For a Riemannian manifold M n with the curvature tensor R, the Jacobi operator RX is defined by RX Y = R(X, Y)X. The manifold M n is called pointwise Osserman if, for every p ∈ M n , the eigenvalues of the Jacobi operator RX do not depend of a unit vector X ∈ TpM n , and is called globally Osserman if they do not depend of the point p either. R. Osserman conjectured that globally Osserman manif...

An algebraic curvature tensor is called Osserman if the eigenvalues of the associated Jacobi operator are constant on the unit sphere. A Riemannian manifold is called conformally Osserman if its Weyl conformal curvature tensor at every point is Osserman. We prove that a conformally Osserman manifold of dimension n 6= 3, 4, 16 is locally conformally equivalent either to a Euclidean space or to a...

Let J be a unitary almost complex structure on a Riemannian manifold (M, g). If x is a unit tangent vector, let π := Span{x, Jx} be the associated complex line in the tangent bundle of M . The complex Jacobi operator and the complex curvature operators are defined, respectively, by J (π) := J (x) + J (Jx) and R(π) := R(x, Jx). We show that if (M, g) is Hermitian or if (M,g) is nearly Kähler, th...

We use the classical results of Baxter and Gollinski-Ibragimov to prove a new spectral equivalence for Jacobi matrices on l(N). In particular, we consider the class of Jacobi matrices with conditionally summable parameter sequences, and find necessary and sufficient conditions on the spectral measure such that ∑ ∞ k=n bk and ∑ ∞ k=n (a k − 1) lie in l 1 ∩ l .

Matrix-valued analogues of the little q-Jacobi polynomials are introduced and studied. For the 2 × 2-matrix-valued little q-Jacobi polynomials explicit expressions for the orthogonality relations, Rodrigues formula, three-term recurrence relation and its relation to matrix-valued q-hypergeometric series and the scalar-valued little q-Jacobi polynomials are presented. The study is based on the m...

Let Hω be a self-adjoint Jacobi operator with a potential sequence {ω(n)}n of independently distributed random variables with continuous probability distributions and let μωφ be the corresponding spectral measure generated by Hω and the vector φ. We consider sets A(ω) which are independent of two consecutive given entries of ω and prove that μωφ(A(ω)) = 0 for almost every ω. This is applied to ...

We present a new oscillation criterion to determine whether the number of eigenvalues below the essential spectrum of a given Jacobi operator is finite or not. As an application we show that Kenser’s criterion for Jacobi operators follows as a special case.

Let x : M → Sn+1(1) be an n-dimensional compact hypersurface with constant scalar curvature n(n − 1)r, r ≥ 1, in a unit sphere Sn+1(1), n ≥ 5, and let Js be the Jacobi operator of M . In 2004, L. J. Aĺıas, A. Brasil and L. A. M. Sousa studied the first eigenvalue of Js of the hypersurface with constant scalar curvature n(n− 1) in Sn+1(1), n ≥ 3. In 2008, Q.-M. Cheng studied the first eigenvalue...

This paper discusses hyperbolic behavior of Jacobi fields along billiard flows on multidimensional Reimannian manifolds. A class of generalized differential operators associated with the impulsive equations (generalized Jacobi equations) are defined using a new Radon measure. We investigate the hyperbolic behavior of functions in the null-space of the operator by applying operator theory.

We use methods from representation theory and invariant theory to compute differential operators invariant under the action of the Jacobi group over a complex quadratic field. This allows us to introduce Maass-Jacobi forms over complex quadratic fields, which are Jacobi forms that are also eigenfunctions of an invariant differential operator. We present explicit examples via Jacobi-Eisenstein s...