test fields on compact space times in this paper some basic aspects of lorentzian field theory on compact lorentz manifolds are studied all compact space times are acausal i e possess closed timelike curves this makes them a useful testbed in analyzing some new notions of causality that will be introduced for more general acausal space times in addition studying compact space times in their own right raises a wide range of fascinating mathematical problems some of which will be explored it will be shown that it is reasonable to expect lorentzian field theory on a compact space time to provide information on the topology of the underlying manifold if this is true then this information is likely to be orthogonal or complementary to the information obtained through the study of euclidean field theory

allometry constants of finite dimensional spaces theory and computations we describe the computations of some intrinsic constants associated to an n dimensional normed space mathcal v namely the n th allometry constants kappa infty n mathcal v inf t cdot t quad t ell infty n to mathcal v t mathcal v to ell infty n tt rm id mathcal v these are related to banach mazur distances and to several types of projection constants we also present the results of our computations for some low dimensional spaces such as sequence spaces polynomial spaces and polygonal spaces an eye is kept on the optimal operators t and t or equivalently in the case n n on the best conditioned bases in particular we uncover that the best conditioned bases of quadratic polynomials are not symmetric and that the lagrange bases at equidistant nodes are best conditioned in the spaces of trigonometric polynomials of degree at most one and two

global rigidity of holomorphic riemannian metrics on compact complex 3 manifolds we study compact complex 3 manifolds m admitting a locally homogeneous holomorphic riemannian metric g we prove the following i if the killing lie algebra of g has a non trivial semi simple part then it preserves some holomorphic riemannian metric on m with constant sectional curvature ii if the killing lie algebra of g is solvable then up to a finite unramified cover m is a quotient g where is a lattice in g and g is either the complex heisenberg group or the complex sol group

on closed anti de sitter spacetimes this article deals with the structure of the fundamental group of compact anti de sitter spacetimes i e lorentz manifolds with constant negative curvature algebraically such a manifold is the quotient of the universal cover of the homogeneous space so 2 n so 1 n by a discrete group gamma acting properly and co compactly on it this exists if and only if n is even indeed as this was observed by kulkarni u 1 d is contained in so 2 2d and acts properly transitively on so 2 2d so 1 2d it then suffices to take gamma as a co compact lattice in u 1 d the results of the present article give evidence to the question in dimension 3 are all compact anti de sitter spacetimes constructed in this way

affine structures on abelian lie groups abstract the nagano yagi goldmann theorem states that on the torus t 2 every affine or projective structure is invariant or is constructed on the basis of some goldmann rings nagano t yagi k osaka j math 11 1974 181 it shows the interest to study the invariant affine structure on the torus t 2 or on abelian lie groups recently the works of kim j differential geom 24 1986 373 and dekimpe ongenae p a m s 128 11 2000 3191 precise the number of non equivalent invariant affine structures on an abelian lie group in the case these structures are complete in this paper we propose a study of complete and non complete affine structures on abelian lie groups based on the geometry of the algebraic variety of finite dimensional associative algebras

polycyclic by finite groups admit a bounded degree polynomial structure for a polycyclic by finite group gamma of hirsch length h an affine resp polynomial structure is a representation of gamma into rm aff bbb r h resp rm p bbb r h the group of polynomial diffeomorphisms letting gamma act properly discontinuously on bbb r h recently it was shown by counter examples that there exist groups gamma even nilpotent ones which do not admit an affine structure thus giving a negative answer to a long standing question of john milnor we prove that every polycyclic by finite group gamma admits a polynomial structure which moreover appears to be of a special simple type called canonical and as a consequence of this consists entirely of polynomials of a bounded degree the construction of this polynomial structure is a special case of an iterated seifert fiber space construction which can be achieved here because of a very strong and surprising cohomology vanishing theorem

polynomial structures for nilpotent groups if a polycyclic by finite rank k group admits a faithful affine representation making it acting on rk properly discontinuously and with compact quotient we say that admits an affine structure in 1977 john milnor questioned the existence of affine structures for such groups very recently examples have been obtained showing that even for torsion free finitely generated nilpotent groups n affine structures do not always exist it looks natural to consider affine structures as examples of polynomial structures of degree one we introduce the concept of a canonical type polynomial structure for polycyclic by finite groups using the algebraic framework of the seifert fiber space construction and a nice cohomology vanishing theorem we prove the existence and uniqueness up to conjugation of canonical type polynomial structures for virtually finitely generated nilpotent groups applying this uniqueness to a result going back to mal cev it follows that for torsion free finitely generated nilpotent groups each canonical polynomial structure is expressed in polynomials of limited degree the minimal degree needed for obtaining a polynomial structure will determine the affine defect number we prove that the known counterexamples to milnor s question have the smallest possible affine defect i e affine defect number equal to one

the uniqueness of polynomial crystallographic actions let gamma be a polycyclic by finite group it is proved in 8 that gamma admits a polynomial action of bounded degree on mathbb r n which is properly discontinuous and such that the quotient gamma backslash mathbb r n is compact we prove here that such an action is unique up to conjugation by a polynomial transformation of mathbb r n

autour de la conjecture de l markus sur les varietes affines for any subgroupg of n we introduce some integer discg n called thediscompacity ofg this number measures to what extent the closure ofg is not compact the markus conjecture says that a compact affinely flat unimodular manifold is complete our main result called the discompact theorem is that this conjecture is true under the assumption that the linear holonomy i e the parallel transport has discompacity 1 because discso n 1 1 1 this ensures that a compact flat lorentz manifoldm is geodesically complete hence by a previous result of w goldman and y kamishima gk such am is up to finite covering a solvmanifold this achieves the proof of a bieberbach s theorem for compact lorentz flat manifolds

norms of minimal projections it is proved that the projection constants of two and three dimensional spaces are bounded by 43 and 1 5 2 respectively these bounds are attained precisely by the spaces whose unit balls are the regular hexagon and dodecahedron in fact a general inequality for the projection constant of a real or complex n dimensional space is obtained and the question of equality therein is discussed

nonstandard lorentz space forms in their recent paper 8 kulharni and raymond show that a closed 3 manifold which admits a complete lorentz metric of constant curvature 1 henceforth called a complete lorentz structure must be seifert fibered over a hyperbolic base furthermore on every such seifert fibered 3 manifold with nonzero euler class they construct such a lorentz metric moreover the lorentz structure they construct has a rather strong additional property which they call standard a lorentz structure is standard if its causal double cover possesses a timelike killing vector field equivalently it possesses a riemannian metric locally isometric to a left invariant metric on sl 2 r kulkarni and raymond asked if every closed 3 dimensional lorentz structure is standard this paper provides a negative answer to this question theorem 1 and a positive answer to the implicit question raised in 8 7 1 1 theorem 3 theorem 1 let m3 be a closed 3 manifold which admits a homogeneous lorentz structure and satisfies h m r 0 then there exists a nonstandard complete lorentz structure on m in 8 it is shown that the unit tangent bundle of a closed surface admits a homogeneous lorentz structure therefore we obtain corollary 2 there exists a complete lorentz structure on the unit tangent bundle of any closed surface f of genus greater than one which is not standard the homogeneous lorentz structures are all classified in 8 a circle bundle of euler number j over a closed surface f f 0 has a homogeneous structure if and only if j f an analogous statement holds when m has singular fibers i e when f is an orbifold we also show theorem 3 let m3 be a 3 manifold which admits a complete lorentz structure then m3 is not covered by a product f x s fa closed surface theorem 3 implies that the euler class of the seifert fiber structure of m3 is nonzero

margulis spacetimes via the arc complex we study strip deformations of convex cocompact hyperbolic surfaces defined by inserting hyperbolic strips along a collection of disjoint geodesic arcs properly embedded in the surface we prove that any deformation of the surface that uniformly lengthens all closed geodesics can be realized as a strip deformation in an essentially unique way the infinitesimal version of this result gives a parameterization by the arc complex of the moduli space of margulis spacetimes with fixed convex cocompact linear holonomy as an application we provide a new proof of the tameness of such margulis spacetimes m by establishing the crooked plane conjecture which states that m admits a fundamental domain bounded by piecewise linear surfaces called crooked planes the noninfinitesimal version gives an analogous theory for noncompact complete anti de sitter 3 manifolds

topological tameness of margulis spacetimes we show that margulis spacetimes without parabolic holonomy elements are topologically tame a margulis spacetime is the quotient of the 3 dimensional minkowski space by a free proper isometric action of the free group of rank geq 2 we will use our particular point of view that the margulis spacetime is a manifold with boundary with an bbb r sf p 3 structure in an essential way the basic tools are a bordification by a closed bbb r sf p 2 surface with free holonomy group and the work of goldman labourie and margulis on geodesics in the margulis spacetimes and 3 manifold topology

a simple proof of geodesical completeness for compact space times of zero curvature any riemannian metric on a compact manifold is geodesically complete by contrast it is widely known that incomplete compact lorentzian manifolds exist in this article it will be proven that a compact manifold with a smooth lorentz metric must be complete if the metric is flat everywhere this question of completeness which has significant implications for the classification of compact lorentzian space forms has remained unresolved until quite recently when it was finally established by carriere as a special case of an even more general result here a simple alternative proof of this special case of carriere s theorem is given the proof requires minimal mathematical machinery but still involves the use of some intriguing connections between the topology and global geometry of a compact flat space time

classification of simple novikov algebras and their irreducible modules of characteristic 0 abstract in this paper we first present a classification theorem of simple infinite dimensional novikov algebras over an algebraically closed field of characteristic 0 then we classify all the irreducible modules of certain infinite dimensional simple novikov algebras with an idempotent element whose left action is locally finite

some global properties of static spacetimes we show that a static lorentzian manifold satisfying a completeness assumption is geodesically connected in particular such condition is satisfied by all compact static manifolds in the compact case we also prove the existence of a closed geodesic in every free homotopy class determined by a finite number of deck transformations

archimedean superrigidity of solvables arithmetic groups abstract let g be a connected solvable linear algebraic group over a number fieldk letsbe a finite set of places ofkthat contains all the infinite places and let o s be the ring ofs integers ofk we define a certain closed subgroup formula of g s v s g kvthat contains g o s and prove that g o s is a superrigid lattice in formula by which we mean that finite dimensional representations g o s gln r more or less extend to representations of formula the subgroup formula may be a proper subgroup of g sfor only two reasons first it is well known that g o s is not a lattice in g sif g has nontrivialk characters so one passes to a certain subgroup g s 1 second g o s may fail to be zariski dense in g s 1 in an appropriate sense in this sense the subgroup formula is the zariski closure of g o s in g s 1 furthermore we note that a superrigidity theorem for many nonsolvables arithmetic groups can be proved by combining our main theorem with the margulis superrigidity theorem

on the full holonomy group of lorentzian manifolds the classification of restricted holonomy groups of n dimensional lorentzian manifolds was obtained about ten years ago however up to now not much is known about the structure of the full holonomy group in this paper we study the full holonomy group of lorentzian manifolds with a parallel null line bundle based on the classification of the restricted holonomy groups of such manifolds we prove several structure results about the full holonomy we establish a construction method for manifolds with disconnected holonomy starting from a riemannian manifold and a properly discontinuous group of isometries this leads to a variety of examples most of them being quotients of pp waves with disconnected holonomy including a non flat lorentzian manifold with infinitely generated holonomy group furthermore we classify the full holonomy groups of solvable lorentzian symmetric spaces and of lorentzian manifolds with a parallel null spinor finally we construct examples of globally hyperbolic manifolds with complete spacelike cauchy hypersurfaces disconnected full holonomy and a parallel spinor

on the geometry of static spacetimes we review geometrical properties of a static spacetime m g including geodesic completeness causality standard splittings compact m closed geodesics and geodesic connectedness we pay special attention to the critical quadratic behaviour at infinity of the coefficients 1 g k k being k a timelike irrotational killing vector field which essentially control completeness causality and geodesic connectedness recent references are specially discussed

flat compact hermite lorentz manifolds in dimension 4 we give a classification up to finite cover of flat compact complete hermite lorentz manifolds up to complex dimension 4

affine manifolds and orbits of algebraic groups this paper is the sequel to the radiance obstruction and parallel forms on a ffne manifolds lyans amer math soc 286 1984 629 649 which introduced a new family of secondary characteristic classes for affine structures on manifolds the present paper utilizes the representation of these classes in lie algebra cohomology and algebraic group cohomology to deduce new results relating the geometric properties of a compact affine manifold mn to the action on rn of the algebraic hull a r of the affine holonomy group r c aff rn a main technical result of the paper is that if m has a nonzero cohomology class represented by a parallel k form then every orbit of a r has dimension k when m is compact then a r acts transitively provided that m is complete or has parallel volume the converse holds when r is nilpotent a 4 dimensional subgroup of aff r3 is exhibited which does not contain the holonomy group of any compact affine 3 manifold when m has solvable holonomy and is complete then m must have parallel volume conversely if m has parallel volume and is of the homotopy type of a solvmanifold then m is complete if m is a compact homogeneous affine manifold or if m possesses a rational riemannian metric then it is shown that the conditions of parallel volume and completeness are equivalent this paper is the sequel to our previous paper 22 in that paper we exploited certain characteristic classes exterior powers of the radiance obstruction to obtain relationships between various properties of affine manifolds our results supported the conjecture first made by l markus 36 that a compact affine manifold is complete if and only if it has parallel volume we shall refer to this as the main conjecture let m be a compact affine manifold with developing map dev m e and affine holonomy representation h 7r h 7r r c aff e here 1r is the group of deck transformations of the universal cover m e is the vector space rn and aff e is the group of affine automorphisms of e the map dev is a locally affine immersion which is equivariant respecting h the linear holonomy homb morphism is the composition a 1r gl e of h with the natural homomorphism l aff e gl e the obstruction to r fixing a point in e is a 1 dimensional cohomology class cm e h1 m ex with coefficients in e twisted by a it comes from a universal class in the group cohomology h1 aff e el which contains the crossed homomorphism u aff e e g g 0 received by the editors december 12 1984 and in revised form may 29 1985 1980 mathematics subject classification 1985 revision primary 57r99 53c05 secondary 53c10 55r25 1 research partially supported by fellowships and grants from the national science foundation a 1986 american mathematical society 0002 9947 86 1 00 25 per page

structure of lorentzian tori with a killing vector field all lorentzian tori with a non discrete group of isometries are characterized and explicitly obtained they can lie into three cases a flat b conformally flat but non flat and c geodesically incomplete a detailed study of many of their properties including results on the logical dependence of the three kinds of causal completeness on geodesic connectedness and on prescribed curvature is carried out the incomplete case is specially analyzed and several known examples and results in the literature are generalized from a unified point of view

global dense multiscale reconstruction for a billion points we present a variational approach for surface reconstruction from a set of oriented points with scale information we focus particularly on scenarios with nonuniform point densities due to images taken from different distances in contrast to previous methods we integrate the scale information in the objective and globally optimize the signed distance function of the surface on a balanced octree grid we use a finite element discretization on the dual structure of the octree minimizing the number of variables the tetrahedral mesh is generated efficiently with a lookup table which allows to map octree cells to the nodes of the finite elements we optimize memory efficiency by data aggregation such that robust data terms can be used even on very large scenes the surface normals are explicitly optimized and used for surface extraction to improve the reconstruction at edges and corners

closed timelike geodesics on montre que chaque classe de t homotopie stable libre de courbes fermees du genre temps dans une variete de lorentz compacte contient une courbe la plus longue qui doit etre une geodesique fermee du genre temps

three dimensional affine crystallographic groups those groups r which act properly discontinuously and aillnely on ii with compact fundamental domain are classified first it is shown that such a group f contains a solvable subgroup of finite index thus establishing a conjecture of auslander in dimension three then unimodular simply transitive altine actions on ir are classified this leads to a classification of attine crystallographic groups acting on ir3 a characterization of which abstract groups admit such an action is given moreover it is proved that every isomorphism between virtually solvable atline crystallographic groups respectively simply transitive afline groups is induced by conjugation by a polynomial automorphism of the affrne space a characterization is given of which closed 3 manifolds can be represented as quotients of ir by groups of afftne transformations a closed 3 manifold m admits a complete atline structure if and only if m has a finite covering homeomorphic or homotopy equivaient to a 2 torus bundle over the circle

on the zariski closure of the linear part of a properly discontinuous group of affine transformations let be a subgroup of the group of affine transformations of the affine space r 2n 1 suppose acts properly discontinuously on r 2n 1 the paper deals with the question which subgroups of gl 2n 1 r occur as zariski closure of the linear part of such a group the two main results of the paper say that so n 1 n does occur as of such a group if n is odd but does not if n is even

proper affine actions and geodesic flows of hyperbolic surfaces let to c 0 2 1 be a schottky group and let h2 to be the corresponding hyperbolic surface let s denote the space of unit length geodesic currents on s the cohomology group if1 to v parametrizes equivalence classes of affine deformations tu of to acting on an irreducible representation v of 0 2 1 we define a continuous biaffine map i s x hl to v u which is linear on the vector space h1 fo v an affine deformation tu acts properly if and only if x u 7 0 for all jl e s consequently the set of proper affine actions whose linear part is a schottky group identifies with a bundle of open convex cones in hl to v over the fricke teichmiiller space of e

adaptively merging large scale range data with reflectance properties in this paper we tackle the problem of geometric and photometric modeling of large intricately shaped objects typical target objects we consider are cultural heritage objects when constructing models of such objects we are faced with several important issues that have not been addressed in the past issues that mainly arise due to the large amount of data that has to be handled we propose two novel approaches to efficiently handle such large amounts of data a highly adaptive algorithm for merging range images and an adaptive nearest neighbor search to be used with the algorithm we construct an integrated mesh model of the target object in adaptive resolution taking into account the geometric and or photometric attributes associated with the range images we use surface curvature for the geometric attributes and laser reflectance values for the photometric attributes this adaptive merging framework leads to a significant reduction in the necessary amount of computational resources furthermore the resulting adaptive mesh models can be of great use for applications such as texture mapping as we will briefly demonstrate additionally we propose an additional test for the k d tree nearest neighbor search algorithm our approach successfully omits back tracking which is controlled adaptively depending on the distance to the nearest neighbor since the main consumption of computational cost lies in the nearest neighbor search the proposed algorithm leads to a significant speed up of the whole merging process in this paper we present the theories and algorithms of our approaches with pseudo code and apply them to several real objects including large scale cultural assets

isometry groups of lorentz manifolds soit v une variete de lorentz a n dimensions si v g est compacte analytique reelle simplement convexe alors le groupe d isometrie is v g est compact

about the parabolicity of trapped submanifolds in standard static spacetimes abstract in this work we introduce the concept of totally trapped submanifolds immersed in a lorentzian manifold moreover we obtain some uniqueness or non existence results for p parabolic submanifolds immersed in standard static spacetimes ssst in particular we derive some rigidity results for this type of submanifolds which guarantee that under appropriate hypothesis the only ones are those contained in slices in order to obtain these results we study the height function associated to the submanifold using the p laplacian for a function associated to the warping function of the ssst

the radiance obstruction and parallel forms on affine manifolds a manifold m is affine if it is endowed with a distinguished atlas whose coordinate changes are locally affine when they are locally linear m is called radiant the obstruction to radiance is a one dimensional class cm with coefficients in the flat tangent bundle of m exterior powers of cm give information on the existence of parallel forms on m especially parallel volume forms as applications various kinds of restrictions are found on the holonomy and topology of compact affine manifolds introduction an affine manifold m is a manifold with a distinguished maximal atlas of charts all of whose coordinate changes are locally affine on such a manifold there is an intrinsic notion of a parallel tensor one whose components in any affine chart are constants more generally there is the notion of a polynomial tensor field of given degree in 1962 l markus conjectured in mk that a compact orientable affine n dimensional manifold has parallel volume form if and only if it is complete meaning that its universal covering is affinely isomorphic to euclidean n space rn the problem of constructing a parallel volume form determines an n dimensional twisted real cohomology class originally studied by j smillie sm2 in this paper we express this class as the nth exterior power of a twisted one dimensional real class which we call the radiance obstruction cm by computing cm in various cohomology theories cech singular de rham and others we are able to exploit cm in several ways to yield more information on the structure of affine manifolds some of these results will appear in a subsequent paper gh3 the basic tool used is a formula proved in 2 6 which expresses the cohomology class of a parallel exterior k form in terms of the k th exterior power of the radiance obstruction akcvf min 2 7 the special case of a parallel volume form on a compact n dimensional manifold m is examined the existence of such a form implies that ancm 0 in 2 8 we show that the affine holonomy group f of such an m cannot preserve a proper affine subspace in gh3 this result will be improved by showing that r preserves no proper semialgebraic subset of rn in 2 10 we show that there tend to be plenty of parallel forms on a compact affine manifold with nilpotent affine holonomy received by the editors may 26 1983 1980 mathematics subject classification primary 57r99 53c05 secondary 53c10 55r25 4c1984 amencan mathematical society 0002 9947 84 1 00 25 per page

on the uniqueness of some affinely flat infra nilmanifolds i p irn irk u p irn irk since pi irn irk pi irn ir k each element of i 1 ir irk is completely determined by k polynomials of total degree at most i in n variables let g be a group and suppose we have a morphism 4 g g l k ir and a homomorphism p g aff ir we can provide a g module structure for p ir f k in the following way for g e g and pepi ir irk we define

on the nonexistence of closed timelike geodesics in flat lorentz 2 step nilmanifolds the main purpose of this paper is to prove that there are no closed timelike geodesics in a compact or noncompact flat lorentz 2 step nilmanifold n where n is a simply connected 2 step nilpotent lie group with a flat left invariant lorentz metric and a discrete subgroup of n acting on n by left translations for this purpose we shall first show that if n is a 2 step nilpotent lie group endowed with a flat left invariant lorentz metric g then the restriction of g to the center z of n is degenerate we shall then determine all 2 step nilpotent lie groups that can admit a flat left invariant lorentz metric we show that they are trivial central extensions of the three dimensional heisenberg lie group h 3 if n g is one such group we prove that no timelike geodesic in n g can be translated by an element of n by the way we rediscover that the heisenberg lie group h 2k 1 admits a flat left invariant lorentz metric if and only if k 1

decomposition theorems for lorentzian manifolds with nonpositive curvature soit m g un espace temps globalement hyperbolique de dimension 2 avec partout des courbures sectionnelles du genre temps non positives k 0 qui contient une ligne du genre temps complete m g alors m g est isometrique a un produit r h dt 2 h ou h h est une variete de riemann complete le facteur r dt 2 est represente par et h h est represente par un ensemble de niveau d une fonction de busemann associee a

the fundamental group of a compact flat lorentz space form is virtually polycyclic a flat lorentz space form is a geodesically complete lorentzian manifold of zero curvature it is well known see auslander markus 3 that such a space m may be represented as a quotient r where r is an dimensional minkowski space n equals the dimension of m and is a group of lorentz isometries acting properly discontinuously and freely on r in particular the universal covering of m is isometric to r and the fundamental group m is isomorphic to theorem let m be a compact flat lorentz space form then m is virtually poly cyclic recall that a group is virtually polycyclic if it can be built by iterated extensions from finitely many finite groups and cyclic groups this result affirms a conjecture of milnor 13 in a special case for discussion of this conjecture as well as another special case we refer to fried goldman 8 the importance of this result is that it reduces the classification of compact flat lorentz space forms to fairly elementary problems concerning lie algebras and lattices in solvable lie groups we hope to pursue this classification in a future publication for a description of this reduction and the classification in dimension 3 see fried goldman 8 in dimension 4 the classification is worked out in fried 7 one immediate consequence of the structure theory developed in fried goldman 8 1 and kamishima 12 is the following corollary let m be the compact flat lorentz space form then m has a finite covering which is diffeomorphic to a solvmanifold the outline of this paper is as follows in the first section we collect some basic facts about the group e n 1 1 of isometries of dimensional minkowski space in the second section we prove the theorem in the special

global topology of hyperbolic components cantor circle case we prove that in the moduli space md of degree d 2 rational maps any hyperbolic component in the disconnectedness locus and of cantor circle type is a finite quotient of r4d 4 n tn where n is determined by dynamics the proof uses some ideas from riemann surface theory abel s theorem dynamical system and algebraic topology

an advancing front delaunay triangulation algorithm designed for robustness abstract a new algorithm is described for generating an unstructured mesh about an arbitrary two dimensional configuration mesh points are generated automatically by the algorithm in a manner which ensures a smooth variation of elements and the resulting triangulation constitutes the delaunay triangulation of these points the algorithm combines the mathematical elegance and efficiency of delaunay triangulation algorithms with the desirable point placement features boundary integrity and robustness that are traditionally associated with advancing front type mesh generation strategies the method offers increased robustness over previous algorithms in that it cannot fail regardless of the initial boundary point distribution and the prescribed cell size distribution throughout the flow field

on the existence of closed timelike geodesics in compact spacetimes in 19 tipler has shown that a compact spacetime having a regular globally hyperbolic covering space with compact cauchy surfaces necessarily contains a closed timelike geodesic the restriction to compact spacetimes with just regular globally hyperbolic coverings i e the cauchy surfaces are not required to be compact is still an open question here we shall answer this question negatively by providing examples of compact flat lorentz space forms without closed timelike geodesics and shall give some criterion for the existence of such geodesics more generally we will show that in a compact spacetime having a regular globally hyperbolic covering each free timelike homotopy class determined by a central deck transformation must contain a closed timelike geodesic whether or not a compact flat spacetime contains closed nonspacelike geodesics is as far as we know an open question we shall answer this question affirmatively we shall also introduce the notion of timelike injectivity radius for a spacetime relative to a free timelike homotopy class and shall show that it is finite whenever the corresponding deck transformation is central

three dimensional manifolds kleinian groups and hyperbolic geometry 1 a conjectural picture of 3 manifolds a major thrust of mathematics in the late 19th century in which poincare had a large role was the uniformization theory for riemann surfaces that every conformai structure on a closed oriented surface is represented by a riemannian metric of constant curvature for the typical case of negative euler characteristic genus greater than 1 such a metric gives a hyperbolic structure any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions the exceptional cases the sphere and the torus have spherical and euclidean structures three manifolds are greatly more complicated than surfaces and i think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 or more except perhaps for the fact that so many 3 manifolds are beautiful the situation has changed so that i feel fairly confident in proposing the

periodic geodesics and geometry of compact lorentzian manifolds with a killing vector field we study the geometry and the periodic geodesics of a compact lorentzian manifold that has a killing vector field which is timelike somewhere using a compactness argument for subgroups of the isometry group we prove the existence of one timelike non self intersecting periodic geodesic if the killing vector field is nowhere vanishing then there are at least two distinct periodic geodesics as a special case compact stationary manifolds have at least two periodic timelike geodesics we also discuss some properties of the topology of such manifolds in particular we show that a compact manifold m admits a lorentzian metric with a nowhere vanishing killing vector field which is timelike somewhere if and only if m admits a smooth circle action without fixed points

superrigidity of lattices in solvable lie groups let be a closed cocompact subgroup of a simply connected solvable lie groupg such that adg has the same zariski closure as adg if gln is any finite dimensional representation of we show that virtually extends to a representation ofg by combining this with work of margulis on lattices in semisimple groups we obtain a similar result for lattices in many groups that are neither solvable nor semisimple furthermore we show that if is isomorphic to a closed cocompact subgroup of another simply connected solvable lie groupg then any isomorphism from to extends to a crossed isomorphism fromg tog in the same vein we prove a more concrete form of mostow s theorem that compact solvmanifolds with isomorphic fundamental groups are diffeomorphic

affine transformations and parallel lightlike vector fields on compact lorentzian 3 manifolds we describe the compact lorentzian 3 manifolds admitting a parallel lightlike vector field the classification of compact lorentzian 3 manifolds admitting non isometric affine diffeomorphisms follows together with the complete description of these morphisms such a lorentzian manifold is in some sense an equivariant deformation of a flat one