cohomogeneity one manifolds with positive ricci curvature one of the central problems in riemannian geometry is to determine how large the classes of manifolds with positive nonnegative sectional ricci or scalar curvature are see gr for scalar curvature the situation is fairly well understood by comparison special surgery constructions as in sy wr and bundle constructions as in na have resulted in a large number of interesting manifolds with positive ricci curvature so far the only known obstructions to have positive ricci curvature come from obstructions to have positive scalar curvature see li and rs and from the classical bonnet myers theorem which implies that a closed manifold with positive ricci curvature must have finite fundamental group it is well known that among homogeneous manifolds g h this is also a sufficient condition see e g the proof of corollary 3 5 or br in this paper we prove that this is true as well when the manifold admits an action by a compact lie group g with orbits of codimension one

inequalities between the chern numbers of a singular fiber in a family of algebraic curves chern numbers of a singular fiber in a family of curves are the local contributions of the fiber to the global chern numbers of the total space our first purpose of this paper is to find the best inequalities between the chern numbers of a singular fiber our second purpose is to try to give a new approach to the classification of singular fibers of genus g we know that when g is big there are too many singular fibers of genus g to classify completely see 5 7 8 16 in order to get the local global relations between the invariants one possible way is to classify singular fibers according to their contributions to the global invariants to explain this approach we will classify singular fibers with big or small chern numbers and give some applications see the survey 2 for the background of the study on the local global properties for families of curves a family of curves of genus g over c is a fibration f x c whose general fibers f are smooth curves of genus g where x is a complex smooth projective surface the family is called semistable if all of the singular fibers are reduced nodal curves if x f c and f is just the second projection to c then we call f a trivial family if all of the smooth fibers of f are isomorphic to each other equivalently f becomes trivial under a finite base change c c then f is called isotrivial we always assume that f is relatively minimal i e there is no 1 curve in any singular fiber when g 1 kodaira 6 found the global invariants from the singular fibers the first chern number c1 x is always zero the second chern number c2 x is equal to 12 ox by noether s formula and

six dimensional nearly kahler manifolds of cohomogeneity one ii let m be a six dimensional manifold endowed with a cohomogeneity one action of g su2 su2 and m rm reg subset m its subset of regular points we show that mreg admits a smooth 2 parameter family of g invariant non isometric strict nearly kahler structures and that a 1 parameter subfamily of such structures smoothly extends over a singular orbit of type s3 this determines a new class of examples of nearly kahler structures on t s3

homogeneous einstein randers metrics on symplectic groups abstract in this paper we prove that lie group sp n n 3 admits at least one family of invariant einstein randers metrics which are not naturally reductive we also give the classification of these metrics under isometries and determine their group of isometries moreover we prove that if homogeneous randers space m f with navigation data h w is berwald then m f is naturally reductive if and only if the riemannian metric m h is naturally reductive

multiple closed geodesics on bumpy finsler n spheres abstract in this paper we prove that for every bumpy finsler metric f on every rationally homological n dimensional sphere s n with n 2 there exist always at least two distinct prime closed geodesics

on dually flat randers metrics abstract the notion of dually flat finsler metrics arise from information geometry in this paper we will study a special class of finsler metrics called randers metrics to be dually flat a simple characterization is provided and some non trivial explicit examples are constructed in particular we will show that the dual flatness of a randers metric always arises from that of some riemannian metric by doing some special deformations

finsler manifolds with nonpositive flag curvature and constant s curvature the flag curvature is a natural extension of the sectional curvature in riemannian geometry and the s curvature is a non riemannian quantity which vanishes for riemannian metrics there are incomplete non riemannian finsler metrics on an open subset in rn with negative flag curvature and constant s curvature in this paper we are going to show a global rigidity theorem that every finsler metric with negative flag curvature and constant s curvature must be riemannian if the manifold is compact we also study the nonpositive flag curvature case

positively curved homogeneous metrics on spheres the study of manifolds with positive sectional curvature has a long history and can be considered as the beginning of global riemannian geometry nevertheless there are few general theorems concerning this class of manifolds which do not require further geometric assumptions apart from obstructions which already hold for manifolds with non negative sectional curvature there are only restrictions on the fundamental group due to the classical theorems of bonnet meyers and synge there are also very few known examples they all arise as quotients of a compact lie group endowed with a left invariant metric by a subgroup of isometries acting freely they consist apart from the rank one symmetric spaces of certain homogeneous spaces in dimensions 6 7 12 13 and 24 due to berger be wallach wa and aloff wallach aw and of biquotients in dimensions 6 7 and 13 due to eschenburg e1 e2 and bazaikin ba

identification of non linear systems using multi scale ridges and skeletons of the wavelet transform abstract a new procedure of non linear system identification is presented the procedure employs the slowly varying time dependent amplitude and phase functions of the impulse response of the system these instantaneous characteristics are obtained from the ridges and skeletons of the wavelet transform the ridge extraction procedure uses the modulus of the transform and involves a combinatorial optimization algorithm based on simulated annealing the method is illustrated using two simple simulated examples it is shown that the procedure can be used for multi degree of freedom systems due to the frequency localization property of the continuous grossman morlet wavelets

damping identification using a continuous wavelet transform application to real data a continuous wavelet transform cwt based on the gabor wavelet function is used to identify the damping of a multi degree of freedom system the common procedures are already known especially the identification with a morlet cwt this study gives special attention to the following a description of the instantaneous noise the edge effect of the cwt the frequency shift of the cwt the bandwidth of the wavelet function and the selection of the parameter of the gabor wavelet function of the cwt the procedures are demonstrated using several numerical examples and on signals acquired from the lateral vibration of a uniform beam the study demonstrates the advantages of using the amplitude and phase methods both of which provide information about the instantaneous noise the procedures presented are appropriate for automating the identification process

on dually flat α β metrics abstract the dual flatness for riemannian metrics in information geometry has been extended to finsler metrics the aim of this paper is to study the dual flatness of the so called metrics in finsler geometry by doing some special deformations we will show that the dual flatness of an metric always arises from that of some riemannian metric in dimensional n 3

ridges reconstruction based on inverse wavelet transform ridges always carry important physical information in practice ridge reconstruction based on inverse wavelet transform is studied in this article it is emphasized that the points within the neighbor area have to be added in the reconstruction to avoid energy lost however no discussion has been proposed about how to choose the neighbor area up to now although it is the key to achieve good performance a formula is given in this article to calculate the width of the neighbor area which varies with scale and the signal itself some examples are given in this article to show the performance of the proposed method which manifests that the method works very well for signals with multiple components as long as there is no cross section between them it also works well when the signal to noise ratio is not very low

homogeneous einstein metrics and simplicial complexes on compact homogeneous spaces we investigate the hilbert action restricted to the space of homogeneous metrics of volume one based on a detailed understanding of the high energy levels of this action we assign to a compact homogeneous space a simplicial complex whose non contractibility is a sufficient condition for the existence of homogeneous einstein metrics

finsler geodesics in the presence of a convex function and their applications in this paper we obtain a result about the existence of only a finite number of geodesics between two fixed non conjugate points in a finsler manifold endowed with a convex function we apply it to randers and zermelo metrics as a by product we also get a result about the finiteness of the number of lightlike and timelike geodesics connecting an event to a line in a standard stationary spacetime

time minimal control of dissipative two level quantum systems the generic case the objective of this article is to complete preliminary results from the work of bonnard and sugny 2009 and sugny et al 2007 concerning the time minimal control of dissipative two level quantum systems whose dynamics is governed by the lindblad equation the extremal system is described by a 3d hamiltonian depending upon three parameters we combine geometric techniques with numerical simulations to deduce the optimal solutions

homogeneous randers spaces with isotropic s curvature and positive flag curvature in this paper we will give a complete classification of homogeneous randers spaces with isotropic s curvature and positive flag curvature this results in a large class of finsler spaces with non constant positive flag curvature at the final part of the paper we prove a rigidity result asserting that a homogeneous randers space with almost isotropic s curvature and negative ricci scalar must be riemannian

sharp uncertainty principles on general finsler manifolds the paper is devoted to sharp uncertainty principles heisenberg pauli weyl caffarelli kohn nirenberg and hardy inequalities on forward complete finsler manifolds endowed with an arbitrary measure under mild assumptions the existence of extremals corresponding to the sharp constants in the heisenberg pauli weyl and caffarelli kohn nirenberg inequalities fully textit characterizes the nature of the finsler manifold in terms of three non riemannian quantities namely its textit reversibility and the vanishing of the textit flag curvature and s textit curvature induced by the measure respectively it turns out in particular that the busemann hausdorff measure is the optimal one in the study of sharp uncertainty principles on finsler manifolds the optimality of our results are supported by randers type finslerian examples originating from the zermelo navigation problem

on the energy functional on finsler manifolds and applications to stationary spacetimes in this paper we first study some global properties of the energy functional on a non reversible finsler manifold in particular we present a fully detailed proof of the palais smale condition under the completeness of the finsler metric moreover we define a finsler metric of randers type which we call fermat metric associated to a conformally standard stationary spacetime we shall study the influence of the fermat metric on the causal properties of the spacetime mainly the global hyperbolicity moreover we study the relations between the energy functional of the fermat metric and the fermat principle for the light rays in the spacetime this allows one to obtain existence and multiplicity results for light rays using the finsler theory finally the case of timelike geodesics with fixed energy is considered

some einstein randers metrics on homogeneous spaces abstract in this paper we study invariant einstein randers metrics on some homogeneous riemannian manifolds of rank 2 by finding out all the invariant killing vector fields on the corresponding homogeneous einstein riemannian manifolds we obtain a complete description of all the invariant einstein randers metrics on those homogeneous manifolds since the underlying riemannian einstein metrics are generically not diagonal this presents a large number of non riemannian homogeneous einstein randers spaces of nonconstant flag curvature

minimal surfaces in a randers sphere with the rotational killing vector field abstract the minimal surfaces in finsler geometry with respect to the busemann hausdorff measure and the holmes thompson measure are called bh minimal and ht minimal surfaces respectively let p 1 p 2 p 3 p 4 be the coordinates of r 4 and s 3 f be a randers sphere of flag curvature k 1 with the navigation data h w where h is the standard sphere metric and w e 0 0 p 4 p 3 0 e 1 is a killing vector field in this paper we study the rotationally invariant minimal surface in s 3 f generated by rotating the curve x s y s z s 0 in the upper half sphere of s 2 around the p 1 p 2 plane s r we first show that such a rotational bh minimal surface in s 3 f is either a great 2 sphere or the catenoid in s 3 h then we give a classification of the rotational ht minimal surfaces where we use the angle data to analyze the solutions of the system of ode that characterizes the ht minimality and prove that such a rotational ht minimal surface must be a great 2 sphere an ht minimal torus or a rotational surface of unduloid type as a special case we obtain a distinguished embedded compact ht minimal torus depending on e the completeness of these surfaces is also studied

s curvature of cohomogeneity one randers spaces abstract in this paper we study s curvature of cohomogeneity one randers spaces with orbit space s 1 in particular we present an explicit coordinate free formula of s curvature in terms of some quantities which is determined by the metric and the structure of the manifold as an application we give a sufficient and necessary condition for a cohomogeneity one randers space to have almost isotropy s curvature furthermore we give some simple criterions for a cohomogeneity one randers space to be a berwald space or a douglas space

projectively flat general α β metrics with constant flag curvature abstract in this paper we study the flag curvature of a new class of finsler metrics called general metrics which are defined by a riemannian metric and a 1 form the classification of such metrics with constant flag curvature are completely determined under some suitable conditions which make them locally projectively flat as a result we construct many new projectively flat finsler metrics with flag curvature 1 0 and 1 in section 9 all of which are of singularity at some directions

on locally and globally symmetric berwald spaces in this paper we generalize cartan s work on riemannian locally and globally symmetric spaces to locally and globally symmetric berwald spaces we prove that a berwald space is locally symmetric if and only if the flag curvature is invariant under parallel displacements and a locally symmetric berwald space is locally isometric to a globally symmetric berwald space

on the flag curvature of a class of randers metric generated from the navigation problem abstract let f x y be a randers metric of douglas type on a manifold m and v be a conformal vector field on m f let f x y be a randers metric generated from f v by solving the navigation problem f x y f x y v x 1 in this paper we establish the relationship among the flag curvature of f the flag curvature of f and the conformal vector field v see theorem 1 1

characterization of signals by the ridges of their wavelet transforms the characterization and the separation of amplitude and frequency modulated signals is a classical problem of signal analysis and signal processing we present a couple of new algorithmic procedures for the detection of ridges in the modulus of the continuous wavelet transform of one dimensional 1 d signals these detection procedures are shown to be robust to additive white noise we also derive and test a new reconstruction procedure the latter uses only information from the restriction of the wavelet transform to a sample of points from the ridge this provides a very efficient way to code the information contained in the signal

projectively flat randers metrics with constant flag curvature we classify locally projectively flat randers metrics with constant ricci curvature and obtain a new family of randers metrics of negative constant flag curvature

real killing spinors and holonomy we give a description of all complete simply connected riemannian manifolds carrying real killing spinors furthermore we present a construction method for manifolds with the exceptional holonomy groupsg2 and spin 7

on general α β metrics with isotropic s curvature abstract the s curvature is an important non riemannian quantity in finsler geometry in this paper we study a class of finsler metrics called general metrics which are defined by a riemannian metric and a 1 form we characterize all of general metrics with isotropic s curvature

invariant randers metrics on homogeneous riemannian manifolds this paper studies randers metrics on homogeneous riemannian manifolds it turns out that we can give a complete description of the invariant randers metrics on a homogeneous riemannian manifold as well as the geodesics the flag curvatures this result provides a convenient method to construct globally defined berwald space which is neither riemannian nor locally minkowskian and gives another explanation of the example of bao et al 1999 an introduction to riemannian finsler geometry berlin springer

asymptotic wavelet and gabor analysis extraction of instantaneous frequencies the behavior of the continuous wavelet and gabor coefficients in the asymptotic limit using stationary phase approximations are investigated in particular it is shown how under some additional assumptions these coefficients allow the extraction of some characteristics of the analyzed signal such as frequency and amplitude modulation laws applications to spectral line estimations and matched filtering are briefly discussed

time optimal control of a self propelled particle in a spatiotemporal flow field abstractwe address a minimum time problem that constitutes an extension of the classical zermelo navigation problem in higher dimensions in particular we address the problem of steering a self propelled particle to a prescribed terminal position with free terminal velocity in the presence of a spatiotemporal flow field furthermore we assume that the norm of the rate of change of the particle s velocity relative to the flow is upper bounded by an explicit upper bound to address the problem we first employ pontryagin s minimum principle to parameterise the set of candidate time optimal control laws in terms of a parameter vector that belongs to a compact set subsequently we develop a simple numerical algorithm for the computation of the minimum time to come function that is tailored to the particular parametrisation of the set of the candidate time optimal control laws of our problem the proposed approach bypasses the task of converting the optimal control problem to a parameter optimisation proble

optimal pinching constants of odd dimensional homogeneous spaces in this article we compute the pinching constants of all invariant riemannian metrics on the berger space b13 su 5 sp 2 2s1 and of all invariant u 2 biinvariant riemannian metrics on the aloff wallach space w7 1 1 su 3 s1 1 1 we prove that the optimal pinching constants are precisely in both cases so far b13 and w7 1 1 were only known to admit riemannian metrics with pinching constants we also investigate the optimal pinching constants for the invariant metrics on the other aloff wallach spaces w7 k l su 3 s1 k l our computations cover the cone of invariant t2 biinvariant riemannian metrics this cone contains all invariant riemannian metrics unless k l 1 it turns out that the optimal pinching constants are given by a strictly increasing function in k l 0 1 thus all the optimal pinching constants are in order to determine the extremal values of the sectional curvature of an invariant riemannian metric on w7 k l we employ a systematic technique which can be applied to other spaces as well the computation of the pinching constants for b13 is reduced to the curvature computation for two proper totally geodesic submanifolds one of them is diffeomorphic to 3 2 and inherits an sp 2 invariant riemannian metric and the other is w7 1 1 embedded as recently found by taimanov this approach explains in particular the coincidence of the optimal pinching constants for w7 1 1 and the berger space b13

zermelo navigation on riemannian manifolds in this paper we study zermelo navigation on riemannian manifolds and use that to solve a long standing problem in finsler geometry namely the complete classification of strongly convex randers metrics of constant flag curvature

twistor geometry of a pair of second order odes we discuss the twistor correspondence between path geometries in three dimensions with vanishing wilczynski invariants and anti self dual conformal structures of signature 2 2 we show how to reconstruct a system of odes with vanishing invariants for a given conformal structure highlighting the ricci flat case in particular using this framework we give a new derivation of the wilczynski invariants for a system of odes whose solution space is endowed with a conformal structure we explain how to reconstruct the conformal structure directly from the integral curves and present new examples of systems of odes with point symmetry algebra of dimension four and greater which give rise to anti self dual structures with conformal symmetry algebra of the same dimension some of these examples are 2 2 analogues of plane wave space times in general relativity finally we discuss a variational principle for twistor curves arising from the finsler structures with scalar flag curvature

solution to the quantum zermelo navigation problem the solution to the problem of finding a time optimal control hamiltonian to generate a given unitary gate in an environment in which there exists an uncontrollable ambient hamiltonian e g a background field is obtained in the classical context finding the time optimal way to steer a ship in the presence of a background wind or current is known as the zermelo navigation problem whose solution can be obtained by working out geodesic curves on a space equipped with a randers metric the solution to the quantum zermelo problem which is shown here to take a remarkably simple form is likewise obtained by finding explicit solutions to the geodesic equations of motion associated with a randers metric on the space of unitary operators the result reveals that the optimal control in a sense goes along with the wind

ricci quadratic homogeneous randers spaces abstract a finsler space is called ricci quadratic if its ricci curvature r i c x y is quadratic in y it is called a berwald space if its chern connection defines a linear connection directly on the underlying manifold m in this article we prove that a homogeneous randers space is ricci quadratic if and only if it is of berwald type

homogeneous einstein randers metrics on spheres abstract in this paper we study the homogeneous einstein randers metrics on spheres it turns out that we can find out all the homogeneous non riemannian einstein randers metrics on spheres furthermore we obtain a complete classification of such metrics under isometries using this we present a large number of homogeneous einstein randers metrics of non constant flag curvature

einstein randers metrics on some homogeneous manifolds abstract in this paper we get some new examples of non riemannian einstein randers metrics on homogeneous manifolds g h when g is either the compact simple lie group f 4 or g 2 based on the discussions associated with kahler c spaces we obtain some new einstein riemannian metrics on some homogeneous manifolds and prove that they admit non riemannian einstein randers metrics

einstein riemannian metrics and einstein randers metrics on a class of homogeneous manifolds abstract in this paper we get a class of homogeneous manifolds admitting einstein riemannian metrics moreover we prove that they admit non riemannian einstein randers metrics

discrete inverses for nonorthogonal wavelet transforms discrete nonorthogonal wavelet transforms play an important role in signal processing by offering finer resolution in time and scale than their orthogonal counterparts the standard inversion procedure for such transforms is a finite expansion in terms of the analyzing wavelet while this approximation works quite well for many signals it fails to achieve good accuracy or requires an excessive number of scales for others this paper proposes several algorithms that provide more adequate inversion and compares them in the case of morlet wavelets in the process both practical and theoretical issues for the inversion of nonorthogonal wavelet transforms are discussed

time minimal control of dissipative two level quantum systems the integrable case the objective of this article is to apply recent developments in geometric optimal control to analyze the time minimum control problem of dissipative two level quantum systems whose dynamics is governed by the lindblad equation we focus our analysis on the case where the extremal hamiltonian is integrable

einstein metrics on s 3 r 3 and r 4 bundles starting from a 4n dimensional quaternionic kahler base space we construct metrics of cohomogeneity one in 4n 3 dimensions whose level surfaces are thes 2 bundle space of almost complex structures on the base manifold we derive the conditions on the metric functions that follow from imposing the einstein equation and obtain solutions both for compact and non compact 4n 3 dimensional spaces included in the non compact solutions are two ricci flat 7 dimensional metrics withg 2 holonomy we also discuss two other ricci flat solutions one on ther 4 bundle overs 3 and the other on anr 4 bundle overs 4 these haveg 2 and spin 7 holonomy respectively

homogeneous einstein α β metrics on compact simple lie groups and spheres abstract in this paper we study homogeneous einstein metrics on compact lie groups and spheres we first show that any left invariant einstein metric on a connected compact simple lie groups except su 2 with vanishing s curvature must be a randers metric secondly we prove that any sp n 1 invariant einstein metric on s 4 n 3 n n with vanishing s curvature is either a randers metric or su 2 n 2 invariant finally we give a complete description of su n 1 invariant einstein finsler metrics on s 2 n 1 n 2

on uniqueness of conformally compact einstein metrics with homogeneous conformal infinity abstract in this paper we show that for a berger metric g on s 3 the non positively curved conformally compact einstein metric on the 4 ball b 1 0 with s 3 g as its conformal infinity is unique up to isometries and it is the metric constructed by pedersen 21 in particular since in 18 we proved that if the yamabe constant of the conformal infinity y s 3 g is close to that of the round sphere then any conformally compact einstein manifold filled in must be negatively curved and simply connected therefore if g is a berger metric on s 3 with y s 3 g close to that of the round metric the conformally compact einstein metric filled in is unique up to isometries

zermelo navigation in the quantum brachistochrone we analyse the optimal times for implementing unitary quantum gates in a constrained finite dimensional controlled quantum system the family of constraints studied is that the permitted set of time dependent hamiltonians is the unit ball of a norm induced by an inner product on we also consider a generalization of this to arbitrary norms we construct a randers metric by applying a theorem of shen on zermelo navigation the geodesics of which are the time optimal trajectories compatible with the prescribed constraint we determine all geodesics and the corresponding time optimal hamiltonian for a specific constraint on the control i e for any given value of some of the results of carlini et al are re derived using alternative methods a first order system of differential equations for the optimal hamiltonian is obtained and shown to be of the form of the euler poincar equations we illustrate that this method can form a methodology for determining which physical substrates are effective at supporting the implementation of fast quantum computation

curvatures of homogeneous randers spaces abstract we study curvatures of homogeneous randers spaces after deducing the coordinate free formulas of the flag curvature and ricci scalar of homogeneous randers spaces we give several applications we first present a direct proof of the fact that a homogeneous randers space is ricci quadratic if and only if it is a berwald space we then prove that any left invariant randers metric on a non commutative nilpotent lie group must have three flags whose flag curvature is positive negative and zero respectively this generalizes a result of j a wolf on riemannian metrics we prove a conjecture of j milnor on the characterization of central elements of a real lie algebra in a more generalized sense finally we study homogeneous finsler spaces of positive flag curvature and particularly prove that the only compact connected simply connected lie group admitting a left invariant finsler metric with positive flag curvature is su 2

nontrivial minimal surfaces in a class of finsler spheres of vanishing s curvature abstract the minimal surface theory in finsler geometry deserves to be well developed as that in riemannian geometry in this paper we derive the mean curvature of submanifolds in a general manifold by considering the busemann hausdorff measure and holmes thompson measure respectively we then study the rotationally invariant minimal surfaces in the finsler 3 sphere endowed with an metric f k k k k k 1 where is a smooth function s 3 k is the berger sphere e 4 k 1 and k is a killing one form of constant length along the hopf fibers of s 3 we define the energy of the minimal surfaces and by using the volume ratio function introduced by the author and y b shen we give the explicit local expressions of the rotationally invariant bh minimal and ht minimal surfaces in such sphere respectively as a special case we give a detailed study of the rotationally invariant ht minimal surfaces in the 3 sphere with square metric

deformations and hilbert s fourth problem in this paper a special class of finsler metrics defined by a riemannian metric and an 1 form is studied the projectively flat metrics in dimension n ge 3 are classified by a new class of metric deformations in riemann geometry the results show that the projective flatness of such finsler metrics always arises from that of some riemannian metric

geodesics in randers spaces of constant curvature geodesies in randers spaces of constant curvature are classified randers metrics have received much attention lately as solutions to zermelo s problem of navigation largely because this navigation structure provides the frame work for a complete classification of constant flag curvature randers spaces flag curvature is the finslerian analog of riemannian sectional curvature see br04 briefly a randers metric is of constant flag curvature if and only if it solves zer melo s problem of navigation on a riemannian manifold of constant sectional cur vature under the influence of an infinitesimal homothety w see subsection 1 1 for a sketch of the navigation problem and theorem 3 for an explicit statement of the classification result the aim of this paper is to develop a geometric description of the geodesies in these spaces of constant curvature intuitively these paths minimize travel time across a riemannian landscape under windy conditions presently we will show that these curves are given by composing geodesies of the riemannian metric with the flow generated by w this claim is formalized by theorem 2 geodesies on surfaces of constant nonpositive curvature are illustrated in section 3 we then turn in section 4 to the constant flag curvature k 1 randers metrics on sn the case of the sphere is especially interesting it is possible to endow this closed manifold with a metric whose geodesies display distinctly non riemannian behaviors for example 1 a metric is projectively flat if every point admits coordinates in which the geodesies are straight lines belt r ami s theorem assures us that a riemannian metric is of constant sectional curvature if and only if it is projectively flat in contrast few randers spaces of constant flag curvature are projectively flat there are infinitely many nonisometric randers metrics of constant positive