Curvature invariants, differential operators and local homogeneity

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

     

Die Erzeugung und Bewegung einzelner Versetzungen in Germaniumkristallen

Topographic X-ray pictures are presented showing dislocation sources in plastically deformed germanium crystals. Both internal Frank-Read sources as well as sources at the crystal surface have been observed. Dislocation velocities from 10^?6 to 10^?4 cm/sec have been evaluated.     

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

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.     

On the spectrum and the geometry of the spherical space forms,I

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 , where the sum runs through all spec(M).     

Geodesic spheres and two-point homogeneous spaces

In the Osserman conjecture and in the isoparametric conjecture it is stated that two-point homogeneous spaces may be characterized via the constancy of the eigenvalues of the Jacobi operator or the shape operator of geodesic spheres, respectively. These conjectures remain open, but in this paper we give complete positive results for similar statements about other symmetric endomorphism fields on small geodesic spheres. In addition, we derive more characteristic properties for this class of spaces by using other properties of small geodesic spheres. In particular, we study Riemannian manifolds with (curvature) homogeneous geodesic spheres. Supported by the Akademie der Naturforscher Leopoldina.     

Studium von Einzelversetzungen in nahezu idealen Kristallen mit einer Röntgenmethode

The integrated intensities of X-ray diffraction for theLaue and theBragg case have been calculated for both an ideal and a mosaic crystal. The ratio of both intensities is thought to be a measure for the contrast of topographic X-ray work. Concerning to the Laue case, in thin crystals the distorted regions around dislocations have a higher diffraction intensity than the undistorted regions. In thick crystals the contrast is reverted. That is the case of anomalous transmission in ideal crystals. In the region of anomalous transparency some experiments have been made with slices of germanium crystals containing only few dislocations. Topographic pictures have been made by a simple method which is described. Secondary effects have been observed besides the normal contrast effects between distorted and undistorted regions. They can be explained assuming that the direction of propagation of the transmitting wave field is shifted in regions containing a long range lattice distorsion.     

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.