On the Finite Spectral Triple of an Almost-Commutative Geometry

In this short communication, we examine the relevance of the signature of the space-time metric in the construction of the product of a pseudo-Riemannian spectral triple with a finite triple describing the internal geometry. We obtain arguments favoring the appearance of SU(2) and U(1) as gage groups in the standard model.     

Jacobi vector fields and the volume of tubes about curves in Sasakian space forms

Summary We solve explicitly the Jacobi equation on a Sasakian space form and use the solutions to consider explicit formulas for the volume of a geodesic sphere and the volume of a tube about a curve in such a manifold. We obtain some generalizations of H. Weyl's result on the volume of tubes.     

Homogeneity and Curvatures of Geodesic Spheres

We study some scalar curvature invariants on geodesic spheres and use them to characterize several kinds of Riemannian manifolds such as homogenous manifolds and in particular, the two-point homogeneous spaces and the Damek-Ricci spaces.     

ω-Kongruenzen von Vincensini und Verallgemeinerungen der Kongruenzen mit einer Minimalfläche als Mittelhüllfläche

Ohne Zusammenfassung     

On disk-homogenous symmetric spaces

We prove a classification theorem for disk-homogeneous locally symmetric spaces.     

Minimal and Harmonic Unit Vector Fields in and Its Dual Space

 The complex two-plane Grassmannian carries a K?hler structure J and also a quaternionic K?hler structure ?. For we consider the classes of connected real hypersurfaces (M, g) with normal bundle such that and are invariant under the action of the shape operator. We prove that the corresponding unit Hopf vector fields on these hypersurfaces always define minimal immersions of (M, g), and harmonic maps from (M, g), into the unit tangent sphere bundle with Sasaki metric . The radial unit vector fields corresponding to the tubular hypersurfaces are also minimal and harmonic. Similar results hold for the dual space . (Received 27 August 1999; in revised form 18 November 1999)