Let M n be a Riemannian manifold and R its curvature tensor. For a point p ∈ M n and a unit vector X ∈ TpM n , the Jacobi operator is defined by RX = R(X, ·)X. The manifold M n is called pointwise Osserman if, for every p ∈ M n , the spectrum of the Jacobi operator does not depend of the choice of X, and is called globally Osserman if it depends neither of X, nor of p. Osserman conjectured that...

We obtain a finite-sum representation for the general solution of the equation ∆ (p(n− 1)∆u(n − 1)) + q(n)u(n) = λr(n)u(n) in terms of a nonvanishing solution corresponding to some fixed value of λ = λ0. Applications of this representation to some results on the boundedness of solutions are given as well as illustrating examples.

We construct a family of pseudo-Riemannian manifolds so that the skew-symmetric curvature operator, the Jacobi operator, and the Szabó operator have constant eigenvalues on their domains of definition. This provides new and non-trivial examples of Osserman, Szabó, and IP manifolds. We also study when the associated Jordan normal form of these operators is constant. Subject Classification: 53B20.

Contents 1. Introduction 1 2. Geometrical setting near a dilated catenoid 6 3. Jacobi-Toda system on the Catenoid 9 4. Jacobi operator and the linear Jacobi-Toda operator on the catenoid. 16 5. Approximation of the solution of the theorem 1 23 6. Proof of theorem 1. 30 7. gluing reduction and solution to the projected problem.

We present a technique for reconstructing a semi-infinite Jacobi operator in the limit circle case from the spectra of two different self-adjoint extensions. Moreover, we give necessary and sufficient conditions for two real sequences to be the spectra of two different self-adjoint extensions of a Jacobi operator in the limit circle case. ∗Mathematics Subject Classification(2000): 47B36, 49N45,...

Let M be a compact (two-sided) minimal hypersurface in a Riemannian manifold M n+1 . It is a simple fact that if M has positive Ricci curvature then M cannot be stable (i. e. its Jacobi operator L has index at least one). If M = S(1) is the unit sphere and L has index one, then it is known that M must be a totally geodesic equator. We prove that if M is the real projective space RP = Sn+1(1)/{±...

We exhibit Walker manifolds of signature (2, 2) with various commutativity properties for the Ricci operator, the skew-symmetric curvature operator, and the Jacobi operator. If the Walker metric is a Riemannian extension of an underlying affine structure A, these properties are related to the Ricci tensor of A.

In this paper we give a non-existence theorem for Hopf hypersurfaces in the complex two-plane Grassmannian G2(C) with recurrent normal Jacobi operator R̄N .

We provide the mathematical foundation for the Xm-Jacobi spectral theory. Namely, we present a self-adjoint operator associated to the differential expression with the exceptional Xm-Jacobi orthogonal polynomials as eigenfunctions. This proves that those polynomials are indeed eigenfunctions of the self-adjoint operator (rather than just formal eigenfunctions). Further, we prove the completenes...