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Nathalie Ollat

Jean-Marc Touzard

LONG TERM IMPACTS OF CLIMATE CHANGE AND ADAPTATION FOR THE FRENCH WINE INDUSTRY

I. The LACCAVE project : a systemic approach of adaptation for the wine industry

-Economic importance -Perennial system for an annual production -Marker of climatic conditions

"The set of actions and processes which societies must utilize to limit the negative impacts of changes and maximize their beneficial effects"

Focus

  scientific network with research and applied objectives 23 french laboratories : climatology, agronomy, oenology genetic, plant physiology, pathology, human sciences, economic, mathematic 90 scientists and students 7 Ph-D A website : https://www6.inra.fr/laccave An international conference with Proceedings : ClimWine2016 -Research activity coordination -Shared knowlegde about impacts and adaptation -Targeted disciplinary and interdisciplinary research studies -Transfer of knowledge to the industry and civil society -Awareness raising -Foresight exercise as adaptation strategy building tool


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Functional investigation of the human oocyte-expressed gene bcar4 using domestic animal models and genome editing.

Rozenn Dalbies-Tran, Maud Peyny, Peggy Jarrier-Gaillard, Véronique

Cadoret, Laurent Boulanger, Nathalie Daniel, Véronique Duranthon,

Sébastien Lavillatte

To cite this version: We have previously characterized BCAR4 (Breast Cancer Anti-estrogen Resistant 4) as a gene preferentially expressed in human oocytes as compared to ovarian somatic cells or non-pathologic nonreproductive tissues. BCAR4 is conserved in primates and in various domestic species such as cow, pig, dog, horse or rabbit, but it is not found in the genome of rodents. This pattern coincides with a delayed major activation of the embryonic genome, i.e. after several cleavages. BCAR4 is also an oncogene overexpressed in a subset of breast tumors and other cancers. Altogether, the restricted expression, proliferative properties and phylogeny suggested that BCAR4 may be involved in early embryonic divisions.We have analyzed BCAR4 expression in the cow. While RNA is already transcribed in the oocyte of preovulatory follicles, protein synthesis is delayed until late maturation, peaks in early cleaving embryos, persists until the morula stage to become undetectable in blastocysts. Microinjecting BCAR4targeting small-interfering RNA significantly decreased BCAR4 expression and compromised in vitro blastocyst development, demonstrating that BCAR4 is a maternal-effect gene.To investigate BCAR4 function in vivo, the rabbit model was chosen for genome editing. Rabbits carrying an altered BCAR4 gene were produced using a transcription activator-like effector nuclease (TALEN). Wild-type, heterozygous and homozygous carrier animals were born following the expected mendelian ratio. They were viable and appeared healthy, as expected for animals with an altered maternal effect-gene. Efficiency of the genetic alteration was evaluated by reverse-transcription coupled to PCR. It showed that BCAR4 RNA expression was abolished in follicles from homozygous carriers as compared to their heterozygous and wild-type littermates. These females will be phenotyped onto various reproductive parameters to assess the role of BCAR4 in fertility in vivo.

02733780 FUNCTIONAL INVESTIGATION OF THE HUMAN OOCYTE-EXPRESSED GENE BCAR4 USING DOMESTIC ANIMAL MODELS AND GENOME EDITING

	Dalbies-Tran, Rozenn	1 ; Peyny, Maud	1 ; Jarrier-Gaillard, Peggy	1 ; Cadoret, Véronique	2,1 ; Boulanger, Laurent	3 ;
	Daniel, Nathalie	3 ; Duranthon, Véronique	3 ; Lavillatte, Sebastien	4
	1 UMR Physiologie de la Reproduction et des Comportements, INRA, CNRS, IFCE, Université de Tours,
	37380 Nouzilly, France,	2 CHU Bretonneau, Médecine et Biologie de la Reproduction-CECOS, 37044 Tours,
	France,	3 UMR Biologie du Développement et Reproduction, INRA-ENVA, 78352 Jouy-en-Josas, France,
	4 PlateForme d'Infectiologie Expérimentale, INRA, 37380 Nouzilly, France

Rozenn Dalbies-Tran, Maud Peyny, Peggy Jarrier-Gaillard, Véronique Cadoret, Laurent Boulanger, et al.. Functional investigation of the human oocyte-expressed gene bcar4 using domestic animal models and genome editing.. Ovarian Club XI meeting, Nov 2018, Paris, France. 2018. hal-


Salone Jean-Jacques
email: jean-jacques.salone@univ-mayotte.fr

The cooperative works, a teacher training system open to the local social network

Keywords: Cooperative works, master MEEF, questioning the world, Mayotte, primary school teachers

an initial training curriculum at master's level is set up for primary school teachers in Mayotte, a French department in the southern hemisphere. It is based on a specific training system, the cooperative works, which favors the improvement of professional skills and more specifically a competence of contextualization. In this paper we describe this training system with concepts from the anthropological theory of didactics such as the Herbartian schema and we expose research results about how it allows the questioning of the Mahoran heritage and society.

Introduction

For Mayotte, a French island since 1843 and department since 2011, the professionalization of teachers is an important issue. One reason is that the School of the Republic in Mayotte educates a growing number of pupils, with a school population exceeding the symbolic hurdle of 100,000 students in [START_REF]Diaporama du Conseil de l'Education Nationale de Mayotte du 31[END_REF][START_REF] Salone | La contextualisation, une compétence professionnelle au centre du master Meef 1er degré de Mayotte[END_REF](Vice-Rectorat of Mayotte, 2018, p. 2). It is in this social context that since September 2017, the Centre Universitaire de Formation et de Recherche (CUFR) of Mayotte, in partnership with the Vice-Rectorat of Mayotte, the University of La Reunion and the Ecole Supérieure du Professorat et de l'Education of La Reunion, has offered an initial training curriculum at master level for Primary School Teachers.

Beyond the context of higher education, Mayotte also presents a particular socio-cultural context. In the first place, the schooling language, French, is not the native language of the island. As established by [START_REF] Laroussi | Mayotte, une île plurilingue en mutation[END_REF], it coexists with two other vehicular languages: Shimaore, a Bantu language, belonging to the Swahili area and Kibushi, a Malagasy variety of the Malayo-Polynesian family, the only one spoken outside of Madagascar. This linguistic difference is driven forward on a more generally cultural level, with a heritage that has its roots in the East African space. Furthermore, [START_REF] Blanchy-Dorel | La vie quotidienne à Mayotte[END_REF] describes a society in which: the worldview conveyed by the culture of Mayotte to its members is essentially a Muslim vision; but in most social circles, it is a popularized Islam that is mixed with non-Muslim elements coming as much from the pre-Islamic Arab animist traditions as from African (Bantu) traditions. 199"0,p. 112) In particular, in the village where he is raised, the young child goes to the Koranic school" (Blanchy-Dorel, 1990, p. 55).

One of the peculiarities of Mayotte also lies in its natural insular and tropical heritage. "Mayotte has been identified internationally as part of the 238 Ecoregions of the World Fund by nature, high places of biodiversity" (Amann, Amann, Arhel, Guiot and Marquet, 2008, p. 30).

But this heritage, cultural or natural, is strongly threatened. Its preservation has become a major issue, and school is no exception. The project put forward for primary and secondary schools suggests "developing the concept of living well together: Recognizing cultural otherness and making civilization by coexisting in different cultures and languages" (Vice-Rectorat of Mayotte, 2016, p. 1).

In the present paper we describe the training system in terms from the Anthropological Theory of Didactics (ATD) and we report global results on how it enforces the students' abilities in questioning the world.

Theoretical framework

Our focus in the graduate master courses we propose is thus placed on the competence of contextualization, the "ability to integrate different contextual spheres which partly drives the nature of the processes at work in the management of this professional situation of teaching" (Sauvage-Luntadi & Tupin, 2012, p. 106). According to [START_REF] Raynaud | Le contexte est-il un concept légitime de l'explication sociologique ? L'Explication sociologique. Quels sont les niveaux d'abstraction légitimes[END_REF], we will not try to differentiate the notion of context from those of environment or situation. We prefer to adopt a systemic approach: "rather than causally depending or referring action to "context" or "environment", we can consider that the individual accesses information through the arcs of his social network" (Raynaud 2005, p. 17).

According to ATD (Chevallard, 1985(Chevallard, /1991;;[START_REF] Chevallard | Passé et présent de la théorie anthropologique du didactique. Sociedad, escuela y matemáticas[END_REF]Bosch and Gascon, 2006;[START_REF] Winslow | Anthopological theory of didactic phenomena : Some examples and principles of its use in the study of mathematics education[END_REF], intentional human activity is always motivated by tasks whose achievements relies on individual or institutional knowledge. These pieces of knowledge, called praxeologies, are constituted by techniques in order to accomplish some types of tasks (the practice block) and by discourses (the logos block) which explain techniques, the technologies, or justify them, the theories. Thus institutions (a person or a socially legitimated group) have their own praxeological equipment. Within a didactic systems S based on didactic relations, some of these institutions Y, teachers for examples, have some other institutions X, the students, to improve their praxeological equipements. Then professional and/or disciplinary questions Q are studied and collective praxeological answers A ♥ are provided with the help of a milieu M. This milieu consists in partial answers A ◊ stamped by some referent institution ◊, among which we find at least students and teachers themselves, and other objects O (other resources for example) and derived questions Q'. The following Herbartian schema summarizes this teaching/learning process:

[ S (X; Y; Q) ! M = {A ◊ , O, Q' } ] " A ♥
In ATD didactic systems are not closed systems. They are imbedded in social networks which condition and constraint their underlying disciplinary and didactic organizations. At least four codeterminacy levels are identified outside classrooms: the pedagogy, which states teaching principles applying to all didactic systems independently from disciplines, the school with its internal capabilities to design and support teaching systems, the society, with its laws, rules or programs, and the civilization, an upper level beyond societies including for example cultural norms and traditions. Didactic systems then appear as open systems at the very core of macro-systems which overlap partially the non ATD concept of context: in each of those upper co-determinacy levels, many a priori a-didactic institutions can be found which all have praxeological equipment and thus which all may become referent institutions. Within this frame, the competence of contextualization may be seen as a professional teaching praxeology aiming at managing the flows of praxeological resources A ◊ or O from the available referent institutions ◊ into the milieu M.

In [START_REF] Salone | Les références praxéologiques dans les systèmes didactiques[END_REF], three types of these praxeological flows are highlighted, all of them spreading through specific social networks : scholar flows, in which scholar knowledge is subject to external and internal didactic transpositions (Chevallard, 1985(Chevallard, /1991)), private flows corresponding to private networks such as students' and teachers' ones, and more general social flows involving others referent institutions whose praxeological equipment are available directly in the nearby or indirectly through websites. These three contextual sub-systems allow teachers to make openings in their classes: activations of systemic relationships in order to influence the didactic systems. Three dimensions are underlined for these class openings: enrichment of the bodies of knowledge to which the class refers, diversification of the roles and topos of actors, widening of the didactic milieu. "In each of these three dimensions, openings can be performed not only internally, in the classroom, but also externally, by integrating non-didactical institutions" (Salone, 2015, p. 332).

Our analyses in this paper adopt this systemic approach with a focus on the society co-determinacy level. More precisely, we will look at the openings allowed by the training system we have designed for Primary School Teachers-in-training (PST) with a main issue: what are the praxeological, social flows between the didactic systems PST are involved in and the Mahoran society?

Description of the cooperative works training system

PST are actually involved in a specific training system: the cooperative works. Its goal is to equip them with professional praxeologies that will enable them to design courses that are both multidisciplinary and contextualized and that abandon the paradigm of visiting works to enter into that of questioning the world. In the cooperative works system, the whole group X of PST is at first part of classical, didactic systems S where it has lectures or helps from the teachers' trainers Y. Some generic questions Q are then asked, such as what is to be done and how, and answers A ♥ are usually produced with the help of some visited theoretical works W ◊ . The Herbartian schema in these phases is:

[ S (X; Y; Q) ! M = {W ◊ } ] " A ♥
PST also freely constitute groups of 3 to 5 people in order to enter a project, the cooperative works stricto sensu, which includes: a contextualized and multidisciplinary inquiry during the first semester, a didactic work and an educational booklet during the second semester and, in semesters 3 and 4 of the second year, teaching sequences for their classes. The results we present in the next section concern exclusively inquiries and didactic works conducted in 2017-2018. In ATD terms, each PST group enter a two years Study and Research Path (SRP) with a generating question Q0 which could be: how can we (our group) design multidisciplinary teaching situations relying on thematic resources about Mayotte's heritage?

The contextualized and multidisciplinary inquiry is a training and teaching technology related to the inquiry-based learning pedagogy [START_REF] Dewey | Logique : la théorie de l'enquête[END_REF] and to a documentary work [START_REF] Gueudet | Ressources vive : Le travail documentaire des professeurs en mathématiques[END_REF]. It articulates two main types of questions: Q1 : what is known about the theme we have chosen in Mayotte's heritage or society; Q2 : which multidisciplinary bodies of academic knowledge could be related to the theme? The inquiry also looks for possible places that can serve as school fieldtrips and for available resource persons, citizens' associations or institutional partners (derived questions Q'). When inquiring, the PST group Gi meet local actors YL who provides some resources A L and who sometimes contribute to their works. It also collects resources O from the available media such books or Internet. Each multidisciplinary survey then leads to a corpus of digital documents C i . The herbartian schema in these inquiries is:

[ S (Gi ; YL ; Q1, Q2) ! M = {A L , O, Q' } ] " C i
In the second semester, the creation of a didactic work W i is part of an educational approach similar to that of the "masterpieces" in the sense of [START_REF] Freinet | Les invariants pédagogiques[END_REF] or Meirieu (2015): trainees have to produce by their own a terminal work, a masterpiece, to show their professional skills and abilities. Theses masterpieces must of course be related to the previous inquiry theme and they always include an aesthetic dimension that here relies mainly on digital and artistic skills. As professional productions, they integrate also a pedagogical intention directed for a general or school-related audience, with for example the transmission of a knowledge or a message to bring awareness to a certain topic. The question for the PST group is then Q3: what knowledge do we want to transmit? And the herbartian schema is:

[ S (Gi ; empty ; Q3) ! M = { C i } ] " W i
After a five months production process, books, youth albums, slideshows or documentary movies are thus created. They are thereafter publicly presented and are evaluated by a jury composed of teachers' trainers, researchers and institutional partners. Finally, the best didactic works are published and disseminated in society, at least in schools, to become teaching resources.

In the context of some disciplinary teaching units such as French or Mathematics, the cooperative groups also produce an educational booklet whose purpose is to propose ways of using in classes the resources thus constituted, inquiry materials and didactic works. Hence these booklets generally contain examples of contextualized disciplinary organizations, with for example activities, problems or exercises statements. In the second year of the master curriculum, when every two weeks PST have their own classes, this transposition process of thematic resources into scholar ones is continued.

Methodological framework

In this paper we present results upon the cooperative works realized during the academic year 2017-2018. Thirty cooperative groups were formed this year. Some of their cooperatives works may be downloaded on our website (http://www.univ-mayotte.fr/fr/formation/sciences-de-l-education/ travaux-des-etudiants.html) and three of them are also described in [START_REF] Salone | La contextualisation, une compétence professionnelle au centre du master Meef 1er degré de Mayotte[END_REF].

In order to answer our research question about the praxeological, social flows induced by the cooperative works training system, we examine in this paper the data coming directly from the contextualized and multidisciplinary inquiries along with the produced didactic works. Three derived questions are answered: what are the themes chosen par the PST? What academic disciplines did they involved? Which socio-economic actors have they met? Indicators are therefore used. For the first question, the indicator is the topic which appears in the titles of the didactic works. We use there a four types categorization derived from [START_REF] Hartog | Temps et patrimoine[END_REF]: natural heritage, tangible cultural heritage, intangible cultural heritage and societal issues. Mixed themes belonging to several categories will be admitted. For the second question, the investigated school disciplines are directly identified in the documents from the inquiries bodies. For the third question, the encountered local actors that we retain are those who are cited or referenced in both the inquiries and the didactic works and we categorize them within three types: institutions, citizens' associations and referent persons.

Results

The chosen contextualized themes are varied. In Figure 1, we can see a cloud of the first-page photographs of the documents gathering the data from the inquiries. Mostly, with 23 works out of 30, they fall under a theme related to intangible cultural heritage, such as traditional games (for example the local awale type sowing game called 'mraha wa tso'), cooking recipes (for example the 'mataba', a typical course from south and central Africa made from mashed cassava leaves), arts and crafts (the countless braiding and uses of coconut leaves for example), lore (the 'msindzano' for example, a yellow, facial mask made of crushed coral, white clay and turmeric used by women) and crafts (such as the manufacturing process of the famous 'laka', the local canoe). 12 themes concern elements of the natural heritage, such as natural protected areas (for example the very known lagoon of Mayotte, or the volcanic lake 'Ziani') or endemic plant species (such as the always green oranges of Mtsamboro village) and 6 deal with societal issues (such as the sorting of waste, the preservation of the coral reef or the tire recycling in wall constructions). 2 themes concern the material cultural heritage (the 'banga', traditional tiny huts made of bamboo and clay, and sugar factories, ruined remains of the island's post-colonial industries, and the maritime transport by barge between the two main islands of Mayotte). Note that many of these themes are of course mixed (for example the 'Ziani' lake is at first a piece of Mayotte's natural heritage, but it is also a place at the very core of many legends and a highly popular tourist site) and that some themes have been addressed by several groups (such as 'msindzano' theme which has been investigated by 3 groups).

With regard to the multidisciplinary, each cooperative group has clearly identified many links between its theme and almost all the academic disciplines. Coming first, Mathematics and French have been investigated by all groups. In Mathematics, 27 groups have retained pieces of knowledge from the geometry domain or from the size and measurement domain which are present in the official programs of cycles 2 and 3 of elementary school (children approximately aged from 6 to 11). In French, all the groups have considered that fluency in written or spoken language can be improved by introducing local texts into the classroom, such as stories, songs, press articles or documents produced by citizens' associations. Visual arts, History, Geography and Moral and civic education are then the next most investigated disciplines. Their scores are respectively 22 groups (out of 30), 17, 16 and 16. Then came Science and technology, Natural sciences, Sport and Music.

The local actors met by the PST and sometimes involved in the cooperative works are numerous, with an average of 5 per group including at least one institutional partner. For example, about 80% of the groups have encountered the staff or the departmental archives of Mayotte and about 60% the team of the MuMa, Mayotte's museum. Among the citizens' associations, two have often been involved: the Mahoran naturalist, an association dealing with environmental and cultural issues, and SHIME, an association of local sociolinguists. Many referent persons in Mayotte, called 'fundi', have also been met. All these partners have been involved in several ways: to provide documentary resources, to answer questionnaires or to participate in interviews, to contribute to the designing of the didactic works. More than fifty institutional, associative or private partners were thus involved by the promotion witch thus woven a real educational network between the CUFR and the Mayotte society.

Conclusion and perspectives

The cooperative works system allows trainee teachers of schools who are engaged in our master curriculum to both strengthen their professional, praxeological equipement and acquire a good knowledge about Mayotte's heritage and society. First of all, through inquiries and systemic openings on the local social networks it fosters the emergence of contextualization skills. It also reinforces professional abilities in active pedagogies and it improves the mastery disciplinary knowledge. Moreover, for most of the PST, questioning the local world in order to create and organize educational activities in their classes has become something possible. Another gain induced by the cooperative works system is the large amount of produced materials: the didactic works (and also the educational booklets) are disseminated in Mayotte to become teaching resources and material coming from the inquiries are shared by researchers and institutional partners.

In terms of research, our team has recently been exploring additional issues about the cooperative works training system. For example, as the vehicular languages are an important part of the cultural heritage, the questions of their place in the didactic works and their uses in classroom are studied. More specifically, questions about the mathematical terms and idioms in these languages and their effects on the concepts formation arise. According to an ethnomathematical approach, other questions are coming to the shore such as the reinvestment in classes of traditional technologies or the use of handicraft artefacts. In a more sociological or ethnological approach, a main research issue also emerges: how, once tenured, will the PST involved in the cooperative works contribute to the development of Mayotte's education system?

Figure 1 :

 1 Figure 1: Cloud of selected themes


C Duval
email: duval@cpt.univ-mrs.fr

Finsler Spinoptics

Keywords: Presymplectic manifolds, Geometrical spinoptics, Finsler structures Mathematics Subject Classification 2000: 78A05, 70G45, 58B20

The objective of this article is to build up a general theory of geometrical optics for spinning light rays in an inhomogeneous and anisotropic medium modeled on a Finsler manifold. The prerequisites of local Finsler geometry are reviewed together with the main properties of the Cartan connection used in this work. Then, the principles of Finslerian spinoptics are formulated on the grounds of previous work on Riemannian spinoptics, and relying on the generic coadjoint orbits of the Euclidean group. A new presymplectic structure on the indicatrix-bundle is introduced, which gives rise to a foliation that significantly departs from that generated by the geodesic spray, and leads to a specific anomalous velocity, due to the coupling of spin and the Cartan curvature, and related to the optical Hall effect.

Introduction

Geometry and optics have maintained a lasting relationship since Euclid's Optics where light rays were first interpreted as oriented straight lines in space (or, put in modern terms, as oriented, non parametrized, geodesics of Euclidean space). One can, withal, trace back the origin of the calculus of variations to Fermat's Principle of least optical path. This principle has served as the basis of geometrical optics in inhomogeneous, isotropic, media and proved a fundamental mathematical tool in the design of optical (and electronic) devices such as mirrors, lenses, etc., and in the understanding of caustics, and optical aberrations. Although Maxwell's theory of wave optics has unquestionably clearly superseded geometrical optics as a bona fide theory of light, the seminal work of Fermat has opened the way to wide branches of mathematics, physics, and mechanics, namely, to the calculus of variations in the large, modern classical (and quantum) field theory, Lagrangian and Hamiltonian or presymplectic mechanics.

It should be stressed that Fermat's Principle has, in essence, a close relationship to modern Finsler geometry, as it rests on a specific "Lagrangian" F (x, y) = n(x) δ ij y i y j

(1.1) where y = (y i ) stands for the "velocity" of light, and n(x) > 0 for the value of the (smooth) refractive index of the medium, at the "location" x in Euclidean space.

(Note that Einstein's summation convention is tacitly understood throughout this article.) As a matter of fact, the function (1.1) is a Finsler metric, namely a positive function, homogeneous of degree one in the velocity, smooth wherever y = 0, and such that the Hessian g ij (x, y) = 1 2 F 2 y i y j is positive-definite (see Section 2.1). Although, this is a very special case of Finsler metric -it actually defines a conformally flat Riemannian metric tensor, viz., g ij (x) = n 2 (x) δ ij -, this fact is worth noting for further generalization. The geodesics of the Fermat-Finsler metric (1.1) are a fairly good mathematical model for light rays in refractive, inhomogeneous, and non dispersive media -provided polarization of light is ignored! It has quite recently been envisaged to consider a general Finsler metric F (x, y) to describe anisotropy of optical media, as the Finsler metric tensor, g ij (x, y), depends, in general non trivially, on the direction of the velocity, y, or "élément de support" in the sense of Cartan [18]. This enables one to account for the fact that [START_REF] Antonelli | The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology[END_REF][START_REF] Ingarden | On physical applications of Finsler geometry[END_REF] In an anisotropic medium, the speed of light depends on its direction, and the unit surface is no longer a sphere. (Finsler, 1969) The Fermat Principle has also been reformulated in the presymplectic framework in [START_REF] Cariñena | On the symplectic structure arising in geometric optics[END_REF], and generalized in [START_REF] Cariñena | Presymplectic geometry and Fermat's principle for anisotropic media[END_REF] to the context of anisotropic media. By the way, the regularity condition imposed, in the latter reference, amounts to demanding a Finsler structure.

One thus takes for granted that oriented Finsler geodesics may describe light rays in anisotropic media.

A Finslerian version of the Fermat Principle now states that the second-order differential equations governing the propagation of light stem from the geodesic spray of a (three-dimensional) Finsler space, (M, F ), given by the Reeb vector field of the contact 1-form

̟ = ω 3 , (1.2)
where ω 3 = F y i dx i is, here, the restriction to the indicatrix-bundle, SM = F -1 (1), of the Hilbert 1-form. See Section 3, and also [START_REF] Foulon | Géométrie des équations différentielles du second ordre[END_REF].

Now, geometrical optics is, from a different standpoint, widely accepted as a semi-classical limit [START_REF] Born | Principles of optics[END_REF] of wave optics with "small parameter" the reduced wavelength λ (or λ/L, where L is some characteristic length of the optical medium, see, e.g., [START_REF] Yu | Non-Abelian evolution of electromagnetic waves in a weakly anisotropic inhomogeneous medium[END_REF]). It has, however, recently been established on experimental grounds that trajectories of light beams in inhomogeneous optical media depart from those predicted by geometrical optics. See, e.g. [START_REF] Yu | Topological spin transport of photons: the optical Magnus Effect and Berry Phase[END_REF][START_REF] Yu | Conservation of Angular Momentum, Transverse Shift, and Spin Hall Effect in Reflection and Refraction of Electromagnetic Wave Packet[END_REF][START_REF] Yu | Geometrical optics of beams with vortices: Berry phase and orbital angular momentum Hall effect[END_REF], and [START_REF] Onoda | Hall Effect of Light[END_REF][START_REF] Onoda | Geometrical Aspects in Optical Wavepackets Dynamics[END_REF], for several approaches to photonic dynamics in terms of a semi-classical limit of the Maxwell equations in inhomogeneous, and isotropic media, highlighting the Berry connection [START_REF] Berry | Quantal phase factors accompanying adiabatic changes[END_REF]. See also [START_REF] Gosselin | Spin Hall effect of Photons in a Static Gravitational Field[END_REF]. The so-called optical Hall effect for polarized light rays, featuring a very small transverse shift, orthogonal to the gradient of the refractive index, has, hence, received a firm theoretical explanation.

From quite a different perspective, a theory of spinning light in arbitrary threedimensional Riemannian manifold has been put forward as an extension of the Fermat Principle to classical, circularly polarized photons, in inhomogeneous, (essentially) isotropic, media. This theory of spinoptics, presented in [START_REF] Duval | Geometrical Spinoptics and the Optical Hall Effect[END_REF], and [START_REF] Duval | Fermat Principle for spinning light and the Optical Hall effect[END_REF],

relies fundamentally on the Euclidean group, E(3), viewed at the same time as the group of isometries of Euclidean space, E 3 , and as the group of symmetries of classical states of free photons represented by Euclidean coadjoint orbits. Straight-forward adaptation of the general relativistic prescription of minimal coupling [START_REF] Künzle | Canonical dynamics of spinning particles in gravitational and electromagnetic fields[END_REF][START_REF] Souriau | Modèle de particule à spin dans le champ électromagnétique et gravitationnel[END_REF][START_REF] Sternberg | On the Role of Field Theories in our Physical Conception of Geometry[END_REF] readily yielded a set of differential equations governing the trajectories of spinning light in inhomogeneous, and isotropic media described by a Riemannian structure. Also did this formalism for spinoptics help to put the optical Hall effect in proper geometrical perspective, in agreement with [START_REF] Onoda | Hall Effect of Light[END_REF].

The main purport of the present article is, as might be expected, to try and provide a fairly natural extension of plain geometrical optics -in non-dispersive, anisotropic, media described in terms of Finsler geodesics -to spinoptics, i.e., to the case of circularly polarized light rays carrying color and helicity in such general optical media. In doing so, one must unavoidably choose a linear Finsler connection from the start (see (1.3)), the crux of the matter being that there is, apart from the special Riemannian case, no canonical Finsler connection at hand. The challenge may, in fact, be accepted once we take seriously the Euclidean symmetry as a guiding principle, a procedure that can be implemented by considering the dipole approximation to ordinary geometrical optics, namely the spinning coadjoint orbits of the Euclidean group. This is the subject of Section 4 which contains the main results of this article, where the Finsler-Cartan connection prevailed definitely over other Finsler connections, as regard to the original, fundamental, Euclidean symmetry of the free model. Let us, however, mention that all resulting expressions, for the foliations we end up with, ultimately depend upon the Finsler metric tensor, the Cartan tensor, and the Chern curvature tensors only.

The hereunder proposed principle of Finsler spinoptics (see Axiom 4.5, which can be understood the prescription of minimal coupling to a Finsler-Cartan connection) thus amounts to consider, instead of (1.2), the following 1-form

̟ = ω 3 + λ ω 12 , (1.3)
where λ is the (signed) wavelength, the ω ab , with a, b = 1, 2, 3, representing the components of the Cartan connection associated with a three-dimensional Finsler manifold (M, F ). The 1-form (1.3) might be considered as providing a deformation of the Hilbert 1-form driven by the wavelength parameter, λ. See Remark 4.6 below.

The characteristic foliation of the novel 2-form σ = d̟ is explicitly calculated, and leads to a drastic deviation from the Finsler geodesic spray, dictated by spincurvature coupling terms which play, in this formalism, quite a significant rôle, as expressed by Theorem 4.12. Of course, the equations of spinoptics in a Riemannian manifold [START_REF] Duval | Geometrical Spinoptics and the Optical Hall Effect[END_REF] are recovered, as special case of those corresponding to a Finsler-Cartan structure. We assert that this foliation can be considered a natural extension of the Finsler geodesics spray to the case of spinoptics in Finsler-Cartan spaces.

The paper is organized as follows.

Section 2 provides a survey of local Finsler geometry. We found it necessary to offer a somewhat technical and detailed introduction of the objects pertaining to Finsler geometry, in particular to the various connections used throughout this article, to make the reading easier to non experts. Emphasis is put on the Chern and Cartan connections, as these turn out to be of central importance in this study.

This section relies essentially on the authoritative Reference [START_REF] Bao | An Introduction to Riemann-Finsler Geometry[END_REF].

In Section 3, we review the principles of geometrical optics, extending Fermat's optics to the area of Finsler structures characterizing anisotropic optical media.

Then, special attention is paid to the Hilbert 1-form in the derivation of the Finsler geodesic spray. The connection of the latter to the Fermat differential equations associated with conformally related Finsler structures is furthermore analyzed.

Section 4 constitutes the major part of the article. It presents, in some details, the basic structures arising in the classification of the SE(3)-homogeneous symplectic spaces, which are interpreted as the seeds of spinoptics, namely the Euclidean coadjoint orbits labeled by color, and spin, according to the classic [START_REF] Souriau | Structure des systèmes dynamiques[END_REF]. The core of our study consists in the choice of a special Finsler connection, namely the Finsler-

T M \ M = {(x, y) ∈ T M | y ∈ T x M \ {0}}
is smooth, and (fiberwise) positively homogeneous of degree one, i.e., F (x, λy) = λF (x, y), for all λ > 0; one furthermore demands that the n × n Hessian matrix with entries

g ij (x, y) = 1 2 F 2 y i y j (2.1)
be positive-definite for all (x, y) ∈ T M \M . The quantities g ij defined in (2.1) are (fiberwise) homogeneous of degree zero, and

g = g ij (x, y)dx i ⊗ dx j (2.2)
defines a sphere's worth of Riemannian metrics [START_REF] Bao | Ricci and Flag Curvatures in Finsler Geometry[END_REF] on each T x M parametrized by the direction of y. We will put (g ij ) = (g ij ) -1 . If π : T M \ M → M stands for the canonical surjection, the metric (or fundamental) "tensor" (2.2) is, actually, a section of the bundle π * (T * M ) ⊗ π * (T * M ) → T M \M .

The distinguished "vector field" (the direction of the the supporting element)

u = u i ∂ ∂x i , where u i (x, y) = y i F (x, y) , (2.3) is, indeed, a section of π * (T M ) → T M \M such that g(u, u) = g ij u i u j = 1.
There is, at last, another tensor specific to Finsler geometry, namely the Cartan tensor, C = C ijk (x, y)dx i ⊗ dx j ⊗ dx k , where C ijk (x, y) = 1 4 F 2 y i y j y k . As in [START_REF] Bao | An Introduction to Riemann-Finsler Geometry[END_REF], we will also use ad lib. the quantities

A ijk = F C ijk , (2.4)
which are totally symmetric, A ijk = A (ijk) , and enjoy the following property, viz.,

A ijk u k = 0. (2.5)
There is a wealth of Finsler structures, apart from the well-known special case of Riemannian structures (M, g) for which F (x, y) = g ij (x)y i y j . See, e.g., [START_REF] Bao | An Introduction to Riemann-Finsler Geometry[END_REF][START_REF] Bao | Ricci and Flag Curvatures in Finsler Geometry[END_REF][START_REF] Shen | Landsberg Curvature, S-Curvature and Riemann Curvature[END_REF] for a survey, and for a list of examples of Finsler structures. We will review below, see (3.11)- (3.13), examples of Finsler structure associated with optical birefringence [START_REF] Antonelli | The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology[END_REF].

Introducing special orthonormal frames

Having chosen a coordinate system (x i ) of M , we denote -with a slight abuse of notation -by ∂/∂x i (resp. dx i ) the so-called transplanted sections of π * (T M ) (resp. π * (T * M )), Accordingly we will denote by ∂/∂y i (resp. dy i ) the standard, vertical, sections of T (T M \M ) (resp. T * (T M \M )).

Introduce now special g-orthonormal frames (e 1 , . . . , e n ) for π * (T M ), such that g(e a , e b ) = δ ab ,

for all a, b = 1, . . . , n, with, as preferred element, the distinguished section

e n = u. (2.7)
Recall that each e a lies in the fiber π * (T M ) (x,y) above (x, y) ∈ T M \M . The local decomposition of these vectors is given by

e a = e i a ∂ ∂x i , (2.8)
for all a = 1, . . . , n, where the matrix (e i a ), defined at (x, y) ∈ T M\M , is nonsingular.

We thus have

e i n = u i .
(2.9)

The dual frames, for π * (T * M ), which we denote by (ω 1 , . . . , ω n ), are such that ω a (e b ) = δ a b , for all a, b = 1, . . . , n. Accordingly, we have the local decomposition

ω a = ω a i dx i , (2.10)
for all a = 1, . . . , n, where (ω a i ) = (e i a ) -1 . The 1-form dual to e n is the Hilbert form

ω H = ω n (2.11)
which, in view of (2.3), reads ω H = u i dx i , with u i = g ij u j = F y i .

The following proposition introduces the principal bundle of orthonormal frames above the slit tangent bundle of a Finsler manifold.

Proposition 2.1. The manifold, SO n-1 (T M \M ), of special g-orthonormal frames for π * (T M ) is endowed with a structure of SO(n -1)-principal bundle over T M\M .

If GL + (M ) stands for the bundle of positively oriented linear frames of M , it is

defined by SO n-1 (T M\M ) = Ψ -1 (0) where Ψ : π * (GL + (M )) → R 1 2 n(n+1) × R n-1 is given by Ψ(x, y, (e a )) = ((g (x,y) (e a , e b ) -δ ab ), y -F (x, y)e n ), where a, b = 1, . . . , n.
The proof is straightforward as is that of the following statement. The pull-back SO n-1 (SM ) = ι * (SO n-1 (T M \M )) is a principal SO(n -1)-bundle over SM .

The non-linear connection

Let us recall that the Finsler metric, F , induces in a canonical fashion a splitting of the tangent bundle of the slit tangent bundle π : T M \M → M of M as follows:

T (T M \M ) = V (T M \M ) ⊕ H(T M \M ), (2.12)
where the vertical tangent bundle is V (T M \M ) = ker π * . The fibers, V (x,y) , of that subbundle are spanned by the vertical local basis vectors ∂/∂y i , with i = 1, . . . , n.

Those, H (x,y) , of the horizontal subbundle H(T M\M ) are spanned by the horizontal

local basis vectors δ δx i = ∂ ∂x i -N j i ∂ ∂y j , (2.13)
with i = 1, . . . , n, where the N j i are the coefficients of a non-linear connection canonically defined by

N i j = 1 2 ∂G i ∂y j (2.14)
in terms of the spray coefficients

G j = 1 2 g jk (F 2 ) y k x l y l -(F 2 ) x k .
The horizontal vectors, δ/δx i , see (2.13), and the vertical vectors, ∂/∂y i , form a local natural basis for T (x,y) (T M \M ) whose dual basis is given by dx i and δy i , where

δy i = dy i + N i j dx j . (2.15)
We now introduce, following (2.8), (2.10) the g-orthonormal frames that will be needed in the sequel.

Definition 2.3. We will call hv-frame any frame for T (T M \M ), compatible with the splitting (2.12), namely [START_REF] Bao | An Introduction to Riemann-Finsler Geometry[END_REF] êa

= e i a δ δx i , êā = e i a F ∂ ∂y i , (2.16)
where ā = a + n, with a = 1, . . . , n. The associated dual basis reads then

ω a = ω a i dx i , ω ā = ω a i δy i F , (2.17)
with a = 1, . . . , n; see (2.10) and (2.15).

Owing to the properties of the non-linear connection (2.14), the metric, F , is

horizontally constant, δF δx i = 0, (2.18)
for all i = 1, . . . , n. This entails that the vertical 1-form ω n is exact,

ω n = d log F. (2.19)

Finsler connections

Unlike the Riemannian case, there is no canonical linear Finsler connection on the bundle π * (T M ), which is required as soon as one needs to differentiate tensor fields, i.e., sections of the bundles π * (T M ) ⊗p ⊗ π * (T * M ) ⊗q .

The Chern connection

A celebrated example, though, is the Chern connection ω i j = Γ i jk (x, y)dx k which is uniquely defined by the following requirements [START_REF] Bao | An Introduction to Riemann-Finsler Geometry[END_REF]: (i) it is symmetric: Γ i jk = Γ i kj , and (ii) it almost transports the metric tensor:

dg ij -ω k i g jk -ω k j g ik = 2 C ijk δy k
, where the δy k are as in (2.15).

The Chern connection coefficients turn out to yield the alternative expression of the non-linear connection, namely,

N i j (x, y) = Γ i jk y k . (2.20)
The covariant derivative

∇ : Γ(π * (T M )) → Γ(T (T M\M ) ⊗ π * (T M )) associated
with the Chern connection is related to the ω i j via ∇ X ∂/∂x j = ω i j (X)∂/∂x i , for all X ∈ Vect(T M \M ). In terms of the special g-orthonormal frames, we can write

∇ X e b = ω a b (X)e b , for all X ∈ Vect(T M \M ) where ω a b = ω a i (de i b + ω i j e j b
) denote the frame components of the Chern connection. The following theorem summarizes the defining properties of the Chern connection.

Theorem 2.4. [START_REF] Bao | An Introduction to Riemann-Finsler Geometry[END_REF] There exists a unique linear connection, (ω a b ), on π * (T M ), named the Chern connection, which is torsionfree and almost g-compatible, namely

Ω a = dω a -ω b ∧ ω a b = 0 (2.21)
and

ω ab + ω ba = -2A abc ω c, (2.22)
where a, b, c = 1, . . . , n. The corresponding connection coefficients are of the form

Γ i jk = 1 2 g il δg kl δx j + δg jl δx k - δg jk δx l , (2.23)
for all i, j, k = 1, . . . , n.

The curvature, (Ω a b ), of the linear connection, (ω a b ), is defined by the structure equations

Ω a b = dω a b -ω c b ∧ ω a c . It retains the form Ω a b = 1 2 R a b cd ω b ∧ ω d + P a b cd ω b ∧ ω d, (2.24)
where

R i j kl = e i a ω b j ω c k ω d l R a b cd , reads R i j kl = δΓ i jl δx k - δΓ i jk δx l + Γ i mk Γ m jl -Γ i ml Γ m jk , (2.25)
and, accordingly,

P i j kl = -F ∂Γ i jk ∂y l . (2.26)
The hv-curvature, P , enjoys the fundamental property

P i j kl u l = 0. (2.27)
Using Cartan's notation [START_REF] Cartan | Les espaces de Finsler[END_REF] (see also [START_REF] Rund | The differential geometry of Finsler spaces[END_REF][START_REF] Bao | An Introduction to Riemann-Finsler Geometry[END_REF]), we write the covariant derivative of, e.g., a section X of π * (T M ) as (∇X) i = dX i + ω i j X j = X i |j dx j + X i j F -1 δy j , where i, j = 1, . . . , n. In particular, it can be easily deduced from (2.20) that the covariant derivative of the unit vector u is given by

(∇u) i = -u i d log F + δy i F , (2.28)
so that (2.18) leads to

u i |j = 0, u i j = δ i j -u i u j , (2.29)
for all i, j = 1, . . . , n.

Let us deduce from (2.28) a classical formula highlighting a special property of the Chern connection.

Proposition 2.5. There holds

ω a n = h a b ω b, (2.30)
where the h ab = δ abδ n a δ n b , with a, b = 1, . . . , n, are the frame-components of the "angular metric".

We end this section by a useful lemma (see Section 3.4 B in [START_REF] Bao | An Introduction to Riemann-Finsler Geometry[END_REF] for a proof).

Lemma 2.6. The first Bianchi identities imply

A ij[k l] = A ij[k u l]
, where the square brackets denote skew-symmetrization. If we define the covariant derivative of the Cartan tensor (2.4) in the direction, u, of of the supporting element by

Ȧijk = A ijk|l u l , (2.31)
then, the same Bianchi identities lead to P ijk = u l P lijk = -Ȧijk .

The Cartan connection

The Cartan connection is another prominent Finsler linear connection on π * (T M ) which is related to the Chern connection in a simple way.

Definition 2.7. [START_REF] Bao | On the Gauss-Bonnet integrand for 4dimensional Landsberg spaces[END_REF][START_REF] Bao | An Introduction to Riemann-Finsler Geometry[END_REF] Let (M, F ) be a Finsler structure with Cartan tensor (A abc ), and let (ω ab ) denote its canonical Chern connection. The frame components of the Cartan connection of (M, F ) are defined by

ω ab = ω ab + A abc ω c, (2.32)
where c = c + n, with a, b, c = 1, . . . , n.

The fundamental virtue of these connection 1-forms is the skewsymmetry

ω ab + ω ba = 0, (2.33)
that guarantees that the fundamental tensor is parallel, ∇g = 0, where ∇ stands for the covariant derivative associated with the Cartan connection. The Cartan connection is not symmetric; its torsion tensor Ω a = dω aω b ∧ ω a b , is nonzero. Indeed, Ω a = Ω aω b ∧ A abc ω c, and, since Ω a = 0, it retains the form

Ω a = -A a bc ω b ∧ ω c. (2.34)
One easily proves the following result.

Theorem 2.8. There exists a unique linear connection, ( ω a b ), on π * (T M ) whose torsion, ( Ω a ), is given by (2.34), and which is g-compatible, ∇g = 0, as expressed by (2.33). This connection is the Cartan connection (2.32).

Other characterizations of the Cartan connection can be found in the literature, e.g, in [START_REF] Abate | Finsler Metrics -A Global Approach[END_REF][START_REF] Antonelli | The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology[END_REF][START_REF] Bejancu | A Geometric Characterization of Finsler Manifolds of Constant Curvature K = 1[END_REF].

Proposition 2.9. The torsion of the Cartan connection is such that

Ω n = 0.
(2.35)

From now on, and whenever possible, we will use frame indices, a, b, c, . . ., rather than local coordinate indices, i, j, k, . . ..

Ω ab = Ω ab + A abc Ω c n + A abd|c ω c ∧ ω d + A e ad A bce ω c ∧ ω d (2.36)
where (Ω a b ) denotes the Chern curvature 2-form (2.24) with (2.25, 2.26), and (A abc ) the Cartan tensor (2.4). One has the decomposition

Ω a b = 1 2 R a b cd ω c ∧ ω d + P a b cd ω c ∧ ω d + 1 2 Q a b cd ω c ∧ ω d, (2.37)
with

R a b cd = R a b cd + A a be R e cd , (2.38)
P a b cd = P a b cd + A a bd|c -A a be Ȧe cd , (2.39)
Q a b cd = 2A a e[c A e d]b , (2.40)
where we use the notation R e cd = R e n cd , and Ȧabc = A abc|n (see (2.31)).

3 Geometrical optics in Finsler spaces

Finsler geodesics

Following Souriau's terminology [START_REF] Souriau | Structure des systèmes dynamiques[END_REF], we call evolution space the indicatrix-bundle

SM = F -1 (1) (3.1)
above M , as it actually hosts the dynamics given by a presymplectic structure; the latter will eventually be inherited from the Finsler metric on T M \M .

Denote, again, by ι : SM ֒→ T M\M the canonical embedding. The fundamental geometric object governing the geodesic spray on SM is the 1-form

̟ = ι * ω H , (3.2)
i.e., the pull-back on SM of Hilbert 1-form ω H (see (2.11)). The direction of this 1-form defines a contact structure on the (2n -1)-dimensional manifold SM , since ̟ ∧ (d̟) n-1 = 0. See [START_REF] Foulon | Géométrie des équations différentielles du second ordre[END_REF][START_REF] Chern | Riemannian geometry as a special case of Finsler geometry[END_REF][START_REF] Chern | Finsler Geometry Is Just Riemannian Geometry without the Quadratic Restriction[END_REF]. The following lemma is classical.

Lemma 3.1. The exterior derivative of the Hilbert 1-form is given by

dω H = δ AB ω Ā ∧ ω B (3.3)
with A, B = 1, . . . , n -1.

Remark 3.2. The exterior derivative of the Hilbert 1-form is independent of the choice of a linear connection; it depends only on the non-linear connection (2.12).

The main result regarding Finsler geodesics can be stated as follows. See also [START_REF] Foulon | Géométrie des équations différentielles du second ordre[END_REF] for a full account on the geometry of second order differential equations.

Theorem 3.3. The geodesic spray of a Finsler structure (M, F ) is the vector field X of SM uniquely defined by

σ(X) = 0, ̟(X) = 1, (3.4)
where σ = d̟.

Proof. Write X ∈ Vect(SM ) in the form X = X A êA + X n ên + X Ā ê Ā + X n ên , see (2.16), using dummy indices A and Ā = A + n, where A = 1, . . . , n -1. Since

X ∈ Vect(T M\M ) is tangent to SM iff X(F ) = ω n(X ) = 0,
(X Ā ω B -X B ω Ā)+λω n = 0, hence X A = X Ā = 0, and λ = 0. Then X = X n ên + X n ên is actually tangent to SM if ω n(X ) = X n = 0, which leads to X ∈ ker(σ) ⇐⇒ X = X n ên (3.5)
for some X n ∈ R. Thus, (SM, σ) is a presymplectic manifold. The quotient SM/ ker(σ) is the set of oriented Finsler geodesics, which (if endowed with a smooth structure) becomes a (2n -2)-dimensional symplectic manifold, see [START_REF] Alvarez Paiva | Some problems on Finsler geometry[END_REF].

We then find that ̟(X) = X n , and the constraints (3.4) express the fact that X is the Reeb vector field, and retains the form X = ên , which, in view of (2.7), we can write

X = u i δ δx i . (3.6)
The vector field (3.6) of SM is the geodesic spray [START_REF] Bao | An Introduction to Riemann-Finsler Geometry[END_REF] of the Finsler structure.

Geometrical optics in anisotropic media

The geodesic spray, X, given by (3.6), integrates to a Finsler geodesic flow, ϕ t , on the bundle SM via the ordinary differential equation dϕ t (x, u)/dt = X(ϕ t ((x, u))

for all t ∈ I ⊂ R. The latter translates as

       dx i dt = u i du i dt = -G i (x, u) (3.7)
where the acceleration components (or spray coefficients) read G i = N i j y j (see [START_REF] Bao | An Introduction to Riemann-Finsler Geometry[END_REF]), for i = 1, . . . , n. The geodesic flow then defines geodesics per se,

x t = π(ϕ t (x, u)),
of the base manifold, M , with initial data (x, u) ∈ SM .

The Fermat Principle

Definition 3.4. [START_REF] Knebelman | Conformal geometry of generalized metric spaces[END_REF] Two Finsler structures (M, F ) and (M, F ) are said to be

conformally related if F (x, y) = n(x)F (x, y) for some n ∈ C ∞ (M, R * + ).
If F is a Riemannian structure, then F is a Riemannian structure conformally related to F , since their metric tensors are such that g ij (x) = n 2 (x) g ij (x). In this case, the geodesics of (M, F ) may be interpreted as the trajectories of light in a medium, modeled on the Riemannian manifold (M, F ), and endowed with a refractive index n. This is, in essence, the Fermat Principle of geometrical optics.

Proposition 3.5. Let (M, F ) and (M, F ) be conformally related Finsler structures, i.e., be such that F (x, y) = n(x)F (x, y) for a given n ∈ C ∞ (M, R * + ), called their relative refractive index. Their geodesic sprays are related as follows:

X = u i δ δx i & X = 1 n u i δ δx i + 1 n 3 (g ij -2u i u j ) ∂n ∂x j ∂ ∂y i , (3.8)
where u i = y i /F , for i = 1, . . . , n. Putting ẋi = n X(x i ), and ẏi = n X(y i ), we obtain the equations of the geodesics of (M, F ) in the following guise:

   ẋi = u i ∇ u (n u) i = g ij ∂n ∂x j (3.9)
where ∇ u is the covariant derivative with reference vector u, defined, for all vector field, v, along the curve with velocity u, by ∇ u (v

) i = vi + Γ i jk (x, u)u j v k .
Proof. The Hilbert 1-forms are related by ω H = n ω H , and their exterior derivatives

by σ = n σ + dn ∧ ω H . In other words σ = n δ AB ω Ā ∧ ω B + n A ω A ∧ ω n , where n A = e i A ∂ i n.
Reproducing the proof of Theorem 3.3, we will decompose X ∈ Vect( SM ) as X = X A êA + X n ên + X Ā ê Ā + X n ên , with the same notation as before. Again

X ∈ ker( σ) iff d ω H ( X) + λ ω n = 0 for some λ ∈ R. (Note that, in view of (2.19),
we have ω n = ω n + dn/n.) This equation readily leaves us with X A = 0, and X Ā = n A X n , for all A = 1, . . . , n -1, together with λ = 0.

At last, X is tangent to SM if ω n( X) = 0, i.e., if X n = -(n n /n) X n . Then, X is the Reeb vector field for F if ω H ( X) = 1, i.e., if X n = 1/n. The geodesic spray of the Finsler structure (M, F ) is thus

X = 1 n ên + n A n ê Ā - n n n ên
while that of the Finsler structure (M, F ) reduces to X = ên by letting n = 1.

We thus recover (3.8) via (2.16) and (2.9), and also by the following fact, viz.,

n A ê Ā -n n ên = (δ AB e i A e j B -e i n e j n )∂ j n F ∂/∂y i = (g ij -2u i u j )(∂ j n/n)∂/∂y i , since F = n F = 1 on SM .
Let us now derive Equations (3.9); we first notice that ẋi = n X(x i ) = u i , and also that ẏi = n X(y i ) = -N i j u j +(F/n)(g ij -2u i u j )∂ j n. Defining, as in, e.g., [START_REF] Bao | An Introduction to Riemann-Finsler Geometry[END_REF], the covariant derivative of the vector field v, with reference vector u, by the expression (2.20), enables us to compute the "geodesic acceleration" ∇ u (n u). Since (n u) i = (n 2 y) i , we get ∇ u (n u) i = 2n ṅ y i + n 2 ∇ u (y) i = 2n u j ∂ j n y i + n 2 ( ẏi + N i j u j ) = 2n u j ∂ j n y i + n 2 (F/n)(g ij -2u i u j )∂ j n = g ij ∂ j n, and we are done.

∇ u (v) i = vi +Γ i jk u j v k = vi +N i j v j /F , see
Remark 3.6. The differential equations (3.9) generalize, to the Finsler framework, the Fermat equations ruling the propagation of light in a Riemannian manifold, through a dielectric medium of refractive index n.

Finsler optics

It has originally been envisioned by Finsler himself (see, e.g., [START_REF] Antonelli | The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology[END_REF][START_REF] Ingarden | On physical applications of Finsler geometry[END_REF]) that the in-

dicatrix S x M = {u ∈ T x M | F (x, u) = 1} of a Finsler structure (M, F ) might
serve as a model for the geometric locus of the "phase velocity" of light waves at a point x ∈ M . The fact that, in anisotropic optical media, the velocity of a (plane) light-wave specifically depends upon the direction of its propagation, prompted him to put forward a classical (as opposed to field-theoretical) model of geometrical optics in anisotropic, non dispersive, media ruled by Finsler structures. Finsler geodesics have therefore consistently received the interpretation of light trajectories in such optical media. Let us mention, among many an example, an application of Finsler optics to dynamical systems engendered by Finsler billiards [START_REF] Gutkin | Billiards in Finsler and Minkowski geometries[END_REF].

When specialized to Riemannian structures, e.g., to Fermat structures presented in Section 3.2.1, Finsler geodesics are nothing but plain Riemannian geodesics, regarded as light rays in (non homogeneous) isotropic media. See, e.g., [START_REF] Cariñena | On the symplectic structure arising in geometric optics[END_REF][START_REF] Cariñena | Presymplectic geometry and Fermat's principle for anisotropic media[END_REF][START_REF] Duval | Geometrical Spinoptics and the Optical Hall Effect[END_REF][START_REF] Duval | Fermat Principle for spinning light and the Optical Hall effect[END_REF]. This justifies the following principle of Finsler geometrical optics. Definition 3.7. The light rays in a non-homogeneous, anisotropic, optical medium described by a Finsler structure (M, F ) are the oriented geodesics associated with the geodesic spray (3.6) of this Finsler structure.

For example, birefringent media (solid or liquid crystalline media) can be described by a pair of Finsler metrics, namely, the ordinary (resp. extraordinary) metric F o (resp. F e ) attached to a (three-dimensional) manifold, M , representing the anisotropic optical medium. Those are respectively given, in the particular case of uniaxial crystals, in terms of a pair of Riemannian metrics a = a ij (x) dx i ⊗ dx j , and b = b ij (x) dx i ⊗ dx j on M , by [START_REF] Antonelli | The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology[END_REF][START_REF] Ingarden | On physical applications of Finsler geometry[END_REF]

F o (x, y) = a ij (x) y i y j , (3.10
)

F e (x, y) = a ij (x) y i y j b ij (x) y i y j . (3.11)
The geodesics of the metric F e are meant to describe extraordinary light rays, whereas those of the Riemannian metric, F o , will merely lead to ordinary rays.

Remark 3.8. Let us emphasize that (M, F e ), where F e is as in (3.11), is a Finsler structure if its fundamental tensor

g e ij = F 2 e 2 a ij a(y, y) - b ij b(y, y) + 4 c i c j b(y, y) 3 , (3.12)
where c i = a(y, y) b ij (x)y jb(y, y) a ij (x)y j , is positive definite. This is, indeed, the case if the quadratic forms a and b verify b/ √ 2 < a < b √ 2, everywhere on T M \M .

The more complex case of biaxial optical media is also studied in [START_REF] Antonelli | The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology[END_REF], and gives rise to a pair of specific Finsler metrics

F ± (x, y) = a ij (x) y i y j b ± ij (x) y i y j (3.13)
where a, b + , and b -are Riemannian metrics characterizing the optical properties of the anisotropic medium. (Let us note that Remark 3.8 applies just as well for the metrics (3.13)).

The example of birefringent solid crystals

Let us review how Finsler metrics of the form (3.13), or (3.10) and (3.11), arise in the particular case of anisotropic solid crystals. To that end, we revisit the original derivation [START_REF] Antonelli | The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology[END_REF] of the Minkowski norms that account for the propagation of light in anisotropic dielectric solids with principal (positive) velocities v 1 , v 2 , v 3 .

In the framework of Maxwell's wave optics, the Fresnel equation of wave normals

û2 1 ( u 2 -v 2 2 )( u 2 -v 2 3 ) + û2 2 ( u 2 -v 2 3 )( u 2 -v 2 1 ) + û2 3 ( u 2 -v 2 1 )( u 2 -v 2 2 ) = 0
expresses the dependence of the phase velocity, u, of a plane wave upon its direction of propagation, û = u/ u , in such a medium; we denote, here, by • the norm on standard Euclidean space (R 3 , • , • ).

-Assuming, e.g., v 1 > v 2 > v 3 , one solves the Fresnel equation for the norm of the phase velocity, viz., u 2 = A + B cos(θ ′ ± θ ′′ ) where θ ′ and θ ′′ are the angles between the direction of propagation, û, and the (oriented) optical axes e ′ and e ′′ ;

the scalars

A = 1 2 (v 2 1 + v 2 3 ), and B = 1 2 (v 2 1 -v 2
3 ), as well as the vectors e ′ , and e ′′ , are characteristic of the crystal [START_REF] Born | Principles of optics[END_REF]. The Minkowski norm, F , associated with each solution of the Fresnel equation is easily found [3] using Okubo's trick that amounts to the replacement u y/F (y), insuring that F (u) = 1. Easy calculation leads us

to u 2 = y 2 /F (y) 2 = A + B y -2 [ e ′
, y e ′′ , y ∓ e ′ × y e ′′ × y ], that is, to

F ± (y) = y 2 A y 2 + B e ′ , y e ′′ , y ∓ e ′ × y e ′′ × y , (3.14)
where × denotes the standard Euclidean cross-product. This expression admits straightforward generalizations to the case of fluid crystals, Faraday-active media, etc., where the quantities A, B, e ′ , and e ′′ become position-dependent; it ultimately leads to the expression (3.13) of a pair of Finsler metric for general biaxial media.

-The case of uniaxial media is treated by assuming, e.g., v 1 = v 2 > v 3 , which implies e ′ = e ′′ (= e). The Minkowski norms (3.14) admit a prolongation to this situation and read -The last case, for which v 1 = v 2 = v 3 , clearly leads to a single Euclidean metric, namely F (y) = y /v 1 , that rules geometrical optics in isotropic media with refractive index n = 1/v 1 .

F -(y) = y v 1 , (3.15)
F + (y) = y 2 v 2 3 y 2 + (v 2 1 -v 2 3 ) e, y 2 . ( 3

Geometrical spinoptics in Finsler spaces

So far, the polarization of light has been neglected in the various formulations of geometrical optics. We contend that spinning light rays do, indeed, admit a clear cut geometrical status allowing for a natural extension of plain geometrical optics to the case of circularly polarized light rays (Euclidean photons) traveling in arbitrary non dispersive optical media.

The touchstone of our viewpoint about geometrical optics for spinning light is the Euclidean symmetry of the manifold of oriented lines in (flat) Euclidean space.

This fundamental symmetry will be taken as a guiding principle to set up a model that could describe the geometry of spinning light rays in quite general, crystalline and liquid, optical media. See [START_REF] Duval | Geometrical Spinoptics and the Optical Hall Effect[END_REF] and [START_REF] Duval | Fermat Principle for spinning light and the Optical Hall effect[END_REF] for a first approach to geometrical spinoptics in inhomogeneous, isotropic, media.

We will therefore start by some elementary facts about the symplectic structure of the space of oriented lines in Euclidean space. The consideration of the generic coadjoint orbits of the Euclidean group will then be justified on physical grounds.

Let us recall that, if we denote by Ad the adjoint action of a Lie group, and by Coad, its coadjoint action, then the orbits of the latter action inherit a canonical structure of symplectic manifolds. These homogeneous symplectic manifolds play a central rôle in mechanics and physics, where some of them may be interpreted as the elementary systems associated with the symmetry group under consideration [START_REF] Souriau | Structure des systèmes dynamiques[END_REF].

The following construction is standard.

Theorem 4.1. Let G be a (finite-dimensional) Lie group G with Lie algebra g. Fix µ 0 ∈ g * and define the following 1-form

̟ µ 0 = µ 0 • ϑ G , (4.1)
where ϑ G is the left-invariant Maurer-Cartan 1-form of G. Then, σ µ 0 = d̟ µ 0 is a presymplectic 2-form on G which is the pull-back of the canonical Kirillov-Kostant-Souriau symplectic 2-form on the G-coadjoint orbit

O µ 0 = {µ = Coad g (µ 0 ) | g ∈ G} ∼ = G/G µ 0 , (4.2)
where G µ 0 is the stabilizer of µ 0 ∈ g * .

Spinoptics and the Euclidean group

From now on we will confine considerations to three-dimensional configuration spaces to comply with the physical principles of geometrical optics.

An oriented straight line, ξ, in Euclidean affine space (E 3 , • , • ) is determined by its direction, a vector u ∈ R 3 of unit length, and an arbitrary point Q ∈ ξ. Having chosen an origin, O ∈ E 3 , we may consider the vector q = Q -O, orthogonal to u.

The set of oriented, non parametrized, straight lines is thus the smooth manifold

M = {ξ = (q, u) ∈ R 3 × R 3 | u, u = 1, u, q = 0}, (4.3)
i.e., the tangent bundle M ∼ = T S 2 of the round sphere S 2 ⊂ R 3 , which has been recognized by Souriau [START_REF] Souriau | Structure des systèmes dynamiques[END_REF] as a coadjoint orbit of the group, E(3), of Euclidean isometries, and inherits, as such, an E(3)-invariant symplectic structure. See also [START_REF] Iglesias | Symétries et moment[END_REF].

Consider the group, SE(3) = SO(3) ⋉ R 3 , of orientation-preserving Euclidean isometries of (E 3 , • , • , vol), viewed as the matrix-group whose elements read

g = R x 0 1 , (4.4)
where R ∈ SO(3), and x ∈ R 3 .

The (left-invariant) Maurer-Cartan 1-form of SE( 3) is therefore given by

ϑ SE(3) = ω ω 0 0 , (4.5)
where ω = R -1 dR, and ω = R -1 dx.

Let µ = (S, P) denote a point in e(3) * where e(3) = o(3) ⋉ R 3 is the Lie algebra of SE(3). We will use the identification o(3

) ∼ = Λ 2 R 3 (resp. (R 3 ) * ∼ = R 3 )
given by S b a = S ab (resp. P a = P a ), where

S ab + S ba = 0, (4.6)
for all a, b = 1, 2, 3. The pairing e(3) * × e(3) → R will be defined by

(S, P) • ( ω, ω) = - 1 2 Tr(S ω) + P, ω (4.7)
= 1 2 S ab ω ab + P a ω a . (4.8)
The coadjoint representation of SE(3), viz., Coad g µ ≡ µ • Ad g -1 , is given by Coad g (S, P) = (R(S + x ∧ P)R -1 , RP). Clearly, C = P, P and C ′ = (S ∧ P)/vol are coadjoint SE(3)-invariants. These are the only invariants of the Euclidean coadjoint representation, and fixing (C, C ′ ) or (C = 0, C ′′ ), where C ′′ = 1 2 S ab S ab , yields a single coadjoint orbit [START_REF] Guillemin | Symplectic techniques in physics[END_REF][START_REF] Marsden | Introduction to Mechanics and Symmetry[END_REF][START_REF] Souriau | Structure des systèmes dynamiques[END_REF].

Colored light rays

Specializing the construction of Theorem 4.1 to the case G = SE(3), with C = p 2 and p > 0 together with C ′ = 0, we can choose µ 0 = (0, P 0 ) and P 0 = (0, 0, p). The invariant p is the color [START_REF] Souriau | Structure des systèmes dynamiques[END_REF] of the chosen coadjoint orbit.

The 1-form (4.1) then associated, via the pairing (4.8), and the Maurer-Cartan 1-form (4.5), with the invariant p is thus ̟ µ 0 = P 0 , ω , or

̟ µ 0 = P a ω a (4.9)
= p ω 3 . (

Straightforward calculation yields ̟ µ 0 = p u i dx i , with u i = δ ij u j , where u = e 3 is the third vector of the orthonormal, positively oriented, basis R = (e 1 , e 2 , e 3 ).

The 1-form (4.10) is, up to an overall multiplicative constant, p, equal to the canonical 1-form (3.2) on the sphere-bundle SE 3 , associated with the trivial Finsler structure (E 3 , F ), with F (x, y) = y, y . Proposition 3.5 just applies, with n = 1, and conforms to Euclid's statement that light, whatever its color, travels in vacuum along oriented geodesics of E 3 . Indeed, the exterior derivative of the 1-form ̟ µ 0 of SE 3 reads σ µ 0 = p du i ∧ dx i . Its kernel, given by

X ∈ ker(σ µ 0 ) ⇐⇒ X = λ u i ∂ ∂x i , (4.11)
with λ ∈ R, yields the (flat Euclidean) geodesic spray (λ = 1). We will resort to generalizations of this particular construct of the geodesic foliation.

The integral invariant, σ µ 0 , descends, as a symplectic 2-form, to the quotient M = SE 3 / ker(σ µ 0 ) described by ξ = (q, u), where q = xu u, x . This is the content of Theorem 4.1 insuring that M ∼ = T S 2 ⊂ e(3) * is endowed with a canonical symplectic structure, namely (M, p du i ∧ dq i ).

We note that the SE(3)-coadjoint orbit O µ 0 is an E(3)-coadjoint orbit.

The spinning and colored Euclidean coadjoint orbits

The generic SE(3)-coadjoint orbits are, in fact, characterized by the Casimir invariants C = p 2 , with p > 0 (color), and C ′ = sp where s = 0 stands for spin. We call helicity the sign of the spin invariant, ε = sign(s).

The origin, µ 0 , of such an orbit can be freely chosen so as to satisfy the constraints S ab P b = 0, for all a = 1, 2, 3, and 1 2 S ab S ab = s 2 , together with P a P a = p 2 . One may posit

S 0 =   0 -s 0 s 0 0 0 0 0   and P 0 = (0, 0, p). (4.12)
The coadjoint orbit, O µ 0 , passing through µ 0 = (S 0 , P 0 ) ∈ e(3) * is, again, diffeomorphic to T S 2 . It is endowed with the symplectic structure coming from the 1-form (4.1) on the group SE(3), which now reads

̟ µ 0 = P a ω a + 1 2 S ab ω ab (4.13) = p ω 3 + s ω 12 , (4.14)
where ω (resp. ω) stand for the flat Levi-Civita connection (resp. soldering) 1-form on the bundle, SE(3) ∼ = SO(E 3 ), of positively oriented, orthonormal frames of E 3 .

Remark 4.2. The 1-form (4.13) is the central geometric object of the present study.

SE(3)

x x r r r r r r r r r r Straightforward computation then leads to ̟ µ 0 = p e 3 , dxs e 1 , de 2 . The exterior derivative of ̟ µ 0 is found [START_REF] Souriau | Structure des systèmes dynamiques[END_REF][START_REF] Guillemin | Symplectic techniques in physics[END_REF][START_REF] Marsden | Introduction to Mechanics and Symmetry[END_REF][START_REF] Duval | Geometrical Spinoptics and the Optical Hall Effect[END_REF] to be given by

O µ 0 ∼ = T S 2 SR 3 o o / / R 3
σ µ 0 = p du i ∧ dx i - s 2 ǫ ijk u i du j ∧ du k , (4.15)
where, again, u = e 3 , and also ǫ ijk stands for the signature of the permutation {1, 2, 3} → {i, j, k}. This 2-form conspicuously descends to SR 3 ∼ = R 3 × S 2 . Its characteristic foliation is, verbatim, given by (4.11): spinning light rays in vacuum are nothing but oriented Euclidean geodesics. As shown in the sequel, things will change dramatically for such light rays in a refractive medium.

Remark 4.3. Actually, "photons" are characterized by |s| = , where is the reduced Planck constant; right-handed photons correspond to s = + , and lefthanded ones to s = -, see [START_REF] Souriau | Structure des systèmes dynamiques[END_REF]. We will, nevertheless, leave the parameter s arbitrary when dealing with "spinning light rays".

The manifold of spinning light rays, O µ 0 = SE 3 / ker(σ µ 0 ) ∼ = T S 2 (see Fig. 1) is, just as before, parametrized by the pairs ξ = (q, u) and endowed with the "twisted" symplectic 2-form ω µ 0 = p du i ∧ dq is 2 ǫ ijk u i du j ∧ du k . Note that the union of two SE(3)-coadjoint orbits defined by the invariants (p, s) and (p, -s) is symplectomorphic to a single E(3)-coadjoint orbit.

Remark 4.4. The SE(3)-coadjoint orbits of spin s, and color p, are symplectomorphic to Marsden-Weinstein reduced massless SE(3, 1) 0 -coadjoint orbits of spin s, at given (positive) energy E = pc, where c stands for the speed of light in vacuum, see [START_REF] Duval | Geometrical Spinoptics and the Optical Hall Effect[END_REF]. This justifies that the color, p, of Euclidean light rays corresponds, via reduction, to the energy of relativistic photons, or to the frequency of their associated monochromatic plane waves [START_REF] Souriau | Structure des systèmes dynamiques[END_REF][START_REF] Guillemin | Symplectic techniques in physics[END_REF][START_REF] Duval | Fermat Principle for spinning light and the Optical Hall effect[END_REF].

Spinoptics in Finsler-Cartan spaces

With these preparations, we formulate the principles of geometrical spinoptics, with the premise that (i) Finsler structures should be considered a privileged geometric background for the description of inhomogeneous, anisotropic, optical media, (ii) the original Euclidean symmetry which pervades geometrical optics should be invoked as a guiding principle in any formulation of spin extensions of geometrical optics.

Minimal coupling to the Cartan connection

̟ µ 0 = µ 0 • ϑ, (4.16)
where ϑ = ( ω, ω) is the "affine" Cartan connection, ω = ( ω a b ) denoting the Cartan connection (2.32), and ω = (ω a ) the coframe (2.10). The characteristic foliation of the 2-form σ µ 0 = d̟ µ 0 yields the differential equations of spinoptics in a medium described by the considered Finsler structure.

The 1-form (4.16) corresponds, mutatis mutandis, to the Euclidean 1-form (4.13).

(In (4.16), and from now on, we simplify the notation and denote by ϑ the pullback ι * ϑ on SO 2 (SM ) of the corresponding 1-form of SO 2 (T M\M ).) In Axiom 4.5, the replacement of the Euclidean group SE(3), see Fig. 1, by the principal bundle SO 2 (SM ), see Fig. 2, and of the Maurer-Cartan 1-form by the affine Cartan connection is akin to the so-called procedure of minimal coupling. We refer to [START_REF] Künzle | Canonical dynamics of spinning particles in gravitational and electromagnetic fields[END_REF] where the minimal coupling of a spinning particle to the gravitational field was originally introduced in the general relativistic framework. See also [START_REF] Souriau | Modèle de particule à spin dans le champ électromagnétique et gravitationnel[END_REF][START_REF] Sternberg | On the Role of Field Theories in our Physical Conception of Geometry[END_REF].

The choice of the Cartan connection is impelled by the fact that the group underlying Finsler-Cartan geometry is the Euclidean group, E(n), which is precisely the fundamental symmetry group of the symplectic model of free photons, for n = 3. Indeed, the "flat" n-dimensional Finsler-Cartan structure defined by both conditions Ω a = 0, and Ω a b = 0, for all a, b = 1, . . . , n, is (locally) isomorphic to the Euclidean space, (E n , • , • ): torsionfreeness yields A abc = 0, see (2.34), hence that (M, F ) is Riemannian; zero curvature then entails local flatness, via (2.37), and (2.25). The Euclidean group, E(n), is then the group of automorphisms of the flat structure.

Let us emphasize that the Cartan connection has been originally referred to as the "connexion euclidienne" in [START_REF] Cartan | Les espaces de Finsler[END_REF]. The 1-form (4.16) of SO 2 (SM ) thus reads

̟ µ 0 = P a ω a + 1 2 S ab ω ab . (4.17)
In view of the choice (4.12) of the moment µ 0 = (S, P) ∈ e(3) * , viz.,

S ab = s ǫ AB δ a A δ b B and P a = p δ 3 a , (4.18)
for all a, b = 1, 2, 3, where ǫ AB = 2δ

[A 1 δ B]
2 , for all A, B = 1, 2, we find See, e.g., [START_REF] Yu | Geometrical optics of beams with vortices: Berry phase and orbital angular momentum Hall effect[END_REF][START_REF] Onoda | Hall Effect of Light[END_REF]. Putting, in (4. [START_REF] Chern | Riemannian geometry as a special case of Finsler geometry[END_REF]), s = ε for photons (where ε = ±1 is helicity), and p = k, where k = ε/λ is the wave number, we observe that s/p = λ, so that Formula (4.19) indeed corresponds to (1.3), up to an overall constant factor.

̟ µ 0 = p ω 3 + s ω 12 . ( 4
Proposition 4.7. The exterior derivative, d̟ µ 0 , of the 1-form (4.17) on SO 2 (SM ) descends to the evolution space, SM , as

σ µ 0 = p h ab ω ā ∧ ω b + 1 2 Ω(S) - 1 2 S ab ω ā ∧ ω b, (4.20)
where the h ab = δ abδ 3 a δ 3 b denote the frame-components of the angular metric, and Ω(S) = Ω ab S ab the spin-curvature coupling 2-form.

Proof. Let us start with the expression (4.19) of the 1-form ̟ µ 0 . We have found, see (3.3), that dω 3 = δ AB ω Ā ∧ ω B . With the help of the structure equations of the Cartan connection, we obtain d ω

12 = Ω 12 + ω a 1 ∧ ω a2 = Ω 12 + ω 3 1 ∧ ω 32 = Ω 12 -ω 1 3 ∧ ω 2 3 ,
using the property (2.33). We then resort to (2.30), and to the property (2.5) of the Cartan tensor (that is A ab3 = 0, for all a, b = 1, 2, 3), to find ω A 3 = ω Ā. This yields

d ω 12 = Ω 12 -ω 1 ∧ ω 2, or, equivalently, d ω AB = Ω AB -ω Ā ∧ ω B , for A, B = 1, 2.
Thus

d̟ µ 0 = p δ AB ω Ā ∧ ω B + 1 2 Ω(S) - s 2 ǫ AB ω Ā ∧ ω B , (4.21)
where Ω(S) = s Ω 12 = 1 2 s ǫ AB Ω AB . In order to prove (4.20), we simply use the frame components, S ab , of the spin tensor given in (4.18).

To complete the proof, it is enough to verify that d̟ µ 0 , given by (4.21), is an integral invariant of the SO(2)-flow generated by

Z = e i 1 ∂ ∂e i 2 -e i 2 ∂ ∂e i 1 . (4.22)
This is, indeed, the case since d̟ µ 0 (Z) = 0.

The Finsler-Cartan spin tensor

Let us now give a construction of the spin tensor on the indicatrix-bundle, SM , that will be useful in the sequel.

Lemma 4.8. The following 3-form of SO 2 (SM ), viz.,

Vol = ω 1 ∧ ω 2 ∧ ω 3 (4.23)
is an integral invariant of the flow generated by the vector field, Z, given by (4.22).

It descends to SM as vol = 1 6 vol ijk (x, u) dx i ∧ dx j ∧ dx k , with

vol ijk (x, u) = det (g lm (x, u)) ǫ ijk , (4.24)
where i, j, k, l, m = 1, 2, 3.

Proof. In view of (2.17), we clearly have Vol(Z) = 0. Using, for example, the Chern connection, we find, with the help of the structure equations (2.21) that dVol = -ω a a ∧ Vol. Then, Equation (2.22) readily implies dVol = A a ab ω b ∧ Vol. Again, (2.17) entails that (dVol)(Z) = 0, thus Z ∈ ker(Vol) ∩ ker(dVol), proving that Vol is an SO(2)-integral invariant.

Locally, we have vol = det(ω a i ) dx 1 ∧ dx 2 ∧ dx 3 = det (g ij ) dx 1 ∧ dx 2 ∧ dx 3 , since (2.6) can be rewritten as ω a i ω b j δ ab = g ij . This proves Equation (4.24).

Let us now regard the S ab as the frame-components of a skew-symmetric tensor, S, on SM , with components S ij = S ab ω a i ω b j . Then, owing to (4.18), we easily find

S ij = s ǫ AB ω A i ω B j = 2s ω 1 [i ω 2 j]
; those turn out to be nothing but the components of the tensor S = s vol(ê 3 ) = s ω 1 ∧ ω 2 . Whence the Lemma 4.9. If we call spin tensor, associated with the moment (4.18), the tensor

S = s vol(û), (4.25)
where û = ê3 , see (2.16), then

S ij = s vol ijk u k , (4.26)
for all i, j = 1, 2, 3, where the vol ijk are as in (4.24).

Proof. Suffice it to note that the Cartan tensor vanishes iff the Finsler structure is Riemannian, hence P ijkl = Q ijkl = 0. The curvature tensor R ijkl in (4.31) then reduces to the Riemann curvature tensor, see (2.25). The proof is completed by noticing that S jk S ki = s 2 (u i u jδ i j ), a direct consequence of (4.26).

Conclusion and outlook

We due to the coupling of spin with the Finsler-Cartan curvature, which also engenders an anomalous velocity. The latter, already present in Riemannian spinoptics [START_REF] Duval | Geometrical Spinoptics and the Optical Hall Effect[END_REF][START_REF] Duval | Fermat Principle for spinning light and the Optical Hall effect[END_REF] has proved crucial in the geometrical interpretation of the brand new optical Hall effect, see, e.g., [START_REF] Yu | Modified geometrical optics of a smoothly inhomogeneous isotropic medium: the anisotropy, Berry phase, and the optical Magnus effect[END_REF][START_REF] Onoda | Hall Effect of Light[END_REF]. The consubstantial nature of this effect with the geometry of Euclidean coadjoint orbits is precisely what prompted the present study, and our endeavor to depart from the case of isotropic media by taking advantage of Finsler-Cartan structures. Although the characteristic foliation (4.31) of the abovementioned presymplectic structure is of a formidable complexity, it is nevertheless a mandatory consequence of a minimal, geometrically justified, modification (1.3) of the Hilbert 1-form (1.2) of central importance in Finsler geometry.

The future perspectives opened by this work are manifold.

It would be desirable to linearize the differential equations of Finsler spinoptics in the case of weakly curved Finsler-Cartan manifolds, to account for weakly anisotropic optical media. This should lead to substantial simplifications, suitable for an explicit calculation of the geodesic deviation in several non trivial examples, such as those given by (3.10), and (3.11). Also, would it be of great importance to compare this linearized set of differential equations with the outcome of the calculations performed by a (short wavelength) semi-classical limit of the Maxwell equations in weakly anisotropic and inhomogeneous media [START_REF] Yu | Non-Abelian evolution of electromagnetic waves in a weakly anisotropic inhomogeneous medium[END_REF].

The Fermat Principle has, most interestingly, been generalized, via a novel variational calculus, to the case of lightlike geodesics in Finsler spacetimes with a Lorentzian signature [START_REF] Perlick | Fermat principle in Finsler spacetimes[END_REF]. It would be worth investigating how that relativistic version of geometrical optics extends to spinoptics in relativistic Finsler spacetimes.

Let us note that Randers Finsler metrics play, as discussed in [START_REF] Caponio | Variational properties of geodesics in non-reversible Finsler manifolds and applications[END_REF], a prominent rôle in such a framework, corresponding to induced (instantaneous) Finsler metrics on the material body of the optical medium.

At last, specific applications of the equations of Finsler-Cartan spinoptics should be explored in a number of other directions such as the Kerr, the Faraday effects, and the Cotton-Mouton effect responsible for plasma birefringence, see, e.g., [START_REF] Segre | Derivation of the pure Faraday and Cotton-Mouton effects when polarimetric effects in a tokamak are large[END_REF],

as well as the photonic Hall effect [START_REF] Van Tiggelen | Transverse diffusion of light in Faraday-Active Media[END_REF] in the presence of a magnetic field.

In truth, the present study of Finsler spinoptics was a challenge, taken up from a purely geometric standpoint; one may, conceivably, expect it will provide further insights into modern trends of geometrical optics of anisotropic media.

Corollary 2 . 2 .

 22 Let SM = F -1 (1) denote the indicatrix-bundle of a Finsler manifold (M, F ), and ι : SM ֒→ T M \M its embedding into the tangent bundle of M .

Proposition 2 .

 2 10. [5] The curvature Ω a b = d ω a bω c b ∧ ω a c of the Cartan connection is given by

. 16 )

 16 Those correspond, respectively, to an ordinary Euclidean metric, F o = F -, and to an extraordinary Minkowski metric, F e = F + , again generalized by (3.10), and(3.11).

Figure 1 :

 1 Figure 1:

Axiom 4 . 5 .

 45 The trajectories of (circularly) polarized light, originating from an Euclidean coadjoint orbit O µ 0 ⊂ e * (3) with color p > 0, and spin s = 0, in a threedimensional Finsler manifold (M, F ), are governed by the following 1-form on the principal bundle SO 2 (SM ) over evolution space SM = F -1 (1), namely

SO 2

 2 Figure 2:

  have proposed a generalization of the Fermat Principle enabling us to describe spinning light rays in a general, non dispersive, optical medium, namely an inhomogeneous and anisotropic medium modeled on a Finsler manifold. The guideline for this extension has been provided by the Euclidean symmetry of the free system, viewed as a generic coadjoint orbit of the Euclidean group, SE(3). Interaction with the optical medium has been justified in terms of a minimal coupling of the model to the (affine) Cartan connection of the Finsler structure; the gist of the procedure lies in the fact that the affine Cartan connection takes, indeed, its values in the Lie algebra e(3) and, thus, couples naturally to the moment µ 0 ∈ e(3) * defining the original coadjoint orbit (the classical states of the free Euclidean photon). The resulting presymplectic structure on (an open submanifold of) the indicatrix-bundle has been investigated. In particular the characteristic foliation of this structure has been worked out, and shown to yield a system of differential equations governing the trajectories of spinning light rays, associated with a vector field departing from the usual Finslerian geodesic spray. The geodesic acceleration of spinning light rays is

Laws of geometrical spinoptics in Finsler-Cartan spaces

We are now ready to determine the explicit expression of the characteristic foliation of the 2-form (4.20) that will provide us with the differential equations governing the trajectories of spinning light in a Finsler-Cartan background. Proof. The frame-components R(S) cd , and P (S) cd , for the Cartan connection exactly match their counterpart for the Chern connection with curvature tensors R i j kl and P i j kl given by (2.25) and (2.26) respectively; indeed, the equations (4.28), and (4.29) are derived, in a straightforward way, using (2.38), and (2.39), together with the total skew-symmetry (resp. symmetry) of the spin tensor, (resp. the Cartan tensor), i.e., S ab = S [ab] (resp. A abc = A (abc) ).

The constitutive equation (Equation (3.4.11) in [START_REF] Bao | An Introduction to Riemann-Finsler Geometry[END_REF]) for the hv-components of the Lemma 4.11. There holds P (S) a3 = P (S) 3a = 0, for all a = 1, 2, 3.

Proof. We have P (S) c3 = 0, because of (2.27). Likewise, the relation P (S) 3a = 0 stems from (4.29) since A a3e = 0 (see (2.5)), and (e 3 ) i |j = 0 (see (2.29)).

We can now proclaim our main result.

Theorem 4.12. The characteristic foliation of the 2-form σ µ 0 of SM , given by (4.20), is expressed as follows, viz.,

for some X 3 ∈ R, where the S i j = s vol i j k u k are as in (4.26), and

The proof of Theorem 3.3 is adapted to the new 2-form (4.34) we are dealing with.

In particular, the vector fields X ∈ Vect(SM ) will be written in the following form,

where λ ∈ R is a Lagrange multiplier associated with the constraint F = 1 defining SM ֒→ T M \M . We find

= p δ A B -

= λ. to readily get

for all A, B = 1, 2. Note that we also have R(S)(S) = s 2 R ABCD ǫ AB ǫ CD , and

Plugging (4.39) into (4.36), we easily find

for all A = 1, 2, where

We then find, with the help of (4.40), and (4.35), the following relationship

where

We clearly have Σ AB + Σ BA = 0, hence Σ AB = Σ ǫ AB , and find, with some more effort,

where ∆ is as in (4.41).

Let us point out that Equation (4.37) trivially holds true in view of the skew-

Equation (4.42) then leaves us with

for all A = 1, 2. Let us recall that the latter equation for X A completely determines X Ā, via (4.40), the components X 3 , and X 3 remaining otherwise arbitrary. Now, X ∈ Vect(SM ) if X(F ) = 0, i.e., ω 3(X ) = X 3 = 0. We are, hence, left with only one arbitrary parameter, X 3 , to define the direction ker(σ µ 0 ) wherever ∆ = 0, and Σ = 0. Thus X = X A êA + X 3 êA + X Ā ê Ā, where X A , and X Ā are as in (4.44), and (4.40) respectively, with X 3 ∈ R.

Introducing the unit supporting element, (u a = δ a 3 ), as well as the spin tensor,

, given in (4.18), we find

for some X 3 ∈ R.

To complete the calculation, we express (4.45) in terms of the coordinates u i = e i 3 of the distinguished element u = e 3 , and those, S ij = g ik g jl S kl , of the spin tensor S, see (4.25), where the S ij are as in (4.26). We also bear in mind that êa = e i a δ/δx i , and êā = e i a ∂/∂u i , for all a = 1, 2, 3, as given by (2.16), on the principal bundle SO 2 (SM ) above the evolution space SM . The upshot of the computation is that the characteristic foliation (4.45) can be recast in the form (4.31); Equations (4.32), and (4.33) also provide alternative expressions for (4.41), and (4.43).

At those points (x, u) ∈ SM where ∆ = 0, or Σ = 0, singularities of the foliation (4.45) do occur; they must be discarded to guarantee a well-behaved presymplectic structure of (generic) rank 4. The proof is now complete.

Remark 4.13. Let us choose, e.g., X 3 = 1 in (4.31) to define the generator, X, of the foliation ker(σ µ 0 ). The latter significantly deviates from a spray since the velocity, ẋ, given by the horizontal projection of X, differs from the direction, u, of the supporting element, namely

where ẋi = X(x i ), for all i = 1, 2, 3. The occurrence of this anomalous velocity in the presence of curvature can be classically interpreted (see [START_REF] Duval | Geometrical Spinoptics and the Optical Hall Effect[END_REF][START_REF] Duval | Fermat Principle for spinning light and the Optical Hall effect[END_REF]) as the source of the optical Hall effect. Moreover, the vertical components of the generator X, namely those of the geodesic acceleration, depend linearly on the helicity, ε = sign(s). They, notably, lead to a splitting, à la Stern-Gerlach, of light rays with opposite helicities.

Let us finish with the following corollary of Theorem 4.12 which help us recover the simpler equations of spinoptics in the Riemannian case, derived in [START_REF] Duval | Geometrical Spinoptics and the Optical Hall Effect[END_REF].

Corollary 4.14. If the Finsler structure, (M, F ), is Riemannian, the characteristic foliation of the 2-form σ µ 0 is spanned by the vector field

with

where the R ijkl are the components of the Riemann curvature tensor.