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


• 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


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



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,
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.

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