Friedbert Prüfer

Curvature invariants, differential operators and local homogeneity

We first prove that a Riemannian manifold (M, g) with globally constant additive Weyl invariants is locally homogeneous. Then we use this result to show that a manifold (M, g) whose Laplacian commutes with all invariant differential operators is a locally homogeneous space.

D'Atri spaces

Riemannian symmetric spaces play an important role in many fields of mathematics and physics. They have been studied by several generations of mathematicians and from different viewpoints. Their classification is well-known. Usually they are characterized as Riemannian manifolds whose geodesic symmetries (that is, geodesic reflections with respect to all points) are globally defined isometries. Hence, these geodesic symmetries are also volume-preserving. This observation led D’Atri and Nickerson to the study of Riemannian and pseudo-Riemannian manifolds all of whose (local) geodesic symmetries are volume-preserving (up to sign) or equivalently, which are divergence-preserving. They are introduced in [DN1], [DN2], [D] where the first non-symmetric examples, namely all naturally reductive homogeneous spaces, are found and where various other characterizations are given. Following [VW1] such spaces are called D’Atri spaces.

SYMMETRIC-LIKE RIEMANNIAN MANIFOLDS AND GEODESIC SYMMETRIES

We treat several classes of Riemannian manifolds whose shape operators of geodesic spheres or Jacobi operators share some properties with the ones on symmetric spaces.

On probabilistic commutative spaces

A new characterization is given for the “probabilistic commutative” Riemannian spaces studied originally (in the compact case) by Roberts and Ursell. Moreover, some earlier basic results are generalized to the non-compact case.

A Classification of Special Riemannian 3-Manifolds with Distinct Constant Ricci Eigenvalues

All Riemannian 3-manifolds with distinct constant Ricci eigenvalues and satisfying some additional geometrical conditions are classified in an explicit form. One obtains locally homogeneous spaces and two different classes of locally non-homogeneous spaces in this way.

On the spectrum and the geometry of a spherical space form II

In this paper we study some relations between the spectrum and the lengths of the closed geodesics of a Riemannian manifold of positive constant sectional curvature 1. Our topic is the development of a Poisson formula for such space forms. Further we obtain explicit results for the lengths of the closed geodesics. We conclude the paper with a result concerning the singular support of the distribution Σ\(\Sigma \cos \sqrt {\mu + (n - 1){\raise0.7ex\hbox{

Geodesic spheres and two-point homogeneous spaces

2

On Riemannian 3-manifolds with distinct constant Ricci eigenvalues

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Curvature tensors in dimension four which do not belong to any curvature homogeneous space

4

Totally geodesic submanifolds of symmetric-like Riemannian manifolds

}}} \cdot\), where the sum runs through all μ e spec(M).In this paper certain relations between the numerical coefficients of the Poisson formula of Part I and geometrical data of a spherical space form M, dim M = 2m+1 are investigated. The results yield an explicit relation between the spectrum of M and the Poincaré map of certain closed geodesics of M. Furthermore, explicit formulas for the multiplicities of the eigenvalues of the Laplacian of M are derived by means of the Poisson formula. At the end of the paper the information about M is examined which is contained in a finite part of spec(M). A partial answer is given in the Corollaries 3 and 6.