The existence of light-like homogeneous geodesics in homogeneous Lorentzian manifolds. Sao Paulo, 2013
|
|
- Jesse Andrews
- 5 years ago
- Views:
Transcription
1 The existence of light-like homogeneous geodesics in homogeneous Lorentzian manifolds Zdeněk Dušek Sao Paulo, 2013
2 Motivation In a previous project, it was proved that any homogeneous affine manifold (and in particular any homogeneous pseudo-riemannian manifold) admits a homogeneous geodesic through arbitrary point. In pseudo-riemannian geometry, null homogeneous geodesics are of particular interest. Plane-wave limits (Penrose limits) of homogeneous spacetimes along light-like homogeneous geodesics are studied. However, it was not known whether any homogeneous pseudo-riemannian or Lorentzian manifold admits a null homogeneous geodesic. An example of a 3-dimensional Lie group with an invariant Lorentzian metric which does not admit light-like homogeneous geodesic was described (G. Calvaruso).
3 Results In the present project, the affine method is adapted to the pseudo-riemannian case. We show that any Lorentzian homogeneous manifold of even dimension admits a light-like homogeneous geodesic. In the case of a Lie group G = M with a left-invariant metric, the calculation is particularly easy. As an illustration, we apply the method on an example of a Lie group in dimension 3.
4 Homogeneous geodesics in affine manifolds Lemma Let (M, ) be a homogeneous affine manifold. Then each regular curve which is an orbit of a 1-parameter subgroup g t G on M is an integral curve of an affine Killing vector field on M. Lemma Let (M, ) be a homogeneous affine manifold and p M. There exist n = dim(m) affine Killing vector fields which are linearly independent at each point of some neighbourhood U of p. Lemma The integral curve γ(t) of the Killing vector field Z on (M, ) is geodesic if and only if Zγ(t) Z = k γ Z γ(t) holds along γ. Here k γ R is a constant.
5 Existence of homogeneous geodesics Theorem Let M = (G/H, ) be a homogeneous affine manifold and p M. Then M admits a homogeneous geodesic through p. Proof. Killing vector fields K 1,..., K n independent near p, basis B = {K 1 (p),..., K n (p)} of T p M. Any vector X T p M, X = (x 1,... x n ) in B, determines a Killing vector field X = x 1 K x n K n and an integral curve γ X of X through p. Sphere S n 1 T p M, vectors X = (x 1,..., x n ) with X = 1. Denote v(x ) = X X and t(x ) = v(x ) v(x ), X X, then t(x ) X and X t(x ) defines a vector field on S n 1. If n is odd, according to the Hair-Dressing Theorem for sphere, there is X T p M such that t( X ) = 0. We see v( X ) = k X, hence X X = k X.
6 Existence of homogeneous geodesics We refine the proof to arbitrary dimension: Recall that X t(x ) defines a smooth vector field on S n 1. Assume now that t(x ) 0 everywhere. Putting f (X ) = t(x )/ t(x ), we obtain a smooth map f : S n 1 S n 1 without fixed points. According to a well-known statement from differential topology, the degree of f is odd (integral degree is deg(f ) = ( 1) n ). On the other hand, we have v(x ) = v( X ) and hence f (X ) = f ( X ) for each X. If Y is a regular value of f, then the inverse image f 1 (Y ) consists of even number of elements, hence deg(f ) is even, which is a contradiction. This implies that there is X T p M such that t( X ) = 0 and again, a homogeneous geodesic exists.
7 Homogeneous Lorentzian manifolds Proposition Let φ X (t) be the 1-parameter group of isometries corresponding to the Killing vector field X. For all t R, it holds φ X (t)(p) = γ X (t), φ X (t) (X p ) = X γ X (t). The covariant derivative X X depends only on the values of X along γ X (t). From the invariance of g and, we obtain Proposition Along the curve γ X (t), it holds for all t R g p (X, X ) = g γx (t)(x γ X (t), X γ X (t) ), φ X (t) ( X X p ) = X X γx (t).
8 Proposition Let (M, g) be a homogeneous Lorentzian manifold, p M and X T p M. Then, along the curve γ X (t), it holds X X γx (t) (X γ X (t) ). Proof. We use the basic property g = 0 in the form X g(x, X ) = 2g( X X, X ). (1) According to Proposition 2, the function g(x, X ) is constant along γ X (t). Hence, the left-hand side of the equality (1) is zero and the right-hand side gives the statement.
9 Theorem Let (M, g) be a homogeneous Lorentzian manifold of even dimension n and let p M. There exist a light-like vector X T p M such that along the integral curve γ X (t) of the Killing vector field X it holds where k R is some constant. Proof. X X γx (t) = k X γ X (t), Killing vector fields K 1,... K n such that {K 1 (p),..., K n (p)} is a pseudo-orthonormal basis of T p M with K n (p) timelike. Any airthmetic vector x = (x 1,..., x n ) R n determines the Killing vector field X = n i=1 x i K i. We identify x with X p and R n T p M. We consider x = ( x, 1), where x S n 2 R n 1. For X, we have g p (X p, X p ) = 0 and the vectors x S n 2 determine light-like directions in R n T p M.
10 For x = ( x, 1) R n T p M, we denote Y x = X X p. With respect to the basis B = {K 1 (p),..., K n (p)}, we denote the components of the vector Y x as y(x) = (y 1,..., y n ). Using Proposition, we see that y(x) x. We define the new vector t x as t x = y(x) y n x. Because x is light-like vector, it holds also t x x. In components, we have t x = ( t x, 0), where t x R n 1. We see that t x x, with respect to the positive scalar product on R n 1 which is the restriction of the indefinite scalar product on R n. The assignment x t x defines a smooth tangent vector field on the sphere S n 2. If n is even, it must have a zero value. There exist a vector x S n 2 such that for the corresponding vector x = ( x, 1) it holds t x = 0. For this vector x, it holds y(x) = k x and X X γx (t) = k X γ X (t) is satisfied.
11 Corollary Let (M, g) be a homogeneous Lorentzian manifold of even dimension n and let p M. There exist a light-like homogeneous geodesic through p. Proof. We consider the vector X T p M which satisfies Theorem. The integral curve γ X (t) through p of the corresponding Killing vector field X is homogeneous geodesic.
12 Invariant metric on a Lie group Let M = G be a Lie group with a left-invariant metric g. For any tangent vector X T e M and the corresponding Killing vector field X, we consider the vector function X γ X (t) along the integral curve γ X (t) through e. It can be uniquely extended to the left-invariant vector field L X on G. Hence, along γ X, we have L X γ X (t) = X γ X (t). (2) At general points q G, values of left-invariant vector field L X do not coincide with the values of the Killing vector field X, which is right-invariant. As we are interested in calculations along the curve γ X (t), we can work with respect to the moving frame of left-invariant vector fields and use formula (2).
13 Proposition Let {L 1,..., L n } be a left-invariant moving frame on a Lie group G with a left-invariant pseudo-riemannian metric g and the induced pseudo-riemannian connection. Then it holds Li L j = n γij k L k, i, j = 1,..., n, k=1 where γ k ij are constants. Proof. It follows from the invariance of the affine connection.
14 An example of a Lie group Now we illustrate the affine method of the previous section with an example of the 3-dimensional Lie group E(1, 1) with an inv. Lorentzian metric which has no light-like homogeneous geodesic. We choose one of the examples described by G. Calvaruso using the geodesic lemma for reductive pseudo-riemannian homogeneous manifolds. We construct explicitly the vector field t x, which has no zero value in this case.
15 The group E(1, 1) can be represented by the matrices e w 0 u 0 e w v Hence, M = E(1, 1) can be identified with R 3 [u, v, w]. The left-inv. vector fields are U = e w u, V = e w v, W = w.. We choose the new moving frame {E 1, E 2, E 3 } given as E 1 = U V, E 2 = W, E 3 = 1/2(U + V ). The pseudo-riemannian metric g such that the basis determined by the above frame at any point p M is pseudo-orthonormal basis of T p M with E 3 timelike.
16 The above metric g in coordinates is ds 2 = 1 4 (3e2w du 2 + 3e 2w dv dudv 4dw 2 ) and the nonzero Christoffel symbols are Γ 3 11 = 3 4 e2w, Γ 1 13 = 9 16, Γ2 13 = e2w, Γ 3 22 = 3 4 e 2w, Γ 2 23 = 9 16, Γ1 23 = e 2w. In the frame {E 1, E 2, E 3 }: g(e 1, E 1 ) = g(e 2, E 2 ) = 1, g(e 3, E 3 ) = 1, g(e i, E j ) = 0 and nonzero covariant derivatives (which satisfy the Proposition): E1 E 2 = 3 4 E 3, E1 E 3 = 3 4 E 2, E2 E 3 = 5 4 E 1, E2 E 1 = 5 4 E 3, E3 E 1 = 3 4 E 2, E3 E 2 = 3 4 E 1. We will perform all calculations in this moving frame, or with respect to the corresponding pseudo-orthonormal basis B = {E 1 (e), E 2 (e), E 3 (e)} of the tangent space T e M R 3.
17 Any x = (x 1, x 2, x 3 ) R 3 determines L X = x 1 E 1 + x 2 E 2 + x 3 E 3. Light-like vectors X T e M are x = (sin(ϕ), cos(ϕ), 1), x = (sin(ϕ), cos(ϕ)) S 1. For L X, it holds L X L X = 2 cos(ϕ)e sin(ϕ)e sin(ϕ) cos(ϕ)e 3, y(x) = (2 cos(ϕ), 3 2 sin(ϕ), 1 ) 2 sin(ϕ) cos(ϕ). We see that y(x) x. The projection t x is t x = y(x) 1 sin(ϕ) cos(ϕ) x = [ 2 = 2 1 ] ( ) 2 sin2 (ϕ) cos(ϕ), sin(ϕ), 0. t x x and t x x x t x defines the smooth vector field on S 1, which is nonzero everywhere. There is not any vector X T e G which satisfies Main Theorem.
18 References Dušek, Z.: The existence of homogeneous geodesics in homogeneous pseudo-riemannian and affine manifolds, J. Geom. Phys 60 (2010). Dušek, Z.: On the reparametrization of affine homogeneous geodesics, Differential Geometry, J.A. Álvarez López and E. García-Río (Eds.), World Scientific (2009), Dušek, Z., Kowalski, O., Vlášek, Z.: Homogeneous geodesics in homogeneous affine manifolds, Result. Math. 54 (2009),
The existence of homogeneous geodesics in homogeneous pseudo-riemannian manifolds
The existence of homogeneous geodesics in homogeneous pseudo-riemannian manifolds Zdeněk Dušek Palacky University, Olomouc Porto, 2010 Contents Homogeneous geodesics in homogeneous affine manifolds General
More informationOn homogeneous Randers spaces with Douglas or naturally reductive metrics
On homogeneous Randers spaces with Douglas or naturally reductive metrics Mansour Aghasi and Mehri Nasehi Abstract. In [4] Božek has introduced a class of solvable Lie groups with arbitrary odd dimension.
More informationA local characterization for constant curvature metrics in 2-dimensional Lorentz manifolds
A local characterization for constant curvature metrics in -dimensional Lorentz manifolds Ivo Terek Couto Alexandre Lymberopoulos August 9, 8 arxiv:65.7573v [math.dg] 4 May 6 Abstract In this paper we
More informationLECTURE 10: THE PARALLEL TRANSPORT
LECTURE 10: THE PARALLEL TRANSPORT 1. The parallel transport We shall start with the geometric meaning of linear connections. Suppose M is a smooth manifold with a linear connection. Let γ : [a, b] M be
More informationMath 6455 Nov 1, Differential Geometry I Fall 2006, Georgia Tech
Math 6455 Nov 1, 26 1 Differential Geometry I Fall 26, Georgia Tech Lecture Notes 14 Connections Suppose that we have a vector field X on a Riemannian manifold M. How can we measure how much X is changing
More informationИзвестия НАН Армении. Математика, том 46, н. 1, 2011, стр HOMOGENEOUS GEODESICS AND THE CRITICAL POINTS OF THE RESTRICTED FINSLER FUNCTION
Известия НАН Армении. Математика, том 46, н. 1, 2011, стр. 75-82. HOMOENEOUS EODESICS AND THE CRITICAL POINTS OF THE RESTRICTED FINSLER FUNCTION PARASTOO HABIBI, DARIUSH LATIFI, MEERDICH TOOMANIAN Islamic
More informationKilling fields of constant length on homogeneous Riemannian manifolds
Killing fields of constant length on homogeneous Riemannian manifolds Southern Mathematical Institute VSC RAS Poland, Bedlewo, 21 October 2015 1 Introduction 2 3 4 Introduction Killing vector fields (simply
More informationGEODESIC VECTORS OF THE SIX-DIMENSIONAL SPACES
Steps in Differential Geometry, Proceedings of the Colloquium on Differential Geometry, 25 30 July, 2000, Debrecen, Hungary GEODESIC VECTORS OF THE SIX-DIMENSIONAL SPACES SZILVIA HOMOLYA Abstract. The
More informationCurved Spacetime I. Dr. Naylor
Curved Spacetime I Dr. Naylor Last Week Einstein's principle of equivalence We discussed how in the frame of reference of a freely falling object we can construct a locally inertial frame (LIF) Space tells
More informationOn constant isotropic submanifold by generalized null cubic
On constant isotropic submanifold by generalized null cubic Leyla Onat Abstract. In this paper we shall be concerned with curves in an Lorentzian submanifold M 1, and give a characterization of each constant
More informationMinimal timelike surfaces in a certain homogeneous Lorentzian 3-manifold II
Minimal timelike surfaces in a certain homogeneous Lorentzian 3-manifold II Sungwook Lee Abstract The 2-parameter family of certain homogeneous Lorentzian 3-manifolds which includes Minkowski 3-space and
More informationHomogeneous Lorentzian structures on the generalized Heisenberg group
Homogeneous Lorentzian structures on the generalized Heisenberg group W. Batat and S. Rahmani Abstract. In [8], all the homogeneous Riemannian structures corresponding to the left-invariant Riemannian
More informationUNIVERSITY OF DUBLIN
UNIVERSITY OF DUBLIN TRINITY COLLEGE JS & SS Mathematics SS Theoretical Physics SS TSM Mathematics Faculty of Engineering, Mathematics and Science school of mathematics Trinity Term 2015 Module MA3429
More informationA brief introduction to Semi-Riemannian geometry and general relativity. Hans Ringström
A brief introduction to Semi-Riemannian geometry and general relativity Hans Ringström May 5, 2015 2 Contents 1 Scalar product spaces 1 1.1 Scalar products...................................... 1 1.2 Orthonormal
More informationCurvature-homogeneous spaces of type (1,3)
Curvature-homogeneous spaces of type (1,3) Oldřich Kowalski (Charles University, Prague), joint work with Alena Vanžurová (Palacky University, Olomouc) Zlatibor, September 3-8, 2012 Curvature homogeneity
More informationBrownian Motion and lorentzian manifolds
The case of Jürgen Angst Institut de Recherche Mathématique Avancée Université Louis Pasteur, Strasbourg École d été de Probabilités de Saint-Flour juillet 2008, Saint-Flour 1 Construction of the diffusion,
More informationA CONSTRUCTION OF TRANSVERSE SUBMANIFOLDS
UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XLI 2003 A CONSTRUCTION OF TRANSVERSE SUBMANIFOLDS by J. Szenthe Abstract. In case of Riemannian manifolds isometric actions admitting submanifolds
More informationChoice of Riemannian Metrics for Rigid Body Kinematics
Choice of Riemannian Metrics for Rigid Body Kinematics Miloš Žefran1, Vijay Kumar 1 and Christopher Croke 2 1 General Robotics and Active Sensory Perception (GRASP) Laboratory 2 Department of Mathematics
More informationDIFFERENTIAL GEOMETRY, LECTURE 16-17, JULY 14-17
DIFFERENTIAL GEOMETRY, LECTURE 16-17, JULY 14-17 6. Geodesics A parametrized line γ : [a, b] R n in R n is straight (and the parametrization is uniform) if the vector γ (t) does not depend on t. Thus,
More informationARITHMETICITY OF TOTALLY GEODESIC LIE FOLIATIONS WITH LOCALLY SYMMETRIC LEAVES
ASIAN J. MATH. c 2008 International Press Vol. 12, No. 3, pp. 289 298, September 2008 002 ARITHMETICITY OF TOTALLY GEODESIC LIE FOLIATIONS WITH LOCALLY SYMMETRIC LEAVES RAUL QUIROGA-BARRANCO Abstract.
More informationOn a new class of infinitesimal group actions on pseudo-riemannian manifolds
On a new class of infinitesimal group actions on pseudo-riemannian manifolds Sigbjørn Hervik arxiv:1805.09402v1 [math-ph] 23 May 2018 Faculty of Science and Technology, University of Stavanger, N-4036
More informationIntroduction to Algebraic and Geometric Topology Week 14
Introduction to Algebraic and Geometric Topology Week 14 Domingo Toledo University of Utah Fall 2016 Computations in coordinates I Recall smooth surface S = {f (x, y, z) =0} R 3, I rf 6= 0 on S, I Chart
More informationExact Solutions of the Einstein Equations
Notes from phz 6607, Special and General Relativity University of Florida, Fall 2004, Detweiler Exact Solutions of the Einstein Equations These notes are not a substitute in any manner for class lectures.
More informationON SUBMAXIMAL DIMENSION OF THE GROUP OF ALMOST ISOMETRIES OF FINSLER METRICS.
ON SUBMAXIMAL DIMENSION OF THE GROUP OF ALMOST ISOMETRIES OF FINSLER METRICS. VLADIMIR S. MATVEEV Abstract. We show that the second greatest possible dimension of the group of (local) almost isometries
More informationINTRO TO SUBRIEMANNIAN GEOMETRY
INTRO TO SUBRIEMANNIAN GEOMETRY 1. Introduction to subriemannian geometry A lot of this tal is inspired by the paper by Ines Kath and Oliver Ungermann on the arxiv, see [3] as well as [1]. Let M be a smooth
More informationArchivum Mathematicum
Archivum Mathematicum Zdeněk Dušek; Oldřich Kowalski How many are affine connections with torsion Archivum Mathematicum, Vol. 50 (2014), No. 5, 257 264 Persistent URL: http://dml.cz/dmlcz/144068 Terms
More informationON OSCULATING, NORMAL AND RECTIFYING BI-NULL CURVES IN R 5 2
Novi Sad J. Math. Vol. 48, No. 1, 2018, 9-20 https://doi.org/10.30755/nsjom.05268 ON OSCULATING, NORMAL AND RECTIFYING BI-NULL CURVES IN R 5 2 Kazım İlarslan 1, Makoto Sakaki 2 and Ali Uçum 34 Abstract.
More information1-TYPE AND BIHARMONIC FRENET CURVES IN LORENTZIAN 3-SPACE *
Iranian Journal of Science & Technology, Transaction A, ol., No. A Printed in the Islamic Republic of Iran, 009 Shiraz University -TYPE AND BIHARMONIC FRENET CURES IN LORENTZIAN -SPACE * H. KOCAYIGIT **
More informationThe local geometry of compact homogeneous Lorentz spaces
The local geometry of compact homogeneous Lorentz spaces Felix Günther Abstract In 1995, S. Adams and G. Stuck as well as A. Zeghib independently provided a classification of non-compact Lie groups which
More informationOBSTRUCTION TO POSITIVE CURVATURE ON HOMOGENEOUS BUNDLES
OBSTRUCTION TO POSITIVE CURVATURE ON HOMOGENEOUS BUNDLES KRISTOPHER TAPP Abstract. Examples of almost-positively and quasi-positively curved spaces of the form M = H\((G, h) F ) were discovered recently
More informationTotally quasi-umbilic timelike surfaces in R 1,2
Totally quasi-umbilic timelike surfaces in R 1,2 Jeanne N. Clelland, University of Colorado AMS Central Section Meeting Macalester College April 11, 2010 Definition: Three-dimensional Minkowski space R
More informationMathematical Relativity, Spring 2017/18 Instituto Superior Técnico
Mathematical Relativity, Spring 2017/18 Instituto Superior Técnico 1. Starting from R αβµν Z ν = 2 [α β] Z µ, deduce the components of the Riemann curvature tensor in terms of the Christoffel symbols.
More informationAn introduction to General Relativity and the positive mass theorem
An introduction to General Relativity and the positive mass theorem National Center for Theoretical Sciences, Mathematics Division March 2 nd, 2007 Wen-ling Huang Department of Mathematics University of
More informationResearch Article New Examples of Einstein Metrics in Dimension Four
International Mathematics and Mathematical Sciences Volume 2010, Article ID 716035, 9 pages doi:10.1155/2010/716035 Research Article New Examples of Einstein Metrics in Dimension Four Ryad Ghanam Department
More informationCOMMUTATIVE CURVATURE OPERATORS OVER FOUR-DIMENSIONAL GENERALIZED SYMMETRIC SPACES
Sah Communications in Mathematical Analysis (SCMA) Vol. 1 No. 2 (2014), 77-90. http://scma.maragheh.ac.ir COMMUTATIVE CURVATURE OPERATORS OVER FOUR-DIMENSIONAL GENERALIZED SYMMETRIC SPACES ALI HAJI-BADALI
More informationHolomorphic Geodesic Transformations. of Almost Hermitian Manifold
International Mathematical Forum, 4, 2009, no. 46, 2293-2299 Holomorphic Geodesic Transformations of Almost Hermitian Manifold Habeeb M. Abood University of Basrha, Department of mathematics, Basrah-Iraq
More informationHadamard s Theorem. Rich Schwartz. September 10, The purpose of these notes is to prove the following theorem.
Hadamard s Theorem Rich Schwartz September 10, 013 1 The Result and Proof Outline The purpose of these notes is to prove the following theorem. Theorem 1.1 (Hadamard) Let M 1 and M be simply connected,
More information5.2 The Levi-Civita Connection on Surfaces. 1 Parallel transport of vector fields on a surface M
5.2 The Levi-Civita Connection on Surfaces In this section, we define the parallel transport of vector fields on a surface M, and then we introduce the concept of the Levi-Civita connection, which is also
More informationRigidity of Black Holes
Rigidity of Black Holes Sergiu Klainerman Princeton University February 24, 2011 Rigidity of Black Holes PREAMBLES I, II PREAMBLE I General setting Assume S B two different connected, open, domains and
More informationINTRODUCTION TO GEOMETRY
INTRODUCTION TO GEOMETRY ERIKA DUNN-WEISS Abstract. This paper is an introduction to Riemannian and semi-riemannian manifolds of constant sectional curvature. We will introduce the concepts of moving frames,
More informationLECTURE 9: MOVING FRAMES IN THE NONHOMOGENOUS CASE: FRAME BUNDLES. 1. Introduction
LECTURE 9: MOVING FRAMES IN THE NONHOMOGENOUS CASE: FRAME BUNDLES 1. Introduction Until now we have been considering homogenous spaces G/H where G is a Lie group and H is a closed subgroup. The natural
More informationGravitational Waves: Just Plane Symmetry
Utah State University DigitalCommons@USU All Physics Faculty Publications Physics 2006 Gravitational Waves: Just Plane Symmetry Charles G. Torre Utah State University Follow this and additional works at:
More informationExercises in Geometry II University of Bonn, Summer semester 2015 Professor: Prof. Christian Blohmann Assistant: Saskia Voss Sheet 1
Assistant: Saskia Voss Sheet 1 1. Conformal change of Riemannian metrics [3 points] Let (M, g) be a Riemannian manifold. A conformal change is a nonnegative function λ : M (0, ). Such a function defines
More informationLagrangian Submanifolds with Constant Angle Functions in the Nearly Kähler S 3 S 3
Lagrangian Submanifolds with Constant Angle Functions in the Nearly Kähler S 3 S 3 Burcu Bektaş Istanbul Technical University, Istanbul, Turkey Joint work with Marilena Moruz (Université de Valenciennes,
More informationarxiv: v1 [math.dg] 2 Oct 2015
An estimate for the Singer invariant via the Jet Isomorphism Theorem Tillmann Jentsch October 5, 015 arxiv:1510.00631v1 [math.dg] Oct 015 Abstract Recently examples of Riemannian homogeneous spaces with
More informationESSENTIAL KILLING FIELDS OF PARABOLIC GEOMETRIES: PROJECTIVE AND CONFORMAL STRUCTURES. 1. Introduction
ESSENTIAL KILLING FIELDS OF PARABOLIC GEOMETRIES: PROJECTIVE AND CONFORMAL STRUCTURES ANDREAS ČAP AND KARIN MELNICK Abstract. We use the general theory developed in our article [1] in the setting of parabolic
More informationObserver dependent background geometries arxiv:
Observer dependent background geometries arxiv:1403.4005 Manuel Hohmann Laboratory of Theoretical Physics Physics Institute University of Tartu DPG-Tagung Berlin Session MP 4 18. März 2014 Manuel Hohmann
More informationQuadratic forms. Here. Thus symmetric matrices are diagonalizable, and the diagonalization can be performed by means of an orthogonal matrix.
Quadratic forms 1. Symmetric matrices An n n matrix (a ij ) n ij=1 with entries on R is called symmetric if A T, that is, if a ij = a ji for all 1 i, j n. We denote by S n (R) the set of all n n symmetric
More information3 Parallel transport and geodesics
3 Parallel transport and geodesics 3.1 Differentiation along a curve As a prelude to parallel transport we consider another form of differentiation: differentiation along a curve. A curve is a parametrized
More informationSELECTED SAMPLE FINAL EXAM SOLUTIONS - MATH 5378, SPRING 2013
SELECTED SAMPLE FINAL EXAM SOLUTIONS - MATH 5378, SPRING 03 Problem (). This problem is perhaps too hard for an actual exam, but very instructional, and simpler problems using these ideas will be on the
More informationWICK ROTATIONS AND HOLOMORPHIC RIEMANNIAN GEOMETRY
WICK ROTATIONS AND HOLOMORPHIC RIEMANNIAN GEOMETRY Geometry and Lie Theory, Eldar Strøme 70th birthday Sigbjørn Hervik, University of Stavanger Work sponsored by the RCN! (Toppforsk-Fellesløftet) REFERENCES
More informationClassifications of Special Curves in the Three-Dimensional Lie Group
International Journal of Mathematical Analysis Vol. 10, 2016, no. 11, 503-514 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2016.6230 Classifications of Special Curves in the Three-Dimensional
More informationModuli spaces of Type A geometries EGEO 2016 La Falda, Argentina. Peter B Gilkey
EGEO 2016 La Falda, Argentina Mathematics Department, University of Oregon, Eugene OR USA email: gilkey@uoregon.edu a Joint work with M. Brozos-Vázquez, E. García-Río, and J.H. Park a Partially supported
More informationChanging sign solutions for the CR-Yamabe equation
Changing sign solutions for the CR-Yamabe equation Ali Maalaoui (1) & Vittorio Martino (2) Abstract In this paper we prove that the CR-Yamabe equation on the Heisenberg group has infinitely many changing
More informationAsymptotic Behavior of Marginally Trapped Tubes
Asymptotic Behavior of Marginally Trapped Tubes Catherine Williams January 29, 2009 Preliminaries general relativity General relativity says that spacetime is described by a Lorentzian 4-manifold (M, g)
More informationCurvature homogeneity of type (1, 3) in pseudo-riemannian manifolds
Curvature homogeneity of type (1, 3) in pseudo-riemannian manifolds Cullen McDonald August, 013 Abstract We construct two new families of pseudo-riemannian manifolds which are curvature homegeneous of
More informationSYMMETRIES OF SECTIONAL CURVATURE ON 3 MANIFOLDS. May 27, 1992
SYMMETRIES OF SECTIONAL CURVATURE ON 3 MANIFOLDS Luis A. Cordero 1 Phillip E. Parker,3 Dept. Xeometría e Topoloxía Facultade de Matemáticas Universidade de Santiago 15706 Santiago de Compostela Spain cordero@zmat.usc.es
More informationAn Optimal Control Problem for Rigid Body Motions in Minkowski Space
Applied Mathematical Sciences, Vol. 5, 011, no. 5, 559-569 An Optimal Control Problem for Rigid Body Motions in Minkowski Space Nemat Abazari Department of Mathematics, Ardabil Branch Islamic Azad University,
More informationKilling Vector Fields of Constant Length on Riemannian Normal Homogeneous Spaces
Killing Vector Fields of Constant Length on Riemannian Normal Homogeneous Spaces Ming Xu & Joseph A. Wolf Abstract Killing vector fields of constant length correspond to isometries of constant displacement.
More informationRIGIDITY OF MINIMAL ISOMETRIC IMMERSIONS OF SPHERES INTO SPHERES. Christine M. Escher Oregon State University. September 10, 1997
RIGIDITY OF MINIMAL ISOMETRIC IMMERSIONS OF SPHERES INTO SPHERES Christine M. Escher Oregon State University September, 1997 Abstract. We show two specific uniqueness properties of a fixed minimal isometric
More informationUNIQUENESS OF STATIC BLACK-HOLES WITHOUT ANALYTICITY. Piotr T. Chruściel & Gregory J. Galloway
UNIQUENESS OF STATIC BLACK-HOLES WITHOUT ANALYTICITY by Piotr T. Chruściel & Gregory J. Galloway Abstract. We show that the hypothesis of analyticity in the uniqueness theory of vacuum, or electrovacuum,
More informationCALCULUS ON MANIFOLDS. 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M =
CALCULUS ON MANIFOLDS 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M = a M T am, called the tangent bundle, is itself a smooth manifold, dim T M = 2n. Example 1.
More informationDIFFERENTIAL GEOMETRY HW 7
DIFFERENTIAL GEOMETRY HW 7 CLAY SHONKWILER 1 Show that within a local coordinate system x 1,..., x n ) on M with coordinate vector fields X 1 / x 1,..., X n / x n, if we pick n 3 smooth real-valued functions
More informationGENERALIZED NULL SCROLLS IN THE n-dimensional LORENTZIAN SPACE. 1. Introduction
ACTA MATHEMATICA VIETNAMICA 205 Volume 29, Number 2, 2004, pp. 205-216 GENERALIZED NULL SCROLLS IN THE n-dimensional LORENTZIAN SPACE HANDAN BALGETIR AND MAHMUT ERGÜT Abstract. In this paper, we define
More informationL 2 Geometry of the Symplectomorphism Group
University of Notre Dame Workshop on Innite Dimensional Geometry, Vienna 2015 Outline 1 The Exponential Map on D s ω(m) 2 Existence of Multiplicity of Outline 1 The Exponential Map on D s ω(m) 2 Existence
More informationNotes on the Riemannian Geometry of Lie Groups
Rose- Hulman Undergraduate Mathematics Journal Notes on the Riemannian Geometry of Lie Groups Michael L. Geis a Volume, Sponsored by Rose-Hulman Institute of Technology Department of Mathematics Terre
More informationTHEODORE VORONOV DIFFERENTIAL GEOMETRY. Spring 2009
[under construction] 8 Parallel transport 8.1 Equation of parallel transport Consider a vector bundle E B. We would like to compare vectors belonging to fibers over different points. Recall that this was
More informationThe Commutation Property of a Stationary, Axisymmetric System
Commun. math. Phys. 17, 233 238 (1970) The Commutation Property of a Stationary, Axisymmetric System BRANDON CARTER Institute of Theoretical Astronomy, Cambridge, England Received December 12, 1969 Abstract.
More informationDIFFERENTIAL GEOMETRY HW 4. Show that a catenoid and helicoid are locally isometric.
DIFFERENTIAL GEOMETRY HW 4 CLAY SHONKWILER Show that a catenoid and helicoid are locally isometric. 3 Proof. Let X(u, v) = (a cosh v cos u, a cosh v sin u, av) be the parametrization of the catenoid and
More informationOn twisted Riemannian extensions associated with Szabó metrics
Hacettepe Journal of Mathematics and Statistics Volume 46 (4) (017), 593 601 On twisted Riemannian extensions associated with Szabó metrics Abdoul Salam Diallo, Silas Longwap and Fortuné Massamba Ÿ Abstract
More informationDifferential Topology Solution Set #3
Differential Topology Solution Set #3 Select Solutions 1. Chapter 1, Section 4, #7 2. Chapter 1, Section 4, #8 3. Chapter 1, Section 4, #11(a)-(b) #11(a) The n n matrices with determinant 1 form a group
More informationTHREE-MANIFOLDS OF CONSTANT VECTOR CURVATURE ONE
THREE-MANIFOLDS OF CONSTANT VECTOR CURVATURE ONE BENJAMIN SCHMIDT AND JON WOLFSON ABSTRACT. A Riemannian manifold has CVC(ɛ) if its sectional curvatures satisfy sec ε or sec ε pointwise, and if every tangent
More informationSolutions for Math 348 Assignment #4 1
Solutions for Math 348 Assignment #4 1 (1) Do the following: (a) Show that the intersection of two spheres S 1 = {(x, y, z) : (x x 1 ) 2 + (y y 1 ) 2 + (z z 1 ) 2 = r 2 1} S 2 = {(x, y, z) : (x x 2 ) 2
More informationActa Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica Alena Vanžurová On Metrizable Locally Homogeneous Connections in Dimension Acta Universitatis Palackianae Olomucensis.
More informationSTRUCTURE OF GEODESICS IN A 13-DIMENSIONAL GROUP OF HEISENBERG TYPE
Steps in Differential Geometry, Proceedings of the Colloquium on Differential Geometry, 25 30 July, 2000, Debrecen, Hungary STRUCTURE OF GEODESICS IN A 13-DIMENSIONAL GROUP OF HEISENBERG TYPE Abstract.
More informationReduction of Homogeneous Riemannian structures
Geometric Structures in Mathematical Physics, 2011 Reduction of Homogeneous Riemannian structures M. Castrillón López 1 Ignacio Luján 2 1 ICMAT (CSIC-UAM-UC3M-UCM) Universidad Complutense de Madrid 2 Universidad
More informationGravitation: Tensor Calculus
An Introduction to General Relativity Center for Relativistic Astrophysics School of Physics Georgia Institute of Technology Notes based on textbook: Spacetime and Geometry by S.M. Carroll Spring 2013
More informationHOMEWORK 2 - RIEMANNIAN GEOMETRY. 1. Problems In what follows (M, g) will always denote a Riemannian manifold with a Levi-Civita connection.
HOMEWORK 2 - RIEMANNIAN GEOMETRY ANDRÉ NEVES 1. Problems In what follows (M, g will always denote a Riemannian manifold with a Levi-Civita connection. 1 Let X, Y, Z be vector fields on M so that X(p Z(p
More informationMILNOR SEMINAR: DIFFERENTIAL FORMS AND CHERN CLASSES
MILNOR SEMINAR: DIFFERENTIAL FORMS AND CHERN CLASSES NILAY KUMAR In these lectures I want to introduce the Chern-Weil approach to characteristic classes on manifolds, and in particular, the Chern classes.
More information1 v >, which will be G-invariant by construction.
1. Riemannian symmetric spaces Definition 1.1. A (globally, Riemannian) symmetric space is a Riemannian manifold (X, g) such that for all x X, there exists an isometry s x Iso(X, g) such that s x (x) =
More informationLECTURE 16: CONJUGATE AND CUT POINTS
LECTURE 16: CONJUGATE AND CUT POINTS 1. Conjugate Points Let (M, g) be Riemannian and γ : [a, b] M a geodesic. Then by definition, exp p ((t a) γ(a)) = γ(t). We know that exp p is a diffeomorphism near
More informationzi z i, zi+1 z i,, zn z i. z j, zj+1 z j,, zj 1 z j,, zn
The Complex Projective Space Definition. Complex projective n-space, denoted by CP n, is defined to be the set of 1-dimensional complex-linear subspaces of C n+1, with the quotient topology inherited from
More informationGeometric inequalities for black holes
Geometric inequalities for black holes Sergio Dain FaMAF-Universidad Nacional de Córdoba, CONICET, Argentina. 3 August, 2012 Einstein equations (vacuum) The spacetime is a four dimensional manifold M with
More informationCOHOMOGENEITY ONE HYPERSURFACES OF EUCLIDEAN SPACES
COHOMOGENEITY ONE HYPERSURFACES OF EUCLIDEAN SPACES FRANCESCO MERCURI, FABIO PODESTÀ, JOSÉ A. P. SEIXAS AND RUY TOJEIRO Abstract. We study isometric immersions f : M n R n+1 into Euclidean space of dimension
More informationDIFFERENTIAL GEOMETRY OF CURVES AND SURFACES IN LORENTZ-MINKOWSKI SPACE
International Electronic Journal of Geometry Volume 7 No. 1 pp. 44-107 (014) c IEJG DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES IN LORENTZ-MINKOWSKI SPACE RAFAEL LÓPEZ Dedicated to memory of Proffessor
More informationis constant [3]. In a recent work, T. IKAWA proved the following theorem for helices on a Lorentzian submanifold [1].
ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII AL.I.CUZA IAŞI Tomul XLVI, s.i a, Matematică, 2000, f.2. ON GENERAL HELICES AND SUBMANIFOLDS OF AN INDEFINITE RIEMANNIAN MANIFOLD BY N. EKMEKCI Introduction. A regular
More informationTHE INITIAL VALUE FORMULATION OF GENERAL RELATIVITY
THE INITIAL VALUE FORMULATION OF GENERAL RELATIVITY SAM KAUFMAN Abstract. The (Cauchy) initial value formulation of General Relativity is developed, and the maximal vacuum Cauchy development theorem is
More information1. Geometry of the unit tangent bundle
1 1. Geometry of the unit tangent bundle The main reference for this section is [8]. In the following, we consider (M, g) an n-dimensional smooth manifold endowed with a Riemannian metric g. 1.1. Notations
More informationIsometries, Local Isometries, Riemannian Coverings and Submersions, Killing Vector Fields
Chapter 16 Isometries, Local Isometries, Riemannian Coverings and Submersions, Killing Vector Fields 16.1 Isometries and Local Isometries Recall that a local isometry between two Riemannian manifolds M
More informationLECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI. 1. Maximal Tori
LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI 1. Maximal Tori By a torus we mean a compact connected abelian Lie group, so a torus is a Lie group that is isomorphic to T n = R n /Z n. Definition 1.1.
More informationSpacelike surfaces with positive definite second fundamental form in 3-dimensional Lorentzian manifolds
Spacelike surfaces with positive definite second fundamental form in 3-dimensional Lorentzian manifolds Alfonso Romero Departamento de Geometría y Topología Universidad de Granada 18071-Granada Web: http://www.ugr.es/
More informationarxiv: v2 [gr-qc] 25 Apr 2016
The C 0 -inextendibility of the Schwarzschild spacetime and the spacelike diameter in Lorentzian geometry Jan Sbierski April 26, 2016 arxiv:1507.00601v2 [gr-qc] 25 Apr 2016 Abstract The maximal analytic
More informationLECTURE 15: COMPLETENESS AND CONVEXITY
LECTURE 15: COMPLETENESS AND CONVEXITY 1. The Hopf-Rinow Theorem Recall that a Riemannian manifold (M, g) is called geodesically complete if the maximal defining interval of any geodesic is R. On the other
More informationStability and Instability of Black Holes
Stability and Instability of Black Holes Stefanos Aretakis September 24, 2013 General relativity is a successful theory of gravitation. Objects of study: (4-dimensional) Lorentzian manifolds (M, g) which
More informationFisica Matematica. Stefano Ansoldi. Dipartimento di Matematica e Informatica. Università degli Studi di Udine. Corso di Laurea in Matematica
Fisica Matematica Stefano Ansoldi Dipartimento di Matematica e Informatica Università degli Studi di Udine Corso di Laurea in Matematica Anno Accademico 2003/2004 c 2004 Copyright by Stefano Ansoldi and
More information5 Constructions of connections
[under construction] 5 Constructions of connections 5.1 Connections on manifolds and the Levi-Civita theorem We start with a bit of terminology. A connection on the tangent bundle T M M of a manifold M
More informationMATH DIFFERENTIAL GEOMETRY. Contents
MATH 3968 - DIFFERENTIAL GEOMETRY ANDREW TULLOCH Contents 1. Curves in R N 2 2. General Analysis 2 3. Surfaces in R 3 3 3.1. The Gauss Bonnet Theorem 8 4. Abstract Manifolds 9 1 MATH 3968 - DIFFERENTIAL
More informationA connection between Lorentzian distance and mechanical least action
A connection between Lorentzian distance and mechanical least action Ettore Minguzzi Università Degli Studi Di Firenze Non-commutative structures and non-relativistic (super)symmetries, LMPT Tours, June
More informationIsometries, Local Isometries, Riemannian Coverings and Submersions, Killing Vector Fields
Chapter 15 Isometries, Local Isometries, Riemannian Coverings and Submersions, Killing Vector Fields The goal of this chapter is to understand the behavior of isometries and local isometries, in particular
More informationThe uniformly accelerated motion in General Relativity from a geometric point of view. 1. Introduction. Daniel de la Fuente
XI Encuentro Andaluz de Geometría IMUS (Universidad de Sevilla), 15 de mayo de 2015, págs. 2934 The uniformly accelerated motion in General Relativity from a geometric point of view Daniel de la Fuente
More information