem). Properties of Riemannian submersions with complete total spaces and totally geodesic fibers. HW
  9. Mar 12: SPRING BREAK
  10. Mar 19: Proof of Hermann's theorem for Riemannian submersions with totally geodesic fibers. Reading: see paper. The Hopf fibrartion as a Riemannian submersion with totally geodesic fibers and the fundamental tensors (of the submersion); the unit sphere as a CR sub-manifold. HW
  11. Mar 26: O'Neill and Grey formulas for the Curvatures of a Riemannian submersions. HW
  12. Apr 2: Almost complex structures, Hermitian and Kahler manifolds. The Kahler structure on CPn. HW
  13. Apr 9: Calculus of variations and geodesics. The first variation formula for the energy and length. The second variation formula of the energy. The Bonnet-Myers theorem. Synge's theorem. HW
  14. Apr 16: Jacobi vector fields. Spaces of constant curvature - Riemann and Killing-Hopf theorems. HW
  15. Apr 23: Jacobi fields and singular values of the exponential. The Cartan-Hadamard theorem. The index form and its reations to conjugate points. Basic index lemma. HW
  16. Apr 30: Proof of the basic index lemma. Failure of distance minimizing property of a geodesic past a conjugate point. The conjugate locus.