Riemannian and pseudo-Riemannian symmetric spaces with semisimple transvection group are known and classified for a long time. Contrary to that the description of pseudo-Riemannian symmetric spaces with non-semisimple transvection group is an open problem. In the last years some progress on this problem was achieved. In this article we want to explain these results and some of their applications.

Let M be a compact smooth Riemannian man-ifold without boundary and let D = ad 0 0 + bb 1 d 1 ? E on the space of smooth sections of the cotangent bundle where a and b are positive constants and where E is an endomorphism. We use functorial methods and the pseudo-diierential operator calculus to compute the quadratic term a 4 (D) in the asymptotic expansion of the heat equation trace. MOS class...

In a study of Riemannian manifolds admitting conformal transformation, the Riemannian and Ricci curvatures play an important role for characterizations and classify such a manifold. In this note, we summarize known results about Riemannian manifold admitting conformal transformations and related topics.

The aim of this paper is to show that the holonomy group of a non-Riemannian Finsler manifold of constant curvature with dimension n > 2 cannot be a compact Lie group and hence it cannot occur as the holonomy group of any Riemannian manifold. This result gives a positive answer to the following problem formulated by S. S. Chern and Z. Shen: Is there a Finsler manifold whose holonomy group is no...

We show that mean curvature flow of a compact submanifold in a complete Riemannian manifold cannot form singularity at time infinity if the ambient Riemannian manifold has bounded geometry and satisfies certain curvature and volume growth conditions .

We prove Liouville theorems for Dirac-harmonic maps from the Euclidean space Rn, the hyperbolic space Hn and a Riemannian manifold Sn (n ≥ 3) with the Schwarzschild metric to any Riemannian manifold N .

Due to the growing interest in embeddings of space-time in higher-dimensional spaces we consider a specific type of embedding. After proving an inequality between intrinsically defined curvature invariants and the squared mean curvature, we extend the notion of ideal embeddings from Riemannian geometry to the indefinite case. Ideal embeddings are such that the embedded manifold receives the lea...