We show that for generic Riemannian metrics on a simply-connected closed spin manifold of dimension ≥ 5 the dimension of the space of harmonic spinors is no larger than it must be by the index theorem. The same result holds for periodic fundamental groups of odd order. The proof is based on a surgery theorem for the Dirac spectrum which says that if one performs surgery of codimension ≥ 3 on a ...

On the manifold M(M) of all Riemannian metrics on a compact manifold M one can consider the natural L-metric as decribed first by [10]. In this paper we consider variants of this metric which in general are of higher order. We derive the geodesic equations, we show that they are well-posed under some conditions and induce a locally diffeomorphic geodesic exponential mapping. We give a condition...

Dini derivative on Riemannian manifold setting is studied in this paper. In addition, a characterization for Lipschitz and convex functions defined on Riemannian manifolds and sufficient optimality conditions for constraint optimization problems in terms of the Dini derivative are given.

Spatial structures often constrain the 3D movement of cells or particles in vivo, yet this information is obscured when microscopy data are analyzed using standard approaches. Here, we present methods, called unwrapping and Riemannian manifold learning, for mapping particle-tracking data along unseen and irregularly curved surfaces onto appropriate 2D representations. This is conceptually simil...

This paper is concerned with eigenvalues of the biharmonic operators and the buckling eigenvalue for complete Riemannian manifolds. We are mostly concerned with relating bounds for these eigenvalues to the behavior of the ends of the manifold. Let M be a complete Riemannian manifold. M is called parabolic if every non-positive subharmonic function on M reduces to a constant. By an end E of M we...

A pair of points in a riemannian manifold M is secure if the geodesics between the points can be blocked by a finite number of point obstacles; otherwise the pair of points is insecure. A manifold is secure if all pairs of points in M are secure. A manifold is insecure if there exists an insecure point pair, and totally insecure if all point pairs are insecure. Compact, flat manifolds are secur...

In this paper, we describe the use of Riemannian geometry, and in particular the relationship between the Laplace-Beltrami operator and the graph Laplacian, for the purposes of embedding a graph onto a Riemannian manifold. Using the properties of Jacobi fields, we show how to compute an edge-weight matrix in which the elements reflect the sectional curvatures associated with the geodesic paths ...

1. Statement of the results. 1.1. Let M be a compact Kaehlerian manifold. The underlying Riemannian manifold which we also denote by M is orientable and of even dimension. Let K = K(<r) be the Riemannian curvature of M, considered as a Riemannian manifold. K(a) is a function on the 2planes a tangent to M. The restriction of K to holomorphic 2-planes is called holomorphic curvature and will be d...

In this work the spaces of Riemannian metrics on a closed manifold M are studied. On the space M of all Riemannian metrics on M the various weak Riemannian structures are defined and the corresponding connections are studied. The space AM of associated metrics on a symplectic manifold M,ω is considered in more detail. A natural parametrization of the space AM is defined. It is shown, that AM is...

We derive a formula for the first variation of horizontal perimeter measure for C2 hypersurfaces of completely general sub-Riemannian manifolds, allowing for the existence of characteristic points. When the manifold admits dilations, we establish a sub-Riemannian Minkowski formula. For C2 hypersurfaces in vertically rigid sub-Riemannian manifolds we also produce a second variation formula for v...