DOCSLIB.ORG

The Projective Geometry of the Spacetime Yielded by Relativistic Positioning Systems and Relativistic Location Systems Jacques Rubin

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Projective Geometry of the Spacetime Yielded by Relativistic Positioning Systems and Relativistic Location Systems Jacques Rubin The projective geometry of the spacetime yielded by relativistic positioning systems and relativistic location systems Jacques Rubin To cite this version: Jacques Rubin. The projective geometry of the spacetime yielded by relativistic positioning systems and relativistic location systems. 2014. hal-00945515 HAL Id: hal-00945515 https://hal.inria.fr/hal-00945515 Submitted on 12 Feb 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The projective geometry of the spacetime yielded by relativistic positioning systems and relativistic location systems Jacques L. Rubin (email: [email protected]) Université de Nice–Sophia Antipolis, UFR Sciences Institut du Non-Linéaire de Nice, UMR7335 1361 route des Lucioles, F-06560 Valbonne, France (Dated: February 12, 2014) As well accepted now, current positioning systems such as GPS, Galileo, Beidou, etc. are not primary, relativistic systems. Nevertheless, genuine, relativistic and primary positioning systems have been proposed recently by Bahder, Coll et al. and Rovelli to remedy such prior defects. These new designs all have in common an equivariant conformal geometry featuring, as the most basic ingredient, the spacetime geometry. In a first step, we show how this conformal aspect can be the four-dimensional projective part of a larger five-dimensional geometry. Our aim has been then to explore and collect all of the geometric, physical consequences of this projective geometry in such spacetime context and ask for a physical process, the implementation of which could reveal this fifth dimension. We find that the latter is physically obtained from a fifth time stamp effectively yielded from a new localization protocol that we present jointly with this projective geometry. Based on this fifth supplementary parameter, beside the four usual ones identified with the four basic time stamps broadcast by any positioning system, we deduce the four-dimensional projective geometry governing spacetime. The former is completely detailed, i.e., the projective Cartan connection and its projec- tive curvature are computed. As a result, the Einstein tensor appears to be algebraically remarkable in this projective context. In particular, this leads to a new surprisingly result in the complete inte- grability of the Einstein equations while the well-known incompleteness of their usual non-projective version is unquestioned. Also, it may validate completely or gives naturally true physical grounds to the old model due Yano, Ohgane and Ishihara or the new approach developed by Bradonjić and Stachel similar to the one due to Schouten, Haantjes and van Dantzig unifying electromagnetism and gravitation in which the homogeneous coordinates remained till now unidentified physically. PACS numbers: 02.10.De , 04.20.-q, 04.20.Cv, 45.10.Na, 91.10.Ws, 95.10.Jk Keywords: Causal types; Emission coordinates; Lorentzian metric; Relativistic positioning systems Typeset by REVTEX LIST OF FIGURES I.1 The Ehlers-Pirani-Schild protocol. .............................................. 1 I.2 Figure on the left: the “hexagonal” domain I0J1K1LK2J2I0. Figure on the right: the dashed ± ± lines are the light-like paths of the signals carrying the time stamps with values t1 and t2 by successive echoes. For instance, at the event E2, we have an echo with the reception of the time − − + stamp with value t1 and a sending to M of a pair of time stamps with values (t1 , t2 ). 3 II.1 General scheme of conformal, projective, Weyl, and Riemannian structures (From Fig. 1. in Ehlers-Pirani-Schild paper [EPS72, p.66]) ........................................ 14 III.1 Schema representing the projective frame Ψ = (p0, [0]), (ζ1, [+∞]), (p2, [1]) ≡ [ζ0], [ζ1], [ζ2] of the projective space RP 1 in R2. ..............................................{ } { } 28 IV.1 The Tzitzeica surface Tı2 in R3 with its four strata. ................................. 74 VIII.1 The system of echoes in dimension two. ........................................ 114 VIII.2 The two-dimensional grid. .................................................. 115 VIII.3 The system of echoes with four past null cones. .................................. 116 VIII.4 The three data received and recorded by the user at U, U and U. 117 VIII.5 The change of projective frame at E. ..........................................e b 120 VIII.6 The event e in the four past null cones of the four events E, E, E and E. This event e is observed on their respective celestial spheres C, C, C and C. .e . .b . 130 ′ ′ ′ ˚′ VIII.7 The event E in the five future null cones of the fivee eventsb e, E , E , E and E . 131 ˚′ ˚• ˚◦ ˚∗ ˚ VIII.8 An example of successive events E , E , E and E on the anchoringe b worldline of E broad- casting their coordinates ˚τ5 towards the four events E, E, E and E. 132 VIII.9 The projective disk at the event E associated to the celestialb spheree C of the emitter E. 133 A.1 Protocole de Marzke-Wheeler. ................................................. 169 A.2 The Marzke-Wheeler principle ................................................ 170 C.1 △P (r2) with △P (h1h5) < 0. .................................................. 200 C.2 △P (r2) with △P (h1h5)=0. .................................................. 200 C.3 △P (r2) with △P (h1h5) > 0. .................................................. 201 C.4 The triangle-shaped loop Γ ................................................... 201 i C.5 The Tzitzeica surface ................................................... 202 C.6 A “cheek” at one of the three vertices ........................................... 204 C.7 The three “cheeks” at the vertices of a “hypothetic” triangle-shaped loop . 204 ii CONTENTS List of Figures i I. Introduction - The Coll-Ferrando-Morales-Tarentola protocols 1 II. The metric and the “energyspacetime” – A fundamental exemple 5 A. A fundamental example 5 B. What is projective or conformal, and in which manifold? 12 III. A short review on the projective differential geometry 18 A. Cartan versus Ehresmann approaches 18 B. The projective manifold and the projective group actions 26 C. The projective Cartan connections 32 1. The general case – Definitions and terminologies 32 2. From Euclidean connections to projective connections and their associated covariant derivatives 49 3. General properties 59 4. The projective (co-)tensors versus the Euclidean (co-)tensors – General remarks. 62 5. A fundamental exemple on RP n: the Cartan approach (1924). 66 3 IV. The three-dimensional submanifolds of RPS modeled on RP 70 M 3 A. The RP projective connections on RPS 70 M B. The metric fields on RPS 80 M 1. An examples of metric field on RPS 80 M 2. The metric field g of the spacetime RPS yielded by relativistic positioning M systems 82 V. The projective curvature 2-form Ω 89 A. The general case on RP n 89 B. The torsion-free and normal projective connections on RP n 95 iii VI. The normal projective curvature of the three-dimensional submanifolds of RPS M modeled on RP 3 103 VII. Interpretations – General remarks 108 VIII. A protocol implemented by receivers to localize events 112 A. The protocol of localization in a (1 + 1)-dimensional spacetime 113 M B. The protocol of localization in a (2 + 1)-dimensional spacetime modeled on M RP 3 114 C. The protocol of localization in a (3 + 1)-dimensional spacetime modeled on M RP 4 128 IX. The geometry of modeled on RP 4 139 M A. The metric field defined on 139 M B. The projective Cartan connection and the Yano-Ishihara projecting 1-form on 144 M ” 1. The projective Cartan connection ω 153 2. The projective geodesic equations 155 ” 3. The projective curvature 2-form and the normal projective Cartan connection 158 4. The metric field from the physical tensors 163 X. Conclusion and interpretations 165 Appendices 169 A. The Marzke-Wheeler protocol 169 B. Proofs of the two “factorization theorems” 171 1. Introduction 171 2. The equivalent generic metric fields 173 a. The systems of equations 174 b. The Pfaff systems and the proof of Theorem B.1 179 iv 3. The isometric equivalence 183 4. Conclusion 184 C. The “loop degeneracy” 184 1. Introduction 185 2. Generic normal forms of ssst or ℓℓℓℓ causal types for metrics 187 { } { } a. The generic normal forms G and Gϵ 187 ⊥ b. The coefficients of G and Gϵ 191 ⊥ 3. The systems of algebraic equations 192 a. The algebraic system in dimension 4 192 b. The reduced algebraic system in dimension 3 193 4. The domains of algebraic resolution 194 a. The roots if r2 = 0 195 b. The roots if r2 = h1h5 196 c. The roots if P (r ) ⩾ 0 with r ]0, h h [ 196 △ 2 2 ∈ 1 5 d. The nonnegative domain of P (r ) with r ]0, h h [ 197 △ 2 2 ∈ 1 5 5. Examples and “loop” solutions 199 6. Conclusion 205 D. The Sturm sequence of P (r ) 205 △ 2 E. The univariate polynomial equations of degree 3 or 4 and their positive roots 207 1. – The discriminant and the roots of polynomial equations of degree 3 207 2. – The discriminant and the roots of polynomial equations of degree 4 208 3. – The signs of the real roots 213 F. The space and time “projective” splitting 217 G. Proof of the 1947 Ehresmann’s Theorem 218 H. Proof of the Theorem 1 229 v I. The inverse maps 230 J. The traces of the Riemann tensors 232 K. The Christoffel symbols 235 L. The infinitesimal automorphisms of the Yano-Ihsihara projecting 1-form πˆ 237 M. The Christoffel symbols of gˆ in 239 M N. The projective geodesics on 243 M ” References 249 ” vi Jacques L. RUBIN – February 12, 2014 I.
Load more

Recommended publications
  • Projective Geometry: a Short Introduction
    Projective Geometry: A Short Introduction Lecture Notes Edmond Boyer Master MOSIG Introduction to Projective Geometry Contents 1 Introduction 2 1.1 Objective . .2 1.2 Historical Background . .3 1.3 Bibliography . .4 2 Projective Spaces 5 2.1 Definitions . .5 2.2 Properties . .8 2.3 The hyperplane at infinity . 12 3 The projective line 13 3.1 Introduction . 13 3.2 Projective transformation of P1 ................... 14 3.3 The cross-ratio . 14 4 The projective plane 17 4.1 Points and lines . 17 4.2 Line at infinity . 18 4.3 Homographies . 19 4.4 Conics . 20 4.5 Affine transformations . 22 4.6 Euclidean transformations . 22 4.7 Particular transformations . 24 4.8 Transformation hierarchy . 25 Grenoble Universities 1 Master MOSIG Introduction to Projective Geometry Chapter 1 Introduction 1.1 Objective The objective of this course is to give basic notions and intuitions on projective geometry. The interest of projective geometry arises in several visual comput- ing domains, in particular computer vision modelling and computer graphics. It provides a mathematical formalism to describe the geometry of cameras and the associated transformations, hence enabling the design of computational ap- proaches that manipulates 2D projections of 3D objects. In that respect, a fundamental aspect is the fact that objects at infinity can be represented and manipulated with projective geometry and this in contrast to the Euclidean geometry. This allows perspective deformations to be represented as projective transformations. Figure 1.1: Example of perspective deformation or 2D projective transforma- tion. Another argument is that Euclidean geometry is sometimes difficult to use in algorithms, with particular cases arising from non-generic situations (e.g.
    [Show full text]
  • Andrzej Miernowski, Witold Mozgawa GEOMETRY OF
    DEMONSTRATE MATHEMATICA Vol. XXXV No 2 2002 Andrzej Miernowski, Witold Mozgawa GEOMETRY OF ¿-PROJECTIVE STRUCTURES Abstract. The main purpose of this paper is to define and study t-projective struc- ture on even dimensional manifold M as a certain reduction of the second order frame bundle over M. This t-structure reveals some similarities to the projective structures of Kobayashi-Nagano [KN1] but it is a completely different one. The structure group is the isotropy group of the tangent bundle of the projective space. With the t-structure we associate in a natural way the so called normal Cartan connection and we investigate its properties. We show that t-structures are closely related the almost tangent structures on M. Finally, we consider the natural cross sections and we derive the coefficients of the normal connection of a t-projective structure. 0. Introduction Grassmannians of higher order appeared for the first time in a paper [Sz2] in the context of the Cartan method of moving frames. Recently, A. Szybiak has given in [Sz3] an explicit formula for the infinitesimal action of the second order jet group in dimension n on the standard fiber of the bundle of second order grassmannians on an n-dimensional manifold. In the present paper we introduce an another notion of a grassmannian of higher order in the case of a projective space which is in the natural way a homogeneous space. It is well-known that many interesting geometric structures can be obtained as structures locally modeled on homogeneous spaces. Interesting general approach to the Cartan geometries is developed in the book by R.
    [Show full text]
  • Convex Sets in Projective Space Compositio Mathematica, Tome 13 (1956-1958), P
    COMPOSITIO MATHEMATICA J. DE GROOT H. DE VRIES Convex sets in projective space Compositio Mathematica, tome 13 (1956-1958), p. 113-118 <http://www.numdam.org/item?id=CM_1956-1958__13__113_0> © Foundation Compositio Mathematica, 1956-1958, tous droits réser- vés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utili- sation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Convex sets in projective space by J. de Groot and H. de Vries INTRODUCTION. We consider the following properties of sets in n-dimensional real projective space Pn(n &#x3E; 1 ): a set is semiconvex, if any two points of the set can be joined by a (line)segment which is contained in the set; a set is convex (STEINITZ [1]), if it is semiconvex and does not meet a certain P n-l. The main object of this note is to characterize the convexity of a set by the following interior and simple property: a set is convex if and only if it is semiconvex and does not contain a whole (projective) line; in other words: a subset of Pn is convex if and only if any two points of the set can be joined uniquely by a segment contained in the set. In many cases we can prove more; see e.g.
    [Show full text]
  • Positive Geometries and Canonical Forms
    Prepared for submission to JHEP Positive Geometries and Canonical Forms Nima Arkani-Hamed,a Yuntao Bai,b Thomas Lamc aSchool of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA bDepartment of Physics, Princeton University, Princeton, NJ 08544, USA cDepartment of Mathematics, University of Michigan, 530 Church St, Ann Arbor, MI 48109, USA Abstract: Recent years have seen a surprising connection between the physics of scat- tering amplitudes and a class of mathematical objects{the positive Grassmannian, positive loop Grassmannians, tree and loop Amplituhedra{which have been loosely referred to as \positive geometries". The connection between the geometry and physics is provided by a unique differential form canonically determined by the property of having logarithmic sin- gularities (only) on all the boundaries of the space, with residues on each boundary given by the canonical form on that boundary. The structures seen in the physical setting of the Amplituhedron are both rigid and rich enough to motivate an investigation of the notions of \positive geometries" and their associated \canonical forms" as objects of study in their own right, in a more general mathematical setting. In this paper we take the first steps in this direction. We begin by giving a precise definition of positive geometries and canonical forms, and introduce two general methods for finding forms for more complicated positive geometries from simpler ones{via \triangulation" on the one hand, and \push-forward" maps between geometries on the other. We present numerous examples of positive geome- tries in projective spaces, Grassmannians, and toric, cluster and flag varieties, both for the simplest \simplex-like" geometries and the richer \polytope-like" ones.
    [Show full text]
  • Characterization of Quantum Entanglement Via a Hypercube of Segre Embeddings
    Characterization of quantum entanglement via a hypercube of Segre embeddings Joana Cirici∗ Departament de Matemàtiques i Informàtica Universitat de Barcelona Gran Via 585, 08007 Barcelona Jordi Salvadó† and Josep Taron‡ Departament de Fisíca Quàntica i Astrofísica and Institut de Ciències del Cosmos Universitat de Barcelona Martí i Franquès 1, 08028 Barcelona A particularly simple description of separability of quantum states arises naturally in the setting of complex algebraic geometry, via the Segre embedding. This is a map describing how to take products of projective Hilbert spaces. In this paper, we show that for pure states of n particles, the corresponding Segre embedding may be described by means of a directed hypercube of dimension (n − 1), where all edges are bipartite-type Segre maps. Moreover, we describe the image of the original Segre map via the intersections of images of the (n − 1) edges whose target is the last vertex of the hypercube. This purely algebraic result is then transferred to physics. For each of the last edges of the Segre hypercube, we introduce an observable which measures geometric separability and is related to the trace of the squared reduced density matrix. As a consequence, the hypercube approach gives a novel viewpoint on measuring entanglement, naturally relating bipartitions with q-partitions for any q ≥ 1. We test our observables against well-known states, showing that these provide well-behaved and fine measures of entanglement. arXiv:2008.09583v1 [quant-ph] 21 Aug 2020 ∗ [email protected] † [email protected] ‡ [email protected] 2 I. INTRODUCTION Quantum entanglement is at the heart of quantum physics, with crucial roles in quantum information theory, superdense coding and quantum teleportation among others.
    [Show full text]
  • PROJECTIVE GEOMETRY Contents 1. Basic Definitions 1 2. Axioms Of
    PROJECTIVE GEOMETRY KRISTIN DEAN Abstract. This paper investigates the nature of finite geometries. It will focus on the finite geometries known as projective planes and conclude with the example of the Fano plane. Contents 1. Basic Definitions 1 2. Axioms of Projective Geometry 2 3. Linear Algebra with Geometries 3 4. Quotient Geometries 4 5. Finite Projective Spaces 5 6. The Fano Plane 7 References 8 1. Basic Definitions First, we must begin with a few basic definitions relating to geometries. A geometry can be thought of as a set of objects and a relation on those elements. Definition 1.1. A geometry is denoted G = (Ω,I), where Ω is a set and I a relation which is both symmetric and reflexive. The relation on a geometry is called an incidence relation. For example, consider the tradional Euclidean geometry. In this geometry, the objects of the set Ω are points and lines. A point is incident to a line if it lies on that line, and two lines are incident if they have all points in common - only when they are the same line. There is often this same natural division of the elements of Ω into different kinds such as the points and lines. Definition 1.2. Suppose G = (Ω,I) is a geometry. Then a flag of G is a set of elements of Ω which are mutually incident. If there is no element outside of the flag, F, which can be added and also be a flag, then F is called maximal. Definition 1.3. A geometry G = (Ω,I) has rank r if it can be partitioned into sets Ω1,..., Ωr such that every maximal flag contains exactly one element of each set.
    [Show full text]
  • Arxiv:1810.02861V1 [Math.AG] 5 Oct 2018 CP Oyoileuto N Ae Xeddt Ufcsadhge Dim Using Higher Besides and Equations
    October 9, 2018 ALGEBRAIC HYPERSURFACES JANOS´ KOLLAR´ Abstract. We give an introduction to the study of algebraic hypersurfaces, focusing on the problem of when two hypersurfaces are isomorphic or close to being isomorphic. Working with hypersurfaces and emphasizing examples makes it possible to discuss these questions without any previous knowledge of algebraic geometry. At the end we formulate the main recent results and state the most important open questions. Contents 1. Stereographic projection 2 2. Projective hypersurfaces 4 3. Rational and birational maps 5 4. The main questions 8 5. Rationality of cubic hypersurfaces 10 6. Isomorphism of hypersurfaces 12 7. Non-rationalityoflargedegreehypersurfaces 14 8. Non-rationalityoflowdegreehypersurfaces 17 9. Rigidity of low degree hypersurfaces 18 10. Connections with the classification of varieties 18 11. Openproblemsabouthypersurfaces 19 References 21 Algebraic geometry started as the study of plane curves C R2 defined by a polynomial equation and later extended to surfaces and higher⊂ dimensional sets defined by systems of polynomial equations. Besides using Rn, it is frequently more advantageous to work with Cn or with the corresponding projective spaces RPn and n arXiv:1810.02861v1 [math.AG] 5 Oct 2018 CP . Later it was realized that the theory also works if we replace R or C by other fields, for example the field of rational numbers Q or even finite fields Fq. When we try to emphasize that the choice of the field is pretty arbitrary, we use An to denote affine n-space and Pn to denote projective n-space. Conceptually the simplest algebraic sets are hypersurfaces; these are defined by 1 equation.
    [Show full text]
  • Euclidean Versus Projective Geometry
    Projective Geometry Projective Geometry Euclidean versus Projective Geometry n Euclidean geometry describes shapes “as they are” – Properties of objects that are unchanged by rigid motions » Lengths » Angles » Parallelism n Projective geometry describes objects “as they appear” – Lengths, angles, parallelism become “distorted” when we look at objects – Mathematical model for how images of the 3D world are formed. Projective Geometry Overview n Tools of algebraic geometry n Informal description of projective geometry in a plane n Descriptions of lines and points n Points at infinity and line at infinity n Projective transformations, projectivity matrix n Example of application n Special projectivities: affine transforms, similarities, Euclidean transforms n Cross-ratio invariance for points, lines, planes Projective Geometry Tools of Algebraic Geometry 1 n Plane passing through origin and perpendicular to vector n = (a,b,c) is locus of points x = ( x 1 , x 2 , x 3 ) such that n · x = 0 => a x1 + b x2 + c x3 = 0 n Plane through origin is completely defined by (a,b,c) x3 x = (x1, x2 , x3 ) x2 O x1 n = (a,b,c) Projective Geometry Tools of Algebraic Geometry 2 n A vector parallel to intersection of 2 planes ( a , b , c ) and (a',b',c') is obtained by cross-product (a'',b'',c'') = (a,b,c)´(a',b',c') (a'',b'',c'') O (a,b,c) (a',b',c') Projective Geometry Tools of Algebraic Geometry 3 n Plane passing through two points x and x’ is defined by (a,b,c) = x´ x' x = (x1, x2 , x3 ) x'= (x1 ', x2 ', x3 ') O (a,b,c) Projective Geometry Projective Geometry
    [Show full text]
  • Eduard Čech, 1893–1960
    Czechoslovak Mathematical Journal Bohuslav Balcar; Václav Koutník; Petr Simon Eduard Čech, 1893–1960 Czechoslovak Mathematical Journal, Vol. 43 (1993), No. 3, 567–575 Persistent URL: http://dml.cz/dmlcz/128420 Terms of use: © Institute of Mathematics AS CR, 1993 Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This document has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://dml.cz Czechoslovak Matheшatical Joumal, 43 (118) 1993, Praha NEWS AND NOTICES EDUARD CECH, 1893-1960 BOHUSLAV BALCAR, VÁCLAV KOUTNÍK, PETR SIMON, Praha This year we observe the 100th anniversary of the birthday of Eduard Cech, one of the leading world specialists in topology and differential geometry. To these fields he contributed works of fundamental importance. He was born on June 29, 1893 in Stracov in northeastern Bohemia. During his high school studies in Hradec Kralove he became interested in mathematics and in 1912 he entered the Philosophical Faculty of Charles University in Prague. At that time there were very few opportunities for mathematicians other than to become a high school teacher. For a position of a high school teacher two fields of study were required. Since Cech was not much interested in physics, the standard second subject, he chose descriptive geometry. During his studies at the university he spent a lot of time in the library of the Union of Czech Mathematicians and Physicists and read many mathematical books of his own choice.
    [Show full text]
  • ON HITCHIN's CONNECTION 1. Introduction 1.1. the Aim of This Paper Is to Give an Explicit Expression for Hitchin's Connectio
    JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 11, Number 1, January 1998, Pages 189{228 S 0894-0347(98)00252-5 ON HITCHIN'S CONNECTION BERT VAN GEEMEN AND AISE JOHAN DE JONG 1. Introduction 1.1. The aim of this paper is to give an explicit expression for Hitchin’s connection in the case of rank 2 bundles with trivial determinant over curves of genus 2. We sketch the general situation. Let π : Sbe a family of projective smooth curves. Let p : Sbe the family of moduliC→ spaces of (S-equivalence classes of) rank r bundlesM→ with trivial determinant on over S.On we have a naturally C Mk defined determinant line bundle . The vector bundles p ( ⊗ )onShave a natural (projective) flat connection (suchL a connection also exists∗ forL other structure groups besides SL(r)). This connection first appeared in Quantum Field Theory (conformal blocks) and was subsequently studied by various mathematicians (see [Hi] and [BM] and the references given there). We follow the approach of Hitchin [Hi] who showed that this connection can be constructed along the lines of Welters’ theory of deformations [W]. In case the curves have genus 2 and the rank r = 2, the family of moduli spaces p : S is isomorphic (over S)toaP3-bundle M→ p : = P S. M ∼ −→ Under this isomorphism corresponds to P(1) p∗ for some line bundle on L Ok ⊗ N N S. Giving a projective connection on p ⊗ is then the same as giving a projective ∗L connection on p P(k) ∗O (k) 1 : p P(k) p P ( k ) Ω S .
    [Show full text]
  • On the Imbeddability of the Real Projective Spaces in Euclidean Space
    Pacific Journal of Mathematics ON THE IMBEDDABILITY OF THE REAL PROJECTIVE SPACES IN EUCLIDEAN SPACE WILLIAM SCHUMACHER MASSEY Vol. 9, No. 3 July 1959 ON THE IMBEDDABILITY OF THE REAL PROJECTIVE SPACES IN EUCLIDEAN SPACE1 W. S. MASSEY 1* Introduction. Let Pn denote ^-dimensional real projective space. This paper is concerned with the following question: What is the lowest dimensional Euclidean space in which Pn can be imbedded topologically or differentiably ? Among previous results along this line, we may mention the following; (a) If n is even, then Pn is a non-orientable manifold, and hence cannot be imbedded topologically in (n + l)-dimensional Euclidean space, Rn+1. (b) For any integer n > 1, Pn cannot be imbedded topologically in n+ R \ because its mod 2 cohomology algebra, H*(Pn, Z2), does not satisfy a certain condition given by R. Thorn (see [6], Theorem V, 15). IC 1 fc (c) If 2 ~ ^ n < 2 then Pn cannot be imbedded topologically in Euclidean space of dimension 2k — 1. This result follows from knowledge of the Stiefel-Whitney classes of Pn (see Thorn, loc. cit., Theorem III. 16 and E. Stiefel, [5]; also [4]). In the present paper, we prove the following result: If m = 2fc, k > 0, im then P3w_! cannot be imbedded differentiably in R . For example P5 s ιG cannot be imbedded differentiably in R , nor can Pn be imbedded in R . Of course if n > m, Pn cannot, a fortiori, be imbedded differentiably in R4m. Thus for many values of n our theorem is an improvement over previous results on this subject.1 The proof of this theorem depends on certain general results on the cohomology mod 2 of sphere bundles.
    [Show full text]
  • Vector Spaces and Projective Geometry
    Vector spaces and projective geometry Lou de Boer October 29, 2017 Abstract A relation is established between the duality within projective geometry and the duality between a vector space and its dual space of linear funtions. The action of a linear function on a vector appears to be a cross ratio. 1 Introduction Vector spaces are of fundamental importance, not only in Mathematics, but in many applied sciences as well, especially in Physics. Less familiar is the concept of the dual vector space - the space of linear functions on a vector space - though it is a basic concept in both Mathematics and Theoretical Physics. Projective spaces are - even among many mathematicians - alien or forgotten objects, that one encounters every once in a while, but hardly ever needs to use. We will show in this article that projective spaces are at the heart of Linear Algebra, and that in particular the dual vector space gets a better understanding when seen in the light of projective geometry. 2 Prerequisites The reader is supposed to be familiar with the basics of linear algebra and projective geometry, in particular with the concepts of (number) field, vector space, projective space and cross ratio. In this article we will restrict to finite-dimensional vector spaces. Essentially these are the n-th powers of number fields: every n-dimensional vector space over a field F is isomorphic to the space F n of n-tuples of numbers of F . An n-dimensional projective space over F can be defined in various ways. One way is to extend Euclidean n-space with an (n − 1)-dimensional hyperplane `at infinity'.
    [Show full text]