THE SPECTRAL GEOMETRY OF THE HOPF FIBRATION

We study the spectral geometry of the Hopf fibration S-1 --> S-3 --> S -2 and determine the right invariant metrics on S-3 for which there ex ist eigenforms of the Laplacian on S-2 which pull back to eigenforms o f the Laplacian on S-3. We show that the pull-back of the volume form on S-2 can be an eigenform of the Laplacian on S-3 with non-zero eigen value. We show that if G --> P --> Y is a principal bundle with a bund le metric and that if H-1(G; C) = 0, then eigenvalues cannot change. T hus eigenvalues do not change for the fibrations SO(n - 1) --> SO(n) - -> S-n - 1 and SPIN(n - 1) --> SPIN(n) --> S-n - 1 if n greater than o r equal to 4. We also study the corresponding questions in the complex category for the fibration of the Hopf manifold S-1 x S-3 --> S-2.