Manifolds all of whose geodesics are closed

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Geometric structures on filtered manifolds

Fil: Capriotti, Santiago. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - Bahia Blanca; Argentina. Universidad Nacional del Sur. Departamento de Matematica; Argentina

Unified formalism for Palatini gravity

We develop the theoretical foundations of a network distance that has recently been applied to various subfields of topological data analysis, namely persistent homology and hierarchical clustering. While this network distance has previously appeared in the context of finite networks, we extend the setting to that of compact networks. The main challenge in this new setting is the lack of an easy notion of sampling from compact networks; we solve this problem in the process of obtaining our results. The generality of our setting means that we automatically establish results for exotic objects such as directed metric spaces and Finsler manifolds. We identify readily computable network invariants and establish their quantitative stability under this network distance. We also discuss the computational complexity involved in precisely computing this distance, and develop easily-computable lower bounds by using the identified invariants. By constructing a wide range of explicit examples, we show that these lower bounds are effective in distinguishing between networks. Finally, we provide a simple algorithm that computes a lower bound on the distance between two networks in polynomial time and illustrate our metric and invariant constructions on a database of random networks and a database of simulated hippocampal networks.

Distances and Isomorphism between Networks and the Stability of Network Invariants

Special coordinate systems in pseudo-Finsler geometry and the equivalence principle

We investigate the relation between weighted quasi-metric Spaces and FinslerSpaces. We show that the induced metric of a Randers space with reversiblegeodesics is a weighted quasi-metric space. 1 Introductionand Motivation Riemannian spaces can be represented as metric spaces. Indeed, for a Riemannian space(M,a) we can deﬁne the induced metric space (M,d α ), with the metricd α : M× M→ [0,∞), d α (x,y) := inf γ∈Γ xy Z ba α(γ(t),γ˙(t))dt, (1.1)where Γ xy := {γ: [a,b] → M| γ(piecewise) C ∞ -curve,γ(a) = x,γ(b) = y} is the set ofcurves joining points xand y, ˙γ(t) := dγ(t)dt the tangent vector to γat γ(t), and α(x,X)the Riemannian norm of the vector X∈ T x M. It is easy to see that d α is a metric on M,i.e. it satisﬁes the axioms:1. Positiveness: d α (x,y) >0 if x6= y, d α (x,x) = 0,2. Symmetry: d α (x,y) = d α (y,x),3. Triangle inequality: d α (x,y) ≤ d α (x,z) +d α (z,y),for any x,y,z∈ M.More general structures than Riemannian ones are Finsler structures (see [BCS00],[S01], [MHSS01] for deﬁnitions).Similarly with the Riemannian case, one can deﬁne the induced metric of a Finslerspace (M,F) byd

Metric structures associated to Finsler metrics

We study the behavior of the geodesics of strong Kropina spaces. The global and local aspects of geodesics theory are discussed. Our theory is illustrated with several examples.

Geodesics on strong Kropina manifolds

This paper addresses the problem of existence of generalized Landsberg structures on surfaces using the Cartan–Kahler theorem and a path geometry approach.

On the existence of generalized unicorns on surfaces

We study the relation between an R-Cartan structure an an (I, J, K)-generalized Finsler structure on a 3-manifold showing the difficulty in finding a general transformation that maps to . In some particular cases, the mapping can be uniquely determined by geometrical conditions. Moreover, we are led in this way to a negative answer to our conjecture in [12].

Moving frames on generalized finsler structures

We study the relation between an R-Cartan structure {\alpha} and an (I, J, K)- generalized Finsler structure on a 3-manifold showing the difficulty in finding a general transformation that maps these structures each other. In some particular cases, the mapping can be uniquely determined by geometrical conditions.

Kazuhiro Shibuya

Papers

Metric structures associated to Finsler metrics

Geodesics on strong Kropina manifolds

On the existence of generalized unicorns on surfaces

Moving frames on generalized finsler structures

Moving frames on generalized Finsler structures