In this note, the authors show by example that an isometry between leaf spaces of singular Riemannian foliations need not induce an equality of the basic spectra. If the leaf space isometry preserves the mean curvature vector fields, then it is proved that the basic spectra are equivalent, i.e. that the leaf spaces are isospectral. As a corollary to the main result, the authors identify geometric conditions that ensure preservation of the mean curvature vector fields, and therefore isospectrality of the leaf spaces.
Abstract. In this paper we study half-geodesics, those closed geodesics that minimize on any subinterval of length L/2, where L is the length of the geodesic. For each nonnegative integer n, we construct Riemannian manifolds diffeomorphic to S 2 admitting exactly n half-geodesics. Additionally, we construct a sequence of Riemannian manifolds, each of which is diffeomorphic to S 2 and admits no half-geodesics, yet which converge in the Gromov-Hausdorff sense to a limit space with infinitely many half-geodesics.
Abstract. We consider the G-invariant spectrum of the Laplacian on an orbit space M/G where M is a compact Riemannian manifold and G acts by isometries. We generalize the Sunada-Pesce-Sutton technique to the G-invariant setting to produce pairs of isospectral non-isometric orbit spaces. One of these spaces is isometric to an orbifold with constant sectional curvature whereas the other admits non-orbifold singularities and therefore has unbounded sectional curvature. We conclude that constant sectional curvature and the presence of non-orbifold singularities are inaudible to the G-invariant spectrum.Mathematics Subject Classification (2010). 58J50; 58J53; 22D99; 53C12.
In this paper we study 1/k-geodesics, those closed geodesics that minimize on any subinterval of length L/k, where L is the length of the geodesic. We investigate the existence and behavior of these curves on doubled polygons and show that every doubled regular n-gon admits a 1/2n-geodesic. For the doubled regular p-gons, with p an odd prime, we conjecture that k = 2p is the minimum value for k such that the space admits a 1/k-geodesic.
In this paper we develop a Morse theory for the uniform energy. We use the one-sided directional derivative of the distance function to study the minimizing properties of variations through closed geodesics. This derivative is then used to define a one-sided directional derivative for the uniform energy which allows us to identify gradient-like vectors at those points where the function is not differentiable. These vectors are used to restart the standard negative gradient flow of the Morse energy at its critical points. We illustrate this procedure on the flat torus and demonstrate that the restarted flow improves the minimizing properties of the associated closed geodesics.
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