Given two c-projectively equivalent metrics on a Kähler manifold we show that the canoncially constructed, Poisson-commuting integrals of motion of the geodesic flow, linear and quadratic in momenta, also commute as quantum operators. The methods employed here also provide a proof of a similar statement in the case of projective equivalence. We also investigate the addition of potentials, i.e. the generalization to natural Hamiltonian systems. We show that the commuting operators lead to separation of variables for Schrödinger's equation.
Let $\mathcal{M}$ be a Type $\mathcal{A}$ affine surface. We show that $\mathcal{M}$ is linearly strongly projectively flat. We use the quasi-Einstein equation together with the condition that $\mathcal{M}$ is strongly projectively flat to examine to examine the geodesic completeness of $\mathcal{M}$.
We proof that on a surface of negative Euler characteristic, two real-analytic Finsler metrics have the same unparametrized oriented geodesics, if and only if they differ by a scaling constant and addition of a closed 1-form.
We use the results of Moser and Knörrer on relations between geodesics on quadrics and solutions of the classical Neumann system to describe explicitly the geodesic scattering on hyperboloids. We explain the relation of Knörrer's reparametrisation with projectively equivalent metrics on quadrics introduced by Tabachnikov and independently by Matveev and Topalov, giving a new proof of their result. We show that the projectively equivalent metric is regular on the projective closure of hyperboloids and extend Knörrer's map to this closure.
Geodesics, which play an important role in spray-Finsler geometry, are integral curves of a spray vector field on a manifold. Some comparison theorems and rigidity issues are established on the completeness of geodesics of a spray or a Finsler metric. In this paper, projectively flat sprays with weak Ricci constant (eps. constant curvature) are classified at the level of geodesics. Further, a geodesic method is introduced to determine an $n$-dimensional spray based on a family of curves with $2(n-1)$ free constant parameters as geodesics. Finally, it shows that a spray is projectively complete…
This paper combines two classical theories, namely metric projective differential geometry and superintegrability. We study superintegrable systems on 2-dimensional geometries that share the same geodesics, viewed as unparametrized curves. We give a definition of projective equivalence of such systems, which may be considered the projective analog of (conformal) Stäckel equivalence (coupling constant metamorphosis). Then, we discuss the transformation behavior for projectively equivalent superintegrable systems and find that the potential on a projectively equivalent geometry can be reconstruc…
Let $M$ be a differentiable manifold and $TM$ be its tangent bundle. A $C^0$-Finsler structure on $M$ is a continuous function $F: TM \rightarrow \mathbb R$ such that its restriction to each tangent space is a norm. In this work we present a large family of projectively equivalent $C^0$-Finsler manifolds $(\hat M \cong \mathbb R^2,\hat F)$. Their structures $\hat F$ don't have partial derivatives and they aren't invariant by any transformation group of $\hat M$. For every $p,q \in (\hat M,\hat F)$, we determine the unique minimizing path connecting $p$ and $q$. They are line segments parallel …
A Liouville classification of a natural Hamiltonian system on the projective plane with a rotation metric and a linear integral is obtained. All Fomenko--Zieschang invariants (i.e., labeled molecules) of the system are calculated.
In this paper we construct a new class of surfaces whose geodesic flow is integrable (in the sense of Liouville). We do so by generalizing the notion of tubes about curves to 3-dimensional manifolds, and using Jacobi fields we derive conditions under which the metric of the generalized tubular sub-manifold admits an ignorable coordinate. Some examples are given, demonstrating that these special surfaces can be quite elaborate and varied.
In this work we study the geodesic flow on nilmanifolds associated to graphs. We are interested in the construction of first integrals to show complete integrability on some compact quotients. Also examples of integrable geodesic flows and of non-integrable ones are shown.
Consider a smooth manifold $M$ equipped with a bracket generating distribution $D$. Two sub-Riemannian metrics on $(M,D)$ are said to be projectively (resp. affinely) equivalent if they have the same geodesics up to reparameterization (resp. up to affine reparameterization). A sub-Riemannian metric $g$ is called rigid (resp. conformally rigid) with respect to projective/affine equivalence, if any sub-Riemannian metric which is projectively/affinely equivalent to $g$ is constantly proportional to $g$ (resp. conformal to $g$). In the Riemannian case the local classification of projectively and a…
We present a geometric interpretation of integrability of geodesic flow by quadratic integrals in terms of the web theory and construct integrable billiards on surfaces admitting such integrals.
Two geodesically (projectively) equivalent Finsler metrics determine a set of invariant volume forms on the projective sphere bundle. Their proportionality factors are geodesically invariant functions and hence they are first integrals. Being 0-homogeneous functions, the first integrals are common for the entire projective class. In Theorem 1.1 we provide a practical and easy way of computing these first integrals as the coefficients of a characteristic polynomial.
With the help of a generalization of the Fermat principle in general relativity, we show that chains in CR geometry are geodesics of a certain Kropina metric constructed from the CR structure. We study the projective equivalence of Kropina metrics and show that if the kernel distributions of the corresponding 1-forms are non-integrable then two projectively equivalent metrics are trivially projectively equivalent. As an application, we show that sufficiently many chains determine the CR structure up to conjugacy, generalizing and reproving the main result of [J.-H. Cheng, 1988]. The correspond…
Projective connections arise from equivalence classes of affine connections under the reparametrization of geodesics. They may also be viewed as quotient systems of the classical geodesic equation. After studying the link between integrals of the (classical) geodesic flow and its associated projective connection, we turn our attention to 2-dimensional metrics that admit one projective vector field, i.e. whose local flow sends unparametrized geodesics into unparametrized geodesics. We review and discuss the classification of these metrics, introducing special coordinates on the linear space of …
We prove that a surface carries a hexagonal 3-web of geodesics if and only if the geodesic flow on the surface admits a cubic first integral and show that the system of partial differential equations, governing metrics on such surfaces, is integrable by generalized hodograph transform method. We present some new local examples of such metrics, discuss known ones, and establish an analogue of the celebrated Graf and Sauer Theorem for Darboux superintegrable metrics.
We study the magnetic geodesic flows on 2-surfaces having an additional first integral which is independent of the Hamiltonian at a fixed energy level. The following two cases are considered: when there exists a quadratic in momenta integral, and also the case of a rational in momenta integral with a linear numerator and denominator. In both cases certain semi-Hamiltonian systems of PDEs appear. In this paper we construct exact solutions (generally speaking, local ones) to these systems: in the first case via the generalized hodograph method, in the second case via the Legendre transformation …
