Let Γ be a discrete group, and let M be a closed spin manifold of dimension m > 3 with π1(M) = Γ. We assume that M admits a Riemannian metric of positive scalar curvature. We discuss how to use the L-rho invariant ρ(2) and the delocalized eta invariant η associated to the Dirac operator on M in order to get information about the space of metrics with positive scalar curvature. In particular ...

Let Γ be a discrete group, and let M be a closed spin manifold of dimension m > 3 with π1(M) = Γ. We assume that M admits a Riemannian metric of positive scalar curvature. We discuss how to use the L-rho invariant ρ(2) and the delocalized eta invariant η associated to the Dirac operator on M in order to get information about the space of metrics with positive scalar curvature. In particular ...

The Yamabe invariant Y(M) of a smooth compact manifold is roughly the supremum of the scalar curvatures of unit-volume constant-scalarcurvature Riemannian metrics g on M . (To be precise, one only considers those constant-scalar-curvature metrics which are Yamabe minimizers, but this technicality does not, e.g., affect the sign of the answer.) In this article, it is shown that many 4-manifolds ...

Let Γ be a discrete group, and let M be a closed spin manifold of dimension m > 3 with π1(M) = Γ. We assume that M admits a Riemannian metric of positive scalar curvature. We discuss how to use the L-rho invariant ρ(2) and the delocalized eta invariant η associated to the Dirac operator on M in order to get information about the space of metrics with positive scalar curvature. In particular ...

This is an expository article based on the author’s lecture delivered at the conference Lie Theory and Its Applications in March 2011, UCSD. We discuss various notions of positivity and their relations with the study of the Ricci flow, including a proof of the assertion, due to Wolfson and the author, that the Ricci flow preserves the positivity of the complex sectional curvature. We discuss th...

In this note, we show that the examples of non-Berwaldian Landsberg surfaces with vanishing flag curvature, obtained in [5] , are fact Berwaldian. Consequently, Bryant's claim is still unverified.

Systolic complexes were introduced in Januszkiewicz–Świa̧tkowski [12] and, independently, in Haglund [10]. They are simply connected simplicial complexes satisfying a certain condition that we call simplicial nonpositive curvature (abbreviated SNPC). The condition is local and purely combinatorial. It neither implies nor is implied by nonpositive curvature for geodesic metrics on complexes, but ...

Let X : (S, g) → R be a C isometric embedding of a C 4 metric g of non-negative sectional curvature on S into the Euclidean space R. We prove a priori bounds for the trace of the second fundamental form H , in terms of the scalar curvature R of g, and the diameter d of the space (S, g). These estimates give a bound on the extrinsic geometry in terms of intrinsic quantities. They generalize esti...

We study a family of 3-dimensional Lorentz manifolds. Some members of the family are 0-curvature homogeneous, 1-affine curvature homogeneous , but not 1-curvature homogeneous. Some are 1-curvature homogeneous but not 2-curvature homogeneous. All are 0-modeled on indecomposible local symmetric spaces. Some of the members of the family are geodesically complete, others are not. All have vanishing...

In the present paper, we find out necessary and sufficient conditions for a Finsler surface $(M,F)$ to be Landsbregian in terms of Berwald curvature $2$-forms. We study surfaces which satisfy some flag $K$ conditions, viz., $V(K)=0,\,\,V(K)= -\mathcal{I}/F^2$ $V(K)=-\mathcal{I}\,K,$ where $\mathcal{I}$ is Cartan scalar. order do so, investigate geometric objects associated with global distribut...