Bounds on Sectional Curvature and Interactions with Topology

In this survey article we exemplarily illustrate implications of curvature assumptions on the topology of the underlying manifold. We shall mainly focus on sectional curvature and three different kinds of restrictions, namely on non-negative respectively on positive sectional curvature, as well as on two-sided curvature bounds.We shall see that there are various implications on the side of topology, namely, for example, geometry having an impact on elementary invariants like the Euler characteristic or Betti numbers as well as on concepts from rational homotopy theory or index theory, and that there are connections to K-theory.On our way of making these connections we shall draw on certain simplifications and tools like group actions or metrics with additional properties like geometric formality.

On geometric formality of rationally elliptic manifolds in dimensions 6 and 7

We discuss the question of geometric formality for rationally elliptic manifolds of dimension 6 and 7. We prove that a geometrically formal six-dimensional biquotient with b2 = 3 has the real cohomology of a symmetric space. We also show that a rationally hyperbolic six-dimensional manifold with b2 ? 2 and b3 = 0 can not be geometrically formal. As it follows from their real homotopy classification, the seven-dimensional geometrically formal rationally elliptic manifolds have the real cohomology of symmetric spaces as well.

On formal Riemannian metrics

Formal Riemannian metrics are characterized by the property that all products of harmonic forms are again harmonic. They have been studied over the last ten years and there are still many interesting open conjectures related to geometric formality.The existence of a formal metric implies Sullivan's formality of the manifold, and hence formal metrics can exist only in presence of a very restricted topology.In this paper we give an overview over the present state of research on geometrically formal manifolds, with emphasis on the recent results obtained by the author together with Liviu Ornea in [11]. We are mainly interested in the topological obstructions to the existence of formal metrics. Moreover, we discuss natural constructions of formal metrics starting from known ones