Rimeannian Geometry and Smooth Metric Measure Spaces
Student No.:50
Time:Fri 10:10-12:00, 2015-03-06~ 2015-05-22(except for May 1)
Instructor:Vlad Moraru  [MSC, Tsinghua University]
Place:Conference Room 3, Floor 2, Jin Chun Yuan West Building
Starting Date:2015-3-6
Ending Date:2015-5-22
The course will address some advanced topics in Riemannina geometry and their connections with measure theory. Emphasis will be put on some, by now, classical theorems concerning manifolds with lower bounds on the Ricci curvature, their applications to general relativity and recent generalisations to smooth metric measure spaces.
In the first part of the course we will deal with the Riemannian and Lorentzian setting, whereas in the second part we will study smooth metric measure spaces with Bakry-Emery Ricci tensor bounded from below. Some results from the first part of the course will be generalised in this setting.
Topics will include: Laplace comparison and applications; Bishop-Gromov volume comparison; Myer's theorem, Hawking and Penrose singularity theorems, and Cheng's maximal diameter rigidity; Cheeger-Gromoll splitting theorem for Riemannian and Lorentzian manifolds; Smooth metric measure spaces with Bakry-Emery Ricci curvature bounded below and generalisation of some of the theorems mentioned above; Singularity theorems for Bakry-Émery Lorentzian manifolds.
1)      A basic course in Riemannian geometry (such as, for example, J. M. Lee “Riemannian Geometry”, or M. do Carmo “Riemannian Geometry” especially Chapters 0 -11)
2)      A basic course in measure theory.
1) P. Petersen, “Riemannian Geometry”, especially Chapters 7 and 9.
2) Zhu, “The Comparison Geometry of Ricci Curvature” in Comparison Geometry MSRI Publications, Vol. 30, 1997
3) Wei & Wylie, “Comparison for the Bakry-Emery Ricci Tensor”, J. Diff. Geom. 83 (2009) 377-405
Additional references will be given during the lectures.