Rigidity and geometric inverse problems in Riemannian geometry
Lecturer: Marco Mazzucchelli
During the winter semester, 2021/22 Marco Mazzucchelli from the University of Lyon will give a course on "Rigidity and geometric inverse problems in Riemannian geometry"
- Monday 10:15 - 11:45, room IA1/75
- Tuesday 10:15 - 11:00, room IA1/71
Description:
• Background from Riemannian geometry: convex Riemannian manifolds, symplectic geometry of the
tangent bundle, geodesic flows, curvature, conjugate points.
• Hopf theorem: Riemannian 2-tori without conjugate points are flat.
• Rigidity problems involving geodesics: boundary rigidity, lens rigidity, length spectrum rigidity, the
geodesic X-ray transform
• Santalo formula and Croke theorem: boundary rigidity of simple Riemannian n-balls within a
conformal class of Riemannian metrics.
• Croke-Otal theorems: boundary rigidity for negatively curved Riemannian 2-balls, marked-length
spectrum rigidity for negatively curved closed surfaces.
• Pestov identity and Paternain-Salo-Uhlmann theorem: injectivity of the X-ray transform of simple
Riemannian 2-balls.
• The Dirichlet-to-Neumann map and the Calderon problem in dimension 2
• Hilbert transform, and Pestov-Uhlmann theorem: boundary rigidity for simple Riemannian 2-balls
Syllabus:
| October 11: | Introduction to inverse problems: hearing the shape of a drum The boundary rigidity problem |
|---|---|
| October 12: | The X-ray transform on symmetric tensors |
| October 18: | Kernel of the X-ray transform and boundary rigidity by deformation |
| October 19: | Determination of the boundary jet of a metric from the boundary distance |
| October 25: | The lens rigidity problem Symplectic and Riemannian geometry of tangent bundles |
| October 26: | More symplectic and Riemannian geometry of tangent bundles |
| November 2: | The Santalò formula |
| November 8: | Boundary rigidity within a conformal class of simple Riemannian metrics |
| November 9: | The unit tangent bundle of Riemannian surfaces Commutator relations |
| November 22: | Pestov identity |
| November 23: | Injectivity of the X-ray transform on 0-tensors |
| November 29: | Injectivity of the X-ray transform on 1-forms |
| November 30: | The Laplace-Beltrami operator |
| December 6: | The Dirichlet-to-Neumann map The Hilbert transform Pestov-Uhlmann identity |
| December 7: | The dual X-ray transform |
| December 13: | Proof of Pestov-Uhlmann theorem |
Literatur:
C. Croke. Rigidity for surfaces of nonpositive curvature. Comment. Math.
Helv. 65 (1990), no. 1, 150-169.
J.-P. Otal. Sur les longueurs des geodesiques d'une metrique a courbure negative dans le disque.
Comment. Math. Helv. 65 (1990), no. 2, 334-347.
G. P. Paternain. Geodesic flows. Progress in Mathematics, 180. Birkhäuser Boston, Inc., Boston, MA,
1999.
G. P. Paternain, M. Salo, G. Uhlmann. Tensor tomography on surfaces. Invent. Math. 193 (2013), no.
1, 229-247.
L. Pestov, G. Uhlmann. Two dimensional compact simple Riemannian manifolds are boundary
distance rigid. Ann. of Math. (2) 161 (2005), no. 2, 1093-1110.
A. Wilkinson. Lectures on Marked Length Spectrum Rigidity. IAS/Park City Mathematics Series
Volume 21, 2012.
