On Nye's Lattice Curvature Tensor 1 Introduction
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Ricci, Levi-Civita, and the Birth of General Relativity Reviewed by David E
BOOK REVIEW Einstein’s Italian Mathematicians: Ricci, Levi-Civita, and the Birth of General Relativity Reviewed by David E. Rowe Einstein’s Italian modern Italy. Nor does the author shy away from topics Mathematicians: like how Ricci developed his absolute differential calculus Ricci, Levi-Civita, and the as a generalization of E. B. Christoffel’s (1829–1900) work Birth of General Relativity on quadratic differential forms or why it served as a key By Judith R. Goodstein tool for Einstein in his efforts to generalize the special theory of relativity in order to incorporate gravitation. In This delightful little book re- like manner, she describes how Levi-Civita was able to sulted from the author’s long- give a clear geometric interpretation of curvature effects standing enchantment with Tul- in Einstein’s theory by appealing to his concept of parallel lio Levi-Civita (1873–1941), his displacement of vectors (see below). For these and other mentor Gregorio Ricci Curbastro topics, Goodstein draws on and cites a great deal of the (1853–1925), and the special AMS, 2018, 211 pp. 211 AMS, 2018, vast secondary literature produced in recent decades by the world that these and other Ital- “Einstein industry,” in particular the ongoing project that ian mathematicians occupied and helped to shape. The has produced the first 15 volumes of The Collected Papers importance of their work for Einstein’s general theory of of Albert Einstein [CPAE 1–15, 1987–2018]. relativity is one of the more celebrated topics in the history Her account proceeds in three parts spread out over of modern mathematical physics; this is told, for example, twelve chapters, the first seven of which cover episodes in [Pais 1982], the standard biography of Einstein. -
Toponogov.V.A.Differential.Geometry
Victor Andreevich Toponogov with the editorial assistance of Vladimir Y. Rovenski Differential Geometry of Curves and Surfaces A Concise Guide Birkhauser¨ Boston • Basel • Berlin Victor A. Toponogov (deceased) With the editorial assistance of: Department of Analysis and Geometry Vladimir Y. Rovenski Sobolev Institute of Mathematics Department of Mathematics Siberian Branch of the Russian Academy University of Haifa of Sciences Haifa, Israel Novosibirsk-90, 630090 Russia Cover design by Alex Gerasev. AMS Subject Classification: 53-01, 53Axx, 53A04, 53A05, 53A55, 53B20, 53B21, 53C20, 53C21 Library of Congress Control Number: 2005048111 ISBN-10 0-8176-4384-2 eISBN 0-8176-4402-4 ISBN-13 978-0-8176-4384-3 Printed on acid-free paper. c 2006 Birkhauser¨ Boston All rights reserved. This work may not be translated or copied in whole or in part without the writ- ten permission of the publisher (Birkhauser¨ Boston, c/o Springer Science+Business Media Inc., 233 Spring Street, New York, NY 10013, USA) and the author, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and re- trieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. (TXQ/EB) 987654321 www.birkhauser.com Contents Preface ....................................................... vii About the Author ............................................. -
Jhep01(2020)007
Published for SISSA by Springer Received: March 27, 2019 Revised: November 15, 2019 Accepted: December 9, 2019 Published: January 2, 2020 Deformed graded Poisson structures, generalized geometry and supergravity JHEP01(2020)007 Eugenia Boffo and Peter Schupp Jacobs University Bremen, Campus Ring 1, 28759 Bremen, Germany E-mail: [email protected], [email protected] Abstract: In recent years, a close connection between supergravity, string effective ac- tions and generalized geometry has been discovered that typically involves a doubling of geometric structures. We investigate this relation from the point of view of graded ge- ometry, introducing an approach based on deformations of graded Poisson structures and derive the corresponding gravity actions. We consider in particular natural deformations of the 2-graded symplectic manifold T ∗[2]T [1]M that are based on a metric g, a closed Neveu-Schwarz 3-form H (locally expressed in terms of a Kalb-Ramond 2-form B) and a scalar dilaton φ. The derived bracket formalism relates this structure to the generalized differential geometry of a Courant algebroid, which has the appropriate stringy symme- tries, and yields a connection with non-trivial curvature and torsion on the generalized “doubled” tangent bundle E =∼ TM ⊕ T ∗M. Projecting onto TM with the help of a natural non-isotropic splitting of E, we obtain a connection and curvature invariants that reproduce the NS-NS sector of supergravity in 10 dimensions. Further results include a fully generalized Dorfman bracket, a generalized Lie bracket and new formulas for torsion and curvature tensors associated to generalized tangent bundles. -
Connections on Bundles Md
Dhaka Univ. J. Sci. 60(2): 191-195, 2012 (July) Connections on Bundles Md. Showkat Ali, Md. Mirazul Islam, Farzana Nasrin, Md. Abu Hanif Sarkar and Tanzia Zerin Khan Department of Mathematics, University of Dhaka, Dhaka 1000, Bangladesh, Email: [email protected] Received on 25. 05. 2011.Accepted for Publication on 15. 12. 2011 Abstract This paper is a survey of the basic theory of connection on bundles. A connection on tangent bundle , is called an affine connection on an -dimensional smooth manifold . By the general discussion of affine connection on vector bundles that necessarily exists on which is compatible with tensors. I. Introduction = < , > (2) In order to differentiate sections of a vector bundle [5] or where <, > represents the pairing between and ∗. vector fields on a manifold we need to introduce a Then is a section of , called the absolute differential structure called the connection on a vector bundle. For quotient or the covariant derivative of the section along . example, an affine connection is a structure attached to a differentiable manifold so that we can differentiate its Theorem 1. A connection always exists on a vector bundle. tensor fields. We first introduce the general theorem of Proof. Choose a coordinate covering { }∈ of . Since connections on vector bundles. Then we study the tangent vector bundles are trivial locally, we may assume that there is bundle. is a -dimensional vector bundle determine local frame field for any . By the local structure of intrinsically by the differentiable structure [8] of an - connections, we need only construct a × matrix on dimensional smooth manifold . each such that the matrices satisfy II. -
Geometry of Curves
Appendix A Geometry of Curves “Arc, amplitude, and curvature sustain a similar relation to each other as time, motion and velocity, or as volume, mass and density.” Carl Friedrich Gauss The rest of this lecture notes is about geometry of curves and surfaces in R2 and R3. It will not be covered during lectures in MATH 4033 and is not essential to the course. However, it is recommended for readers who want to acquire workable knowledge on Differential Geometry. A.1. Curvature and Torsion A.1.1. Regular Curves. A curve in the Euclidean space Rn is regarded as a function r(t) from an interval I to Rn. The interval I can be finite, infinite, open, closed n n or half-open. Denote the coordinates of R by (x1, x2, ... , xn), then a curve r(t) in R can be written in coordinate form as: r(t) = (x1(t), x2(t),..., xn(t)). One easy way to make sense of a curve is to regard it as the trajectory of a particle. At any time t, the functions x1(t), x2(t), ... , xn(t) give the coordinates of the particle in n R . Assuming all xi(t), where 1 ≤ i ≤ n, are at least twice differentiable, then the first derivative r0(t) represents the velocity of the particle, its magnitude jr0(t)j is the speed of the particle, and the second derivative r00(t) represents the acceleration of the particle. As a course on Differential Manifolds/Geometry, we will mostly study those curves which are infinitely differentiable (i.e. -
Laplacians in Geometric Analysis
LAPLACIANS IN GEOMETRIC ANALYSIS Syafiq Johar syafi[email protected] Contents 1 Trace Laplacian 1 1.1 Connections on Vector Bundles . .1 1.2 Local and Explicit Expressions . .2 1.3 Second Covariant Derivative . .3 1.4 Curvatures on Vector Bundles . .4 1.5 Trace Laplacian . .5 2 Harmonic Functions 6 2.1 Gradient and Divergence Operators . .7 2.2 Laplace-Beltrami Operator . .7 2.3 Harmonic Functions . .8 2.4 Harmonic Maps . .8 3 Hodge Laplacian 9 3.1 Exterior Derivatives . .9 3.2 Hodge Duals . 10 3.3 Hodge Laplacian . 12 4 Hodge Decomposition 13 4.1 De Rham Cohomology . 13 4.2 Hodge Decomposition Theorem . 14 5 Weitzenb¨ock and B¨ochner Formulas 15 5.1 Weitzenb¨ock Formula . 15 5.1.1 0-forms . 15 5.1.2 k-forms . 15 5.2 B¨ochner Formula . 17 1 Trace Laplacian In this section, we are going to present a notion of Laplacian that is regularly used in differential geometry, namely the trace Laplacian (also called the rough Laplacian or connection Laplacian). We recall the definition of connection on vector bundles which allows us to take the directional derivative of vector bundles. 1.1 Connections on Vector Bundles Definition 1.1 (Connection). Let M be a differentiable manifold and E a vector bundle over M. A connection or covariant derivative at a point p 2 M is a map D : Γ(E) ! Γ(T ∗M ⊗ E) 1 with the properties for any V; W 2 TpM; σ; τ 2 Γ(E) and f 2 C (M), we have that DV σ 2 Ep with the following properties: 1. -
The Simplicial Ricci Tensor 2
The Simplicial Ricci Tensor Paul M. Alsing1, Jonathan R. McDonald 1,2 & Warner A. Miller3 1Information Directorate, Air Force Research Laboratory, Rome, New York 13441 2Insitut f¨ur Angewandte Mathematik, Friedrich-Schiller-Universit¨at-Jena, 07743 Jena, Germany 3Department of Physics, Florida Atlantic University, Boca Raton, FL 33431 E-mail: [email protected] Abstract. The Ricci tensor (Ric) is fundamental to Einstein’s geometric theory of gravitation. The 3-dimensional Ric of a spacelike surface vanishes at the moment of time symmetry for vacuum spacetimes. The 4-dimensional Ric is the Einstein tensor for such spacetimes. More recently the Ric was used by Hamilton to define a non-linear, diffusive Ricci flow (RF) that was fundamental to Perelman’s proof of the Poincar`e conjecture. Analytic applications of RF can be found in many fields including general relativity and mathematics. Numerically it has been applied broadly to communication networks, medical physics, computer design and more. In this paper, we use Regge calculus (RC) to provide the first geometric discretization of the Ric. This result is fundamental for higher-dimensional generalizations of discrete RF. We construct this tensor on both the simplicial lattice and its dual and prove their equivalence. We show that the Ric is an edge-based weighted average of deficit divided by an edge-based weighted average of dual area – an expression similar to the vertex-based weighted average of the scalar curvature reported recently. We use this Ric in a third and independent geometric derivation of the RC Einstein tensor in arbitrary dimension. arXiv:1107.2458v1 [gr-qc] 13 Jul 2011 The Simplicial Ricci Tensor 2 1. -
Selected Papers on Teleparallelism Ii
SELECTED PAPERS ON TELEPARALLELISM Edited and translated by D. H. Delphenich Table of contents Page Introduction ……………………………………………………………………… 1 1. The unification of gravitation and electromagnetism 1 2. The geometry of parallelizable manifold 7 3. The field equations 20 4. The topology of parallelizability 24 5. Teleparallelism and the Dirac equation 28 6. Singular teleparallelism 29 References ……………………………………………………………………….. 33 Translations and time line 1928: A. Einstein, “Riemannian geometry, while maintaining the notion of teleparallelism ,” Sitzber. Preuss. Akad. Wiss. 17 (1928), 217- 221………………………………………………………………………………. 35 (Received on June 7) A. Einstein, “A new possibility for a unified field theory of gravitation and electromagnetism” Sitzber. Preuss. Akad. Wiss. 17 (1928), 224-227………… 42 (Received on June 14) R. Weitzenböck, “Differential invariants in EINSTEIN’s theory of teleparallelism,” Sitzber. Preuss. Akad. Wiss. 17 (1928), 466-474……………… 46 (Received on Oct 18) 1929: E. Bortolotti , “ Stars of congruences and absolute parallelism: Geometric basis for a recent theory of Einstein ,” Rend. Reale Acc. dei Lincei 9 (1929), 530- 538...…………………………………………………………………………….. 56 R. Zaycoff, “On the foundations of a new field theory of A. Einstein,” Zeit. Phys. 53 (1929), 719-728…………………………………………………............ 64 (Received on January 13) Hans Reichenbach, “On the classification of the new Einstein Ansatz on gravitation and electricity,” Zeit. Phys. 53 (1929), 683-689…………………….. 76 (Received on January 22) Selected papers on teleparallelism ii A. Einstein, “On unified field theory,” Sitzber. Preuss. Akad. Wiss. 18 (1929), 2-7……………………………………………………………………………….. 82 (Received on Jan 30) R. Zaycoff, “On the foundations of a new field theory of A. Einstein; (Second part),” Zeit. Phys. 54 (1929), 590-593…………………………………………… 89 (Received on March 4) R. -
Millennium Prize for the Poincaré
FOR IMMEDIATE RELEASE • March 18, 2010 Press contact: James Carlson: [email protected]; 617-852-7490 See also the Clay Mathematics Institute website: • The Poincaré conjecture and Dr. Perelmanʼs work: http://www.claymath.org/poincare • The Millennium Prizes: http://www.claymath.org/millennium/ • Full text: http://www.claymath.org/poincare/millenniumprize.pdf First Clay Mathematics Institute Millennium Prize Announced Today Prize for Resolution of the Poincaré Conjecture a Awarded to Dr. Grigoriy Perelman The Clay Mathematics Institute (CMI) announces today that Dr. Grigoriy Perelman of St. Petersburg, Russia, is the recipient of the Millennium Prize for resolution of the Poincaré conjecture. The citation for the award reads: The Clay Mathematics Institute hereby awards the Millennium Prize for resolution of the Poincaré conjecture to Grigoriy Perelman. The Poincaré conjecture is one of the seven Millennium Prize Problems established by CMI in 2000. The Prizes were conceived to record some of the most difficult problems with which mathematicians were grappling at the turn of the second millennium; to elevate in the consciousness of the general public the fact that in mathematics, the frontier is still open and abounds in important unsolved problems; to emphasize the importance of working towards a solution of the deepest, most difficult problems; and to recognize achievement in mathematics of historical magnitude. The award of the Millennium Prize to Dr. Perelman was made in accord with their governing rules: recommendation first by a Special Advisory Committee (Simon Donaldson, David Gabai, Mikhail Gromov, Terence Tao, and Andrew Wiles), then by the CMI Scientific Advisory Board (James Carlson, Simon Donaldson, Gregory Margulis, Richard Melrose, Yum-Tong Siu, and Andrew Wiles), with final decision by the Board of Directors (Landon T. -
The Work of Grigory Perelman
The work of Grigory Perelman John Lott Grigory Perelman has been awarded the Fields Medal for his contributions to geom- etry and his revolutionary insights into the analytical and geometric structure of the Ricci flow. Perelman was born in 1966 and received his doctorate from St. Petersburg State University. He quickly became renowned for his work in Riemannian geometry and Alexandrov geometry, the latter being a form of Riemannian geometry for metric spaces. Some of Perelman’s results in Alexandrov geometry are summarized in his 1994 ICM talk [20]. We state one of his results in Riemannian geometry. In a short and striking article, Perelman proved the so-called Soul Conjecture. Soul Conjecture (conjectured by Cheeger–Gromoll [2] in 1972, proved by Perelman [19] in 1994). Let M be a complete connected noncompact Riemannian manifold with nonnegative sectional curvatures. If there is a point where all of the sectional curvatures are positive then M is diffeomorphic to Euclidean space. In the 1990s, Perelman shifted the focus of his research to the Ricci flow and its applications to the geometrization of three-dimensional manifolds. In three preprints [21], [22], [23] posted on the arXiv in 2002–2003, Perelman presented proofs of the Poincaré conjecture and the geometrization conjecture. The Poincaré conjecture dates back to 1904 [24]. The version stated by Poincaré is equivalent to the following. Poincaré conjecture. A simply-connected closed (= compact boundaryless) smooth 3-dimensional manifold is diffeomorphic to the 3-sphere. Thurston’s geometrization conjecture is a far-reaching generalization of the Poin- caré conjecture. It says that any closed orientable 3-dimensional manifold can be canonically cut along 2-spheres and 2-tori into “geometric pieces” [27]. -
Curvature Tensors in a 4D Riemann–Cartan Space: Irreducible Decompositions and Superenergy
Curvature tensors in a 4D Riemann–Cartan space: Irreducible decompositions and superenergy Jens Boos and Friedrich W. Hehl [email protected] [email protected]"oeln.de University of Alberta University of Cologne & University of Missouri (uesday, %ugust 29, 17:0. Geometric Foundations of /ravity in (artu Institute of 0hysics, University of (artu) Estonia Geometric Foundations of /ravity Geometric Foundations of /auge Theory Geometric Foundations of /auge Theory ↔ Gravity The ingredients o$ gauge theory: the e2ample o$ electrodynamics ,3,. The ingredients o$ gauge theory: the e2ample o$ electrodynamics 0henomenological Ma24ell: redundancy conserved e2ternal current 5 ,3,. The ingredients o$ gauge theory: the e2ample o$ electrodynamics 0henomenological Ma24ell: Complex spinor 6eld: redundancy invariance conserved e2ternal current 5 conserved #7,8 current ,3,. The ingredients o$ gauge theory: the e2ample o$ electrodynamics 0henomenological Ma24ell: Complex spinor 6eld: redundancy invariance conserved e2ternal current 5 conserved #7,8 current Complete, gauge-theoretical description: 9 local #7,) invariance ,3,. The ingredients o$ gauge theory: the e2ample o$ electrodynamics 0henomenological Ma24ell: iers Complex spinor 6eld: rce carr ry of fo mic theo rrent rosco rnal cu m pic en exte att desc gredundancyiv er; N ript oet ion o conserved e2ternal current 5 invariance her f curr conserved #7,8 current e n t s Complete, gauge-theoretical description: gauge theory = complete description of matter and 9 local #7,) invariance how it interacts via gauge bosons ,3,. Curvature tensors electrodynamics :ang–Mills theory /eneral Relativity 0oincaré gauge theory *3,. Curvature tensors electrodynamics :ang–Mills theory /eneral Relativity 0oincaré gauge theory *3,. Curvature tensors electrodynamics :ang–Mills theory /eneral Relativity 0oincar; gauge theory *3,. -
Book of Abstracts Ii Contents
Spring workshop on gravity and cosmology Monday, 25 May 2020 - Friday, 29 May 2020 Jagiellonian University Book of Abstracts ii Contents Stochastic gravitational waves from inflaton decays ..................... 1 Approximate Killing symmetries in non-perturbative quantum gravity .......... 1 Disformal transformations in modified teleparallelism .................... 1 Quantum fate of the BKL scenario ............................... 2 Ongoing Efforts to Constrain Lorentz Symmetry Violation in Gravity ........... 2 Proof of the quantum null energy condition for free fermionic field theories . 2 Cosmic Fibers and the parametrization of time in CDT quantum gravity . 3 Stable Cosmology in Generalised Massive Gravity ...................... 3 Constraining the inflationary field content .......................... 3 The (glorious) past, (exciting) present and (foreseeable) future of gravitational wave detec- tors. ............................................. 4 reception ............................................. 4 Testing Inflation with Primordial Messengers ........................ 4 Gravitational waves from inflation ............................... 4 CDT, a theory of quantum geometry. ............................. 4 Cosmic Fibers and the parametrization of time in CDT quantum gravity . 5 Loop-based observables in 4D CDT .............................. 5 Spectral Analysis of Causal Dynamical Triangulations via Finite Element Methods . 5 A consistent theory of D -> 4 Einstein-Gauss-Bonnet gravity ................ 5 Quantum Ostrogradsky theorem