It is a well-known fact that on a bounded spectral interval the Dirac spectrum can described locally by a non-decreasing sequence of continuous functions of the Riemannian metric. In the present article we extend this result to a global version. We think of the spectrum of a Dirac operator as a function Z → R and endow the space of all spectra with an arsinh-uniform metric. We prove that the sp...

We study how convergence of an observer whose state lives in a copy of the given system’s space can be established using a Riemannian metric. We show that the existence of an observer guaranteeing the property that a Riemannian distance between system and observer solutions is nonincreasing implies that the Lie derivative of the Riemannian metric along the system vector field is conditionally n...

In 1923, Eisenhart 1 gave the condition for the existence of a second-order parallel symmetric tensor in a Riemannian manifold. In 1925, Levy 2 proved that a second-order parallel symmetric nonsingular tensor in a real-space form is always proportional to the Riemannian metric. As an improvement of the result of Levy, Sharma 3 proved that any second-order parallel tensor not necessarily symmetr...

We produce new examples of harmonic maps, having as either source or target manifold the tangent bundle TM on a Riemannian manifold (M,g), equipped with a Riemannian g-natural metric G. In particular, we study the harmonicity of the canonical projection π : (TM,G)→ (M,g), and of the identity map (TM,G)→ (TM,gS) and conversely, gS being the Sasaki metric on TM. A corresponding study is made for ...

In the context of mesh adaptation, Riemannian metric spaces have been used to prescribe orientation, density and stretching of anisotropic meshes. But, such structures are only considered to compute lengths in adaptive mesh generators. In this article, a Riemannian metric space is shown to be more than a way to compute a length. It is proven to be a reliable continuous mesh model. In particular...

The set of spatial rigid body motions forms a Lie group known as the special Euclidean group in three dimensions, (3). Chasles’s theorem states that there exists a screw motion between two arbitrary elements of (3). In this paper we investigate whether there exist a Riemannian metric whose geodesics are screw motions. We prove that no Riemannian metric with such geodesics exists and we show tha...

A Riemannian almost product structure on a manifold induces on a submanifold of codimension 1 a structure generalizing the paracontact structures and containing a Riemannain metric and an one form . We show that the pair consisting of this Riemannian metric and one form defines a strongly convex Randers metric on submanifold. We establish some properties of this Randers metric and we provide so...

Hadamard (or complete $CAT(0)$) spaces are complete, non-positive curvature, metric spaces. Here, we prove a nonlinear ergodic theorem for continuous non-expansive semigroup in these spaces as well as a strong convergence theorem for the commutative case. Our results extend the standard non-linear ergodic theorems for non-expansive maps on real Hilbert spaces, to non-expansive maps on Ha...