On geodesically reversible Finsler manifolds

A Finsler manifold is said to be geodesically reversible if the reversed curve of any geodesic remains a geometrical geodesic. Well-known examples of geodesically reversible Finsler metrics are Randers metrics with closed [Formula: see text]-forms. Another family of well-known examples are projectively flat Finsler metrics on the [Formula: see text]-sphere that have constant positive curvature. In this paper, we prove some geometrical and dynamical characterizations of geodesically reversible Finsler metrics, and we prove several rigidity results about a family of the so-called Randers-type Finsler metrics. One of our results is as follows: let [Formula: see text] be a Riemannian–Finsler metric on a closed surface [Formula: see text], and [Formula: see text] be a small antisymmetric potential on [Formula: see text] that is a natural generalization of [Formula: see text]-form (see Sec. 1). If the Randers-type Finsler metric [Formula: see text] is geodesically reversible, and the geodesic flow of [Formula: see text] is topologically transitive, then we prove that [Formula: see text] must be a closed [Formula: see text]-form. We also prove that this rigidity result is not true for the family of projectively flat Finsler metrics on the [Formula: see text]-sphere of constant positive curvature.

Metamaterial Acoustics on the (2 + 1)D Einstein Cylinder

The Einstein cylinder is the first cosmological model for our universe in modern history. Its geometry not only describes a static universe—a universe being invariant under time reversal—but it is also the prototype for a maximally symmetric spacetime with constant positive curvature. As such, it is still of crucial importance in numerous areas of physics and engineering, offering a fruitful playground for simulations and new theories. Here, we focus on the implementation and simulation of acoustic wave propagation on the Einstein cylinder. Engineering such an extraordinary device is the territory of metamaterial science, and we will propose an appropriate tuning of the relevant acoustic parameters in such a way as to mimic the geometric properties of this spacetime in acoustic space. Moreover, for probing such a space, we derive the corresponding wave equation from a variational principle for the underlying curved spacetime manifold and examine some of its solutions. In particular, fully analytical results are obtained for concentric wave propagation. We present predictions for this case and thereby investigate the most significant features of this spacetime. Finally, we produce simulation results for a more sophisticated test model which can only be tackled numerically.

Moduli spaces for Lamé functions and Abelian differentials of the second kind

The topology of the moduli space for Lamé functions of degree [Formula: see text] is determined: this is a Riemann surface which consists of two connected components when [Formula: see text]; we find the Euler characteristics and genera of these components. As a corollary we prove a conjecture of Maier on degrees of Cohn’s polynomials. These results are obtained with the help of a geometric description of these Riemann surfaces, as quotients of the moduli spaces for certain singular flat triangles. An application is given to the study of metrics of constant positive curvature with one conic singularity with the angle [Formula: see text] on a torus. We show that the degeneration locus of such metrics is a union of smooth analytic curves and we enumerate these curves.

Determinant of the Laplacian on Tori of Constant Positive Curvature with one Conical Point

We find an explicit expression for the zeta-regularized determinant of (the Friedrichs extensions of) the Laplacians on a compact Riemann surface of genus one with conformal metric of curvature $1$ having a single conical singularity of angle $4\unicode[STIX]{x1D70B}$ .

Non-Euclidean Extensions

This chapter adapts the foregoing results to present two non-Euclidean theories, both in line with the (semi-)Aristotelian theme of rejecting points, as parts of regions (but working with actual infinity). The first theory is a two-dimensional hyperbolic space, that is, one that has a negative constant curvature. The second theory captures a space of constant positive curvature, a two-dimensional spherical geometry. The task here is to formulate axioms on regions which allow us to prove that (i) there are no infinitesimal regions and (ii) that there are no parallels to any given “line” through any “point” not on the given “line”.