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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 seminar series, we understand the eigenvalues of the Laplacian on Berger spheres. 

The topic of this week is: 

    "Dimension counting for Eigenspaces 2."

We will be using the following resources.

References:

1. Tanno, Shûkichi. The topology of contact Riemannian manifolds.
   Illinois J. Math. 12 (1968), 700–717.


2. Tanno, Shûkichi. The first eigenvalue of the Laplacian on spheres.
   Tohoku Math. J. (2) 31 (1979), no. 2, 179–185.