Title of the course: Berger Spheres in Riemannian Geometry
Instructor: Assoc. Prof. Mustafa Kalafat
Institution: Bonn University
Dates: 16-21 September 2024
Prerequisites: Multivariable Calculus, Linear Algebra, Algebraic Curves, Riemannian Geometry (not a must but preferable)
Level: Graduate, advanced undergraduate
Abstract: Berger spheres are introduced by Marcel Berger. These are odd dimensional spheres, which are squashed in a direction. The shrinking direction can be provided by the circle action on the complex Euclidean space. In other words, it corresponds to shrinking or expanding the Hopf circles in an odd dimensonal sphere. Consequently, the resulting Riemannian metric is different than the round metric. In these lecture series, we understand the eigenvalues of the Laplacian on Berger spheres.
Syllabus:
BS-1: Introduction, metrics on the 2-sphere.
BS-2: Berger sphere metrics.
BS-3: Spectrum of the complex projective space.
BS-4: Jacobi Fields on Spheres.
BS-5: Orthogonal, graded decomposition of the Berger Eigenspace.
BS-6: Nontriviality of Eigenspaces.
BS-7: Dimension counting for Eigenspaces.
BS-8: Dimension counting continued.
BS-9: First eigenvalue of the Laplacian on Berger Spheres.
We will be using the following resources.
References:
1. Kühnel, Wolfgang – Differential geometry. Curves,surfaces,manifolds. Third edition. Translated from the 2013 German edition. American Mathematical Society. 2015.
2. Riemannian Geometry. Sylvestre Gallot, Dominique Hulin, Jacques Lafontaine Springer. 2004.
3. Tanno, Shûkichi. The topology of contact Riemannian manifolds. Illinois J. Math. 12 (1968), 700–717.
4. Tanno, Shûkichi. The first eigenvalue of the Laplacian on spheres. Tohoku Math. J. (2) 31 (1979), no. 2, 179–185.
Language: EN