SHAPE OPERATORS OF EINSTEIN HYPERSURFACES IN INDEFINITE SPACE FORMS

The possible shape operators for an Einstein hypersurface in an indefi- nite space form are classified algebraically. If the shape operator A is not diagonal- izable then either ^2-0or^2 = -?>2Id. Introduction. In (F) A. Fialkow classifies Einstein hypersurfaces in indefinite space forms, if the shape operator is diagonalizable at each point. He calls such an immersion proper (p. 764). This paper investigates what happens if the immersion is improper, i.e., if the shape operator is not diagonalizable at a point. It is possible for such a shape operator to have complex eigenvalues or eigenvectors with zero length. The main tool is Petrov's classification of symmetric operators in an indefinite inner product space (P).