33 JINWEI ZHOU acts on the left of u =(u ::: u N ) 2 F N Let G(N) be the group acting on the right of F N which preserving the standard inner product
|
|
- Magdalene Ball
- 5 years ago
- Views:
Transcription
1 SOOHOW JOURNL OF MTHEMTIS Volume 24, No 4, pp , October 998 THE GEODESIS IN GRSSMNN MNIFOLDS Y JINWEI ZHOU bstract With a suitable basis of Euclidean space, every geodesic in Grassmann manifold can be expressed by (t) = [cos( t)e +sin( t)e n+ ::: cos( nt)e n +sin( nt)e 2n] where 2 n and N;n+ = = n =ifn <2n Introduction Let G F (n N) be the Grassmann manifold formed by all n-subspaces in F N, where F is the set of real numbers, complex numbers or quaternions The manifold G F (n N) is a symmetric space (see [2] or [3]) In this short paper we study the geodesics in Grassmann manifolds We show that every geodesic in Grassmann manifolds lies in a at totally geodesic submanifold which isisometric to a torus In this way, we can write geodesics in Grassmann manifolds explicitly s [4], the results of this paper can be used to study the dierential geometry of Grassmann manifolds s a byproduct, we nd a simple method of the diagonalization of the F -matrices 2 The Geodesics in G F (n N) Let F be the set of real number R, complex numbers or quaternions H For any u 2 F, u is the conjugation of u(u v = v u if u v 2 H) For any 2 F, Received March, 998 MS Subject lassication 5322 Key words Grassmann manifold, geodesic, moving frame 329
2 33 JINWEI ZHOU acts on the left of u =(u ::: u N ) 2 F N Let G(N) be the group acting on the right of F N which preserving the standard inner product (,) on F N If F = R, G(N) =O(N) is the orthogonal group if F =, G(N) =U(N) is the complex unitary group if F = H, G(N) =S p (N) is the symplectic group G(N) G(n)G(N ;n) Let G F (n N) = be the Grassmann manifold formed by all n-subspaces in F N Denote = [e ::: e n ] if 2 G F (n N) is generated by e ::: e n Let e ::: e N be orthonormal frame elds on F N y the method of moving frame, there are local -forms! dened by de = X! e! +! = = ::: N: Restricting the two form = Re( P! i! i ) on G F (n N) denes a Riemannian metric (see []) s is well known, the tangent spacet G F (n N) can be realized as the space of all F valued n (N ; n) matrices Let E i (i = ::: n = n + ::: N) be basis of T G F (n N) dened as follows a curve in G F (n N) Let (t) = [e (t) ::: e n (t)] be omplete e (t) ::: e n (t) to orthonormal frame elds e (t) ::: e n (t) e n+ (t) :::, e N (t) Set a i =( de i dt e ) t= Then the tangent vector of (t) at = () is X = X a i E i : The tanget vector X can also be represented by ann (N ; n) matrix =(a i ) If e ::: e n e n+ ::: e N e e n = is another frame with e e n e n+ e N = D e n+ where e = e () and 2 G(n), D 2 G(N ; n) Then the tangent vector X with this new frame can be represented by t D The tangentspacet G (n N) of complex Grassmann manifold is a complex space of dimension n(n ; n) and G (n N) is a kaehler manifold ut there is no naturally dened quanternion structure on the tangent space T G H (n N) of quaternion Grassmann manifold (in general, t D 6= t D, for 2 H) The following lemma follows from Lemma 22 e N
3 THE GEODESIS IN GRSSMNN MNIFOLDS 33 Lemma 2 For any X 2 T G F (n N), there exists an orthonormal basis e ::: e n ::: e N on F N, such that =[e ::: e n ] and X = nx i E i n+i i= where E i is determined by the frame e ::: e N and 2 n, N ;n+ = = n =if N < 2n Lemma 22 For any F valued n (N ; n) matrix, there are elements g 2 G(n) and g 2 2 G(N ; n) such that g g 2 = g g 2 = N ;n n where n are real numbers if N < 2n if N 2n Proof In the following we assume that N 2n The case of N < 2n can be proved similarly Let f(u) =(u u) = u t u t for any u 2 F n with (u u) = f is a non-negative function and bounded on F n \ S rn;, where r is the real dimension of F ssuming that = f(u ) is the maximum of f for some u 2 F n \ S rn; Then for any u 2 F n \ S rn; with (u u ) =, we have (u u )+(u u )= Then the real part of (u u ) is zero Replace u by u, 2 F, jj =,we can show that (u u )=: Restricting f on fu 2 F n j(u u )=, juj =g, we can get vector u 2 and 2 = f(u 2 ) In this way, we can obtain a G(n)-frame u ::: u n and some non-negative numbers i = f(u i ), n,(u i u j )=ifi 6= j
4 332 JINWEI ZHOU From (u i u j )= i ij,weknow that we canchoose v i 2 F N ;n such that u i = i v i (v i v j )= ij i i = ::: n: omplete v ::: v n to a G(N ;n)-frame v ::: v n ::: v N ;n on F N ;n We have proved that u v = : n Denote g = u u n u n proof of the lemma 2 G(n) andgt 2 = v v N ;n The following proposition can be proved similarly v N ;n 2 G(N ; n) This completes the Proposition 23 Let be a( )-symmetric matrix of order n, thatis t = Then there is a g 2 G(n) such that gg t = where 2 n are real numbers s is well known, the tangentvectors E i n+i (i minfn N ;ng)oft G F (n N) determine a totally geodesic submanifold in G F (n N) which is isometric to n G F ( 2) G F ( 2): ny geodesic with initial tangent vector X = P i E i n+i i 2 R, lies in a at torus of this submanifold Theorem 24 For any geodesic in G F (n N), there is an orthonormal basis e ::: e N on F N such that can be represented by (t) = [cos( t)e +sin( t)e n+ ::: cos( n t)e n +sin( n t)e 2n )] where 2 n, N ;n+ = = n =if N < 2n
5 THE GEODESIS IN GRSSMNN MNIFOLDS 333 s G F ( N) is a projective space, from Theorem 24 we know that the geodesics in projective space are all closed Wong [4] showed that for any, 2 G F (n N), there is an orthonormal basis e ::: e N of F N and numbers ::: n such that and can be written as =[e ::: e n ] =[cos e +sin e n+ ::: cos n e n +sin n e 2n ] where 2 2 n y Theorem 24, the curve dened by (t) = [cos( t)e + sin( t)e n+ ::: cos( n t)e n +sin( n t)e 2n ] is a minimal geodesic joining and Note that, for any real numbers ::: n, if there are integers k ::: k n and real number t such that i t = k i + i Then (t) = [cos( t)e +sin( t)e n+ ::: cos( n t)e n + sin( n t)e 2n ] is also a geodesic through the points, with length qx (ki + i ) 2 : orollary 25 For any, 2 G F (n N), n 2, N 4, therearecountably many geodesics passing through and The geodesics in oriented real Grassmann manifold can be studied similarly The method can also been used to study the geodesics in some homogeneous spaces References [] S S hern, omplex manifolds without potential theory, Springer-Verlag, New York, 979 [2] S Helgason, Dierential geometry, Lie groups, and symmetric spaces, cademic Press, New York, 978 [3] S Kabayashi and K Nomizu, Foundations of dierential geometry, vol 2, Interscience Publishers, New York, 969 [4] Yung-how Wong, Dierential geometry of Grassmann manifolds, Proc Nat cad Sci US, 57 (967), Department of Mathematics, Suzhou University, Suzhou 256, hina
gz(t,,t2) = Re Tr[(I + ZZ*)-lTi(I + Z*Z)-1T2*]. (2) Gn(F"+m) In (2) and (3), we express the curvature tensor and the sectional
SECTIONAL CURVATURES OF GRASSMANN MANIFOLDS BY YUNG-CHOW WONG UNIVERSITY OF HONG KONG Communicated by S. S. Chern, March 6, 1968 (1) Introduction.-Let F be the field R of real numbers, the field C of complex
More informationBott Periodicity. Anthony Bosman Senior Honors Thesis Department of Mathematics, Stanford University Adviser: Eleny Ionel
Bott Periodicity Anthony Bosman Senior Honors Thesis Department of Mathematics, Stanford University Adviser: Eleny Ionel Acknowledgements This paper is being written as a Senior honors thesis. I m indebted
More informationMetric nilpotent Lie algebras of dimension 5
Metric nilpotent Lie algebras of dimension 5 Ágota Figula and Péter T. Nagy University of Debrecen, University of Óbuda 16-17.06.2017, GTG, University of Trento Preliminaries Denition Let g be a Lie algebra
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 informationSOME ASPECTS ON CIRCLES AND HELICES IN A COMPLEX PROJECTIVE SPACE. Toshiaki Adachi* and Sadahiro Maeda
Mem. Fac. Sci. Eng. Shimane Univ. Series B: Mathematical Science 32 (1999), pp. 1 8 SOME ASPECTS ON CIRCLES AND HELICES IN A COMPLEX PROJECTIVE SPACE Toshiaki Adachi* and Sadahiro Maeda (Received December
More informationQUATERNIONS AND ROTATIONS
QUATERNIONS AND ROTATIONS SVANTE JANSON 1. Introduction The purpose of this note is to show some well-known relations between quaternions and the Lie groups SO(3) and SO(4) (rotations in R 3 and R 4 )
More informationON THE CLASSIFICATION OF HOMOGENEOUS 2-SPHERES IN COMPLEX GRASSMANNIANS. Fei, Jie; Jiao, Xiaoxiang; Xiao, Liang; Xu, Xiaowei
Title Author(s) Citation ON THE CLASSIFICATION OF HOMOGENEOUS -SPHERES IN COMPLEX GRASSMANNIANS Fei, Jie; Jiao, Xiaoxiang; Xiao, Liang; Xu, Xiaowei Osaka Journal of Mathematics 5(1) P135-P15 Issue Date
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 informationStratification of 3 3 Matrices
Stratification of 3 3 Matrices Meesue Yoo & Clay Shonkwiler March 2, 2006 1 Warmup with 2 2 Matrices { Best matrices of rank 2} = O(2) S 3 ( 2) { Best matrices of rank 1} S 3 (1) 1.1 Viewing O(2) S 3 (
More informationTHE EULER CHARACTERISTIC OF A LIE GROUP
THE EULER CHARACTERISTIC OF A LIE GROUP JAY TAYLOR 1 Examples of Lie Groups The following is adapted from [2] We begin with the basic definition and some core examples Definition A Lie group is a smooth
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 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 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 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 informationAdjoint Orbits, Principal Components, and Neural Nets
Adjoint Orbits, Principal Components, and Neural Nets Some facts about Lie groups and examples Examples of adjoint orbits and a distance measure Descent equations on adjoint orbits Properties of the double
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 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 informationThe existence of light-like homogeneous geodesics in homogeneous Lorentzian manifolds. Sao Paulo, 2013
The existence of light-like homogeneous geodesics in homogeneous Lorentzian manifolds Zdeněk Dušek Sao Paulo, 2013 Motivation In a previous project, it was proved that any homogeneous affine manifold (and
More informationNON POSITIVELY CURVED METRIC IN THE SPACE OF POSITIVE DEFINITE INFINITE MATRICES
REVISTA DE LA UNIÓN MATEMÁTICA ARGENTINA Volumen 48, Número 1, 2007, Páginas 7 15 NON POSITIVELY CURVED METRIC IN THE SPACE OF POSITIVE DEFINITE INFINITE MATRICES ESTEBAN ANDRUCHOW AND ALEJANDRO VARELA
More informationClifford Algebras and Spin Groups
Clifford Algebras and Spin Groups Math G4344, Spring 2012 We ll now turn from the general theory to examine a specific class class of groups: the orthogonal groups. Recall that O(n, R) is the group of
More informationActa Mathematica Academiae Paedagogicae Nyíregyháziensis 21 (2005), ISSN
Acta Mathematica Academiae Paedagogicae Nyíregyháziensis 21 (2005), 79 7 www.emis.de/journals ISSN 176-0091 WARPED PRODUCT SUBMANIFOLDS IN GENERALIZED COMPLEX SPACE FORMS ADELA MIHAI Abstract. B.Y. Chen
More informationSome aspects of the Geodesic flow
Some aspects of the Geodesic flow Pablo C. Abstract This is a presentation for Yael s course on Symplectic Geometry. We discuss here the context in which the geodesic flow can be understood using techniques
More informationII. DIFFERENTIABLE MANIFOLDS. Washington Mio CENTER FOR APPLIED VISION AND IMAGING SCIENCES
II. DIFFERENTIABLE MANIFOLDS Washington Mio Anuj Srivastava and Xiuwen Liu (Illustrations by D. Badlyans) CENTER FOR APPLIED VISION AND IMAGING SCIENCES Florida State University WHY MANIFOLDS? Non-linearity
More informationHYPERKÄHLER MANIFOLDS
HYPERKÄHLER MANIFOLDS PAVEL SAFRONOV, TALK AT 2011 TALBOT WORKSHOP 1.1. Basic definitions. 1. Hyperkähler manifolds Definition. A hyperkähler manifold is a C Riemannian manifold together with three covariantly
More informationLinear Algebra using Dirac Notation: Pt. 2
Linear Algebra using Dirac Notation: Pt. 2 PHYS 476Q - Southern Illinois University February 6, 2018 PHYS 476Q - Southern Illinois University Linear Algebra using Dirac Notation: Pt. 2 February 6, 2018
More informationProblems in Linear Algebra and Representation Theory
Problems in Linear Algebra and Representation Theory (Most of these were provided by Victor Ginzburg) The problems appearing below have varying level of difficulty. They are not listed in any specific
More informationGeometric Modelling Summer 2016
Geometric Modelling Summer 2016 Exercises Benjamin Karer M.Sc. http://gfx.uni-kl.de/~gm Benjamin Karer M.Sc. Geometric Modelling Summer 2016 1 Dierential Geometry Benjamin Karer M.Sc. Geometric Modelling
More informationREPRESENTATION THEORY WEEK 7
REPRESENTATION THEORY WEEK 7 1. Characters of L k and S n A character of an irreducible representation of L k is a polynomial function constant on every conjugacy class. Since the set of diagonalizable
More informationHilbert Spaces. Contents
Hilbert Spaces Contents 1 Introducing Hilbert Spaces 1 1.1 Basic definitions........................... 1 1.2 Results about norms and inner products.............. 3 1.3 Banach and Hilbert spaces......................
More informationAn introduction to the geometry of homogeneous spaces
Proceedings of The Thirteenth International Workshop on Diff. Geom. 13(2009) 121-144 An introduction to the geometry of homogeneous spaces Takashi Koda Department of Mathematics, Faculty of Science, University
More informationGeometry of almost-product (pseudo-)riemannian manifold. manifolds and the dynamics of the observer. Aneta Wojnar
Geometry of almost-product (pseudo-)riemannian manifolds and the dynamics of the observer University of Wrocªaw Barcelona Postgrad Encounters on Fundamental Physics, October 2012 Outline 1 Motivation 2
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 informationDIFFERENTIAL GEOMETRY HW 12
DIFFERENTIAL GEOMETRY HW 1 CLAY SHONKWILER 3 Find the Lie algebra so(n) of the special orthogonal group SO(n), and the explicit formula for the Lie bracket there. Proof. Since SO(n) is a subgroup of GL(n),
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 informationREGULAR TRIPLETS IN COMPACT SYMMETRIC SPACES
REGULAR TRIPLETS IN COMPACT SYMMETRIC SPACES MAKIKO SUMI TANAKA 1. Introduction This article is based on the collaboration with Tadashi Nagano. In the first part of this article we briefly review basic
More informationThe Erwin Schrodinger International Boltzmanngasse 9. Institute for Mathematical Physics A-1090 Wien, Austria
ESI The Erwin Schrodinger International oltzmanngasse 9 Institute for Mathematical hysics -9 Wien, ustria Solutions of Finite Type of Sine{Gordon Equation Guosong Zhao Vienna, reprint ESI 485 997) September,
More informationHolonomy groups. Thomas Leistner. Mathematics Colloquium School of Mathematics and Physics The University of Queensland. October 31, 2011 May 28, 2012
Holonomy groups Thomas Leistner Mathematics Colloquium School of Mathematics and Physics The University of Queensland October 31, 2011 May 28, 2012 1/17 The notion of holonomy groups is based on Parallel
More informationSub-Riemannian geometry in groups of diffeomorphisms and shape spaces
Sub-Riemannian geometry in groups of diffeomorphisms and shape spaces Sylvain Arguillère, Emmanuel Trélat (Paris 6), Alain Trouvé (ENS Cachan), Laurent May 2013 Plan 1 Sub-Riemannian geometry 2 Right-invariant
More informationTwo-Step Nilpotent Lie Algebras Attached to Graphs
International Mathematical Forum, 4, 2009, no. 43, 2143-2148 Two-Step Nilpotent Lie Algebras Attached to Graphs Hamid-Reza Fanaï Department of Mathematical Sciences Sharif University of Technology P.O.
More informationLECTURE: KOBORDISMENTHEORIE, WINTER TERM 2011/12; SUMMARY AND LITERATURE
LECTURE: KOBORDISMENTHEORIE, WINTER TERM 2011/12; SUMMARY AND LITERATURE JOHANNES EBERT 1.1. October 11th. 1. Recapitulation from differential topology Definition 1.1. Let M m, N n, be two smooth manifolds
More informationHyperbolic Geometry on Geometric Surfaces
Mathematics Seminar, 15 September 2010 Outline Introduction Hyperbolic geometry Abstract surfaces The hemisphere model as a geometric surface The Poincaré disk model as a geometric surface Conclusion Introduction
More informationHeptagonal Triangles and Their Companions
Forum Geometricorum Volume 9 (009) 15 148. FRUM GEM ISSN 1534-1178 Heptagonal Triangles and Their ompanions Paul Yiu bstract. heptagonal triangle is a non-isosceles triangle formed by three vertices of
More information4 Riemannian geometry
Classnotes for Introduction to Differential Geometry. Matthias Kawski. April 18, 2003 83 4 Riemannian geometry 4.1 Introduction A natural first step towards a general concept of curvature is to develop
More informationCHAPTER 1 PRELIMINARIES
CHAPTER 1 PRELIMINARIES 1.1 Introduction The aim of this chapter is to give basic concepts, preliminary notions and some results which we shall use in the subsequent chapters of the thesis. 1.2 Differentiable
More informationNotes on Cartan s Method of Moving Frames
Math 553 σιι June 4, 996 Notes on Cartan s Method of Moving Frames Andrejs Treibergs The method of moving frames is a very efficient way to carry out computations on surfaces Chern s Notes give an elementary
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 informationCitation Osaka Journal of Mathematics. 49(3)
Title ON POSITIVE QUATERNIONIC KÄHLER MAN WITH b_4=1 Author(s) Kim, Jin Hong; Lee, Hee Kwon Citation Osaka Journal of Mathematics. 49(3) Issue 2012-09 Date Text Version publisher URL http://hdl.handle.net/11094/23146
More informationSYMPLECTIC MANIFOLDS, GEOMETRIC QUANTIZATION, AND UNITARY REPRESENTATIONS OF LIE GROUPS. 1. Introduction
SYMPLECTIC MANIFOLDS, GEOMETRIC QUANTIZATION, AND UNITARY REPRESENTATIONS OF LIE GROUPS CRAIG JACKSON 1. Introduction Generally speaking, geometric quantization is a scheme for associating Hilbert spaces
More informationRICCI CURVATURE OF SUBMANIFOLDS IN SASAKIAN SPACE FORMS
J. Austral. Math. Soc. 72 (2002), 27 256 RICCI CURVATURE OF SUBMANIFOLDS IN SASAKIAN SPACE FORMS ION MIHAI (Received 5 June 2000; revised 19 February 2001) Communicated by K. Wysocki Abstract Recently,
More informationMath 215B: Solutions 1
Math 15B: Solutions 1 Due Thursday, January 18, 018 (1) Let π : X X be a covering space. Let Φ be a smooth structure on X. Prove that there is a smooth structure Φ on X so that π : ( X, Φ) (X, Φ) is an
More informationSYMPLECTIC GEOMETRY: LECTURE 5
SYMPLECTIC GEOMETRY: LECTURE 5 LIAT KESSLER Let (M, ω) be a connected compact symplectic manifold, T a torus, T M M a Hamiltonian action of T on M, and Φ: M t the assoaciated moment map. Theorem 0.1 (The
More informationAbstract. Jacobi curves are far going generalizations of the spaces of \Jacobi
Principal Invariants of Jacobi Curves Andrei Agrachev 1 and Igor Zelenko 2 1 S.I.S.S.A., Via Beirut 2-4, 34013 Trieste, Italy and Steklov Mathematical Institute, ul. Gubkina 8, 117966 Moscow, Russia; email:
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 informationFinite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product
Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a pre-hilbert space, or a unitary space) if there is a mapping (, )
More informationJ. Korean Math. Soc. 32 (1995), No. 3, pp. 471{481 ON CHARACTERIZATIONS OF REAL HYPERSURFACES OF TYPE B IN A COMPLEX HYPERBOLIC SPACE Seong Soo Ahn an
J. Korean Math. Soc. 32 (1995), No. 3, pp. 471{481 ON CHARACTERIZATIONS OF REAL HYPERSURFACES OF TYPE B IN A COMPLEX HYPERBOLIC SPACE Seong Soo Ahn and Young Jin Suh Abstract. 1. Introduction A complex
More informationBredon, Introduction to compact transformation groups, Academic Press
1 Introduction Outline Section 3: Topology of 2-orbifolds: Compact group actions Compact group actions Orbit spaces. Tubes and slices. Path-lifting, covering homotopy Locally smooth actions Smooth actions
More informationarxiv: v2 [math.dg] 18 Jul 2014
Hopf fibrations are characterized by being fiberwise homogeneous HAGGAI NUCHI arxiv:1407.4549v2 [math.dg] 18 Jul 2014 Abstract. The Hopf fibrations of spheres by great spheres have a number of interesting
More informationGeometrical study of real hypersurfaces with differentials of structure tensor field in a Nonflat complex space form 1
Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 14, Number 9 (2018), pp. 1251 1257 Research India Publications http://www.ripublication.com/gjpam.htm Geometrical study of real hypersurfaces
More informationTOTALLY REAL SURFACES IN THE COMPLEX 2-SPACE
Steps in Differential Geometry, Proceedings of the Colloquium on Differential Geometry, 25 30 July, 2000, Debrecen, Hungary TOTALLY REAL SURFACES IN THE COMPLEX 2-SPACE REIKO AIYAMA Introduction Let M
More informationCLASSICAL GROUPS DAVID VOGAN
CLASSICAL GROUPS DAVID VOGAN 1. Orthogonal groups These notes are about classical groups. That term is used in various ways by various people; I ll try to say a little about that as I go along. Basically
More informationHyperkähler geometry lecture 3
Hyperkähler geometry lecture 3 Misha Verbitsky Cohomology in Mathematics and Physics Euler Institute, September 25, 2013, St. Petersburg 1 Broom Bridge Here as he walked by on the 16th of October 1843
More informationChapter 4 Euclid Space
Chapter 4 Euclid Space Inner Product Spaces Definition.. Let V be a real vector space over IR. A real inner product on V is a real valued function on V V, denoted by (, ), which satisfies () (x, y) = (y,
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 informationLecture 4 - The Basic Examples of Collapse
Lecture 4 - The Basic Examples of Collapse July 29, 2009 1 Berger Spheres Let X, Y, and Z be the left-invariant vector fields on S 3 that restrict to i, j, and k at the identity. This is a global frame
More informationEinstein H-umbilical submanifolds with parallel mean curvatures in complex space forms
Proceedings of The Eighth International Workshop on Diff. Geom. 8(2004) 73-79 Einstein H-umbilical submanifolds with parallel mean curvatures in complex space forms Setsuo Nagai Department of Mathematics,
More informationMax-Planck-Institut fur Mathematik in den Naturwissenschaften Leipzig Bernstein type theorems for higher codimension by Jurgen Jost and Yuan-Long Xin Preprint-Nr.: 40 1998 BERNSTEIN TYPE THEOREMS FOR
More informationLinear algebra 2. Yoav Zemel. March 1, 2012
Linear algebra 2 Yoav Zemel March 1, 2012 These notes were written by Yoav Zemel. The lecturer, Shmuel Berger, should not be held responsible for any mistake. Any comments are welcome at zamsh7@gmail.com.
More informationCHARACTERIZATION OF TOTALLY GEODESIC SUBMANIFOLDS IN TERMS OF FRENET CURVES HIROMASA TANABE. Received October 4, 2005; revised October 26, 2005
Scientiae Mathematicae Japonicae Online, e-2005, 557 562 557 CHARACTERIZATION OF TOTALLY GEODESIC SUBMANIFOLDS IN TERMS OF FRENET CURVES HIROMASA TANABE Received October 4, 2005; revised October 26, 2005
More informationMany of the exercises are taken from the books referred at the end of the document.
Exercises in Geometry I University of Bonn, Winter semester 2014/15 Prof. Christian Blohmann Assistant: Néstor León Delgado The collection of exercises here presented corresponds to the exercises for the
More informationINNER PRODUCT SPACE. Definition 1
INNER PRODUCT SPACE Definition 1 Suppose u, v and w are all vectors in vector space V and c is any scalar. An inner product space on the vectors space V is a function that associates with each pair of
More informationarxiv: v2 [math.dg] 18 Jul 2014
A surprising fibration of S 3 S 3 by great 3-spheres HAGGAI NUCHI arxiv:1407.4548v2 [math.dg] 18 Jul 2014 Abstract. In this paper, we describe a new surprising example of a fibration of the Clifford torus
More informationDierential Geometry Curves and surfaces Local properties Geometric foundations (critical for visual modeling and computing) Quantitative analysis Algo
Dierential Geometry Curves and surfaces Local properties Geometric foundations (critical for visual modeling and computing) Quantitative analysis Algorithm development Shape control and interrogation Curves
More informationLectures 15: Parallel Transport. Table of contents
Lectures 15: Parallel Transport Disclaimer. As we have a textbook, this lecture note is for guidance and supplement only. It should not be relied on when preparing for exams. In this lecture we study the
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 informationRiemannian geometry of positive definite matrices: Matrix means and quantum Fisher information
Riemannian geometry of positive definite matrices: Matrix means and quantum Fisher information Dénes Petz Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences POB 127, H-1364 Budapest, Hungary
More information1 Linear Algebra Problems
Linear Algebra Problems. Let A be the conjugate transpose of the complex matrix A; i.e., A = A t : A is said to be Hermitian if A = A; real symmetric if A is real and A t = A; skew-hermitian if A = A and
More informationGEOMETRY OF GEODESIC SPHERES IN A COMPLEX PROJECTIVE SPACE IN TERMS OF THEIR GEODESICS
Mem. Gra. Sci. Eng. Shimane Univ. Series B: Mathematics 51 (2018), pp. 1 5 GEOMETRY OF GEODESIC SPHERES IN A COMPLEX PROJECTIVE SPACE IN TERMS OF THEIR GEODESICS SADAHIRO MAEDA Communicated by Toshihiro
More informationA crash course the geometry of hyperbolic surfaces
Lecture 7 A crash course the geometry of hyperbolic surfaces 7.1 The hyperbolic plane Hyperbolic geometry originally developed in the early 19 th century to prove that the parallel postulate in Euclidean
More informationSurfaces in the Euclidean 3-Space
Surfaces in the Euclidean 3-Space by Rolf Sulanke Finished September 24, 2014 Mathematica v. 9, v. 10 Preface In this notebook we develop Mathematica tools for applications to Euclidean differential geometry
More informationOn the dynamics of a rigid body in the hyperbolic space
On the dynamics of a rigid body in the hyperbolic space Marcos Salvai FaMF, Ciudad Universitaria, 5000 Córdoba, rgentina e-mail: salvai@mate.uncor.edu bstract Let H be the three dimensional hyperbolic
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 informationREMARKS ON THE TIME-OPTIMAL CONTROL OF A CLASS OF HAMILTONIAN SYSTEMS. Eduardo D. Sontag. SYCON - Rutgers Center for Systems and Control
REMARKS ON THE TIME-OPTIMAL CONTROL OF A CLASS OF HAMILTONIAN SYSTEMS Eduardo D. Sontag SYCON - Rutgers Center for Systems and Control Department of Mathematics, Rutgers University, New Brunswick, NJ 08903
More informationChapter 3 Least Squares Solution of y = A x 3.1 Introduction We turn to a problem that is dual to the overconstrained estimation problems considered s
Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A. Dahleh George Verghese Department of Electrical Engineering and Computer Science Massachuasetts Institute of Technology 1 1 c Chapter
More informationStatistics of Shape: Eigen Shapes PCA and PGA
Sarang Joshi 10/10/2001 #1 Statistics of Shape: Eigen Shapes PCA and PGA Sarang Joshi Departments of Radiation Oncology, Biomedical Engineering and Computer Science University of North Carolina at Chapel
More informationWorkshop on the occasion of the 70th birthday of. Francesco Mercuri. Universität zu Köln. Classical Symmetric Spaces. Gudlaugur Thorbergsson
Classical I Universität zu Köln Workshop on the occasion of the 70th birthday of Francesco Mercuri Parma, September 19-20, 2016 I We say that a symmetric space M = G/K is classical if the groups in the
More informationThe Symmetric Space for SL n (R)
The Symmetric Space for SL n (R) Rich Schwartz November 27, 2013 The purpose of these notes is to discuss the symmetric space X on which SL n (R) acts. Here, as usual, SL n (R) denotes the group of n n
More informationMatrix Lie groups. and their Lie algebras. Mahmood Alaghmandan. A project in fulfillment of the requirement for the Lie algebra course
Matrix Lie groups and their Lie algebras Mahmood Alaghmandan A project in fulfillment of the requirement for the Lie algebra course Department of Mathematics and Statistics University of Saskatchewan March
More informationLecture 4 Orthonormal vectors and QR factorization
Orthonormal vectors and QR factorization 4 1 Lecture 4 Orthonormal vectors and QR factorization EE263 Autumn 2004 orthonormal vectors Gram-Schmidt procedure, QR factorization orthogonal decomposition induced
More informationH-convex Riemannian submanifolds
H-convex Riemannian submanifolds Constantin Udrişte and Teodor Oprea Abstract. Having in mind the well known model of Euclidean convex hypersurfaces [4], [5] and the ideas in [1], many authors defined
More informationThe Real Grassmannian Gr(2, 4)
The Real Grassmannian Gr(2, 4) We discuss the topology of the real Grassmannian Gr(2, 4) of 2-planes in R 4 and its double cover Gr + (2, 4) by the Grassmannian of oriented 2-planes They are compact four-manifolds
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 informationEigenvalues and Eigenvectors
/88 Chia-Ping Chen Department of Computer Science and Engineering National Sun Yat-sen University Linear Algebra Eigenvalue Problem /88 Eigenvalue Equation By definition, the eigenvalue equation for matrix
More informationLecture Notes LIE GROUPS. Richard L. Bishop. University of Illinois at Urbana-Champaign
Lecture Notes on LIE GROUPS by Richard L. Bishop University of Illinois at Urbana-Champaign LECTURE NOTES ON LIE GROUPS RICHARD L. BISHOP Contents 1. Introduction. Definition of a Lie Group The course
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 informationThe Spinor Representation
The Spinor Representation Math G4344, Spring 2012 As we have seen, the groups Spin(n) have a representation on R n given by identifying v R n as an element of the Clifford algebra C(n) and having g Spin(n)
More information5. Orthogonal matrices
L Vandenberghe EE133A (Spring 2017) 5 Orthogonal matrices matrices with orthonormal columns orthogonal matrices tall matrices with orthonormal columns complex matrices with orthonormal columns 5-1 Orthonormal
More informationHOMOGENEOUS EINSTEIN METRICS
HOMOGENEOUS EINSTEIN METRICS Megan M. Kerr A Dissertation in Mathematics Presented to the Faculties of the University of Pennsylvania in Partial Fulfillment of the Requirements for the Degree of Doctor
More informationCHAPTER 3. Gauss map. In this chapter we will study the Gauss map of surfaces in R 3.
CHAPTER 3 Gauss map In this chapter we will study the Gauss map of surfaces in R 3. 3.1. Surfaces in R 3 Let S R 3 be a submanifold of dimension 2. Let {U i, ϕ i } be a DS on S. For any p U i we have a
More informationTutorials in Optimization. Richard Socher
Tutorials in Optimization Richard Socher July 20, 2008 CONTENTS 1 Contents 1 Linear Algebra: Bilinear Form - A Simple Optimization Problem 2 1.1 Definitions........................................ 2 1.2
More informationDIFFERENTIAL GEOMETRY HW 9
DIFFERENTIAL GEOMETRY HW 9 CLAY SHONKWILER 2. Prove the following inequality on real functions (Wirtinger s inequality). Let f : [, π] R be a real function of class C 2 such that f() f(π). Then f 2 dt
More information