1 1 INTRODUCTION 2 1 Introduction Classical calculus is a basic tool in analysis. We use it so often that we forget that its construction needed considerable time and effort. Especially in the last decade, the progresses made in the field of analysis in metric spaces make us reconsider this calculus. Along this line of thought, all started with the definition of Pansu derivative [24] and its ve...

Abstract The reduced Hamiltonian system on T∗(SU(3)/SU(2)) is derived from a Riemannian geodesic motion on the SU(3) group manifold parameterised by the generalised Euler angles and endowed with a bi-invariant metric. Our calculations show that the metric defined by the derived reduced Hamiltonian flow on the orbit space SU(3)/SU(2) ≃ S5 is not isometric or even geodesically equivalent to the s...

After recalling some background, we define Riemannian metrics and Riemannian manifolds. We analyze the basic tensorial operations that become available in the presence of a Riemannian metric. Then we construct the Levi-Civita connection, which is the basic " new " differential operator coming from such a metric. Background The purpose of this section is two–fold. On the one hand, we want to rel...

We consider a Riemannian spin manifold (M, g) with a fixed spin structure. The zero sets of solutions of generalized Dirac equations on M play an important role in some questions arising in conformal spin geometry and in mathematical physics. In this setting the mass endomorphism has been defined as the constant term in an expansion of Green’s function for the Dirac operator. One is interested ...

In computational anatomy, organ’s shapes are often modeled as deformations of a reference shape, i.e., as elements of a Lie group. To analyze the variability of the human anatomy in this framework, we need to perform statistics on Lie groups. A Lie group is a manifold with a consistent group structure. Statistics on Riemannian manifolds have been well studied, but to use the statistical Riemann...

The geodesic distance vanishes on the group Diffc(M) of compactly supported diffeomorphisms of a Riemannian manifold M of bounded geometry, for the right invariant weak Riemannian metric which is induced by the Sobolev metric Hs of order 0 ≤ s < 1 2 on the Lie algebra Xc(M) of vector fields with compact support.

The concept of locally dually flat Finsler metrics originate from information geometry. As we know, (α, β)-metrics defined by a Riemannian metric α and an 1-form β, represent an important class of Finsler metrics, which contains the Matsumoto metric. In this paper, we study and characterize locally dually flat first approximation of the Matsumoto metric with isotropic S-curvature, which is not ...

In Theorem 1, we generalize the results of Szabó [Sz1, Sz2] for Berwald metrics that are not necessary strictly convex: we show that for every Berwald metric F there always exists a Riemannian metric affine equivalent to F . Further, we investigate geodesic equivalence of Berwald metrics. Theorem 2 gives a system of PDE that has a (nontrivial) solution if and only if the given essentially Berwa...

We determine all Riemannian manifolds for which the tangent sphere bundles, equipped with the Sasaki metric, are local or global Riemannian product manifolds.

Gromov-Hausdorff convergence is an important tool in comparison Riemannian geometry. Given a sequence of Riemannian manifolds of dimension n with Ricci curvature bounded from below, Gromov’s precompactness theorem says that a subsequence will converge in the pointed Gromov-Hausdorff topology to a length space [G-99, Section 5A]. If the sequence has bounded sectional curvature, then the limit wi...