By Peter B Gilkey
A imperative challenge in differential geometry is to narrate algebraic houses of the Riemann curvature tensor to the underlying geometry of the manifold. the whole curvature tensor is generally particularly tricky to house. This ebook provides effects in regards to the geometric outcomes that stick with if a variety of typical operators outlined by way of the Riemann curvature tensor (the Jacobi operator, the skew-symmetric curvature operator, the Szabo operator, and better order generalizations) are assumed to have consistent eigenvalues or consistent Jordan common shape within the applicable domain names of definition. The ebook offers algebraic preliminaries and numerous Schur style difficulties; offers with the skew-symmetric curvature operator within the genuine and intricate settings and gives the type of algebraic curvature tensors whose skew-symmetric curvature has consistent rank 2 and incessant eigenvalues; discusses the Jacobi operator and the next order generalization and offers a unified remedy of the Osserman conjecture and comparable questions; and establishes the consequences from algebraic topology which are priceless for controlling the eigenvalue constructions. an intensive bibliography is equipped. effects are defined within the Riemannian, Lorentzian, and better signature settings, and plenty of households of examples are displayed.
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