DOCSLIB.ORG

LECTURE 14: NORMAL COORDINATES 1. the Normal

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
LECTURE 14: NORMAL COORDINATES 1. the Normal LECTURE 14: NORMAL COORDINATES 1. The Normal Coordinates Now we see for any p 2 M, there exists a neighborhood U ⊂ M of p and a neighborhood V ⊂ TpM of 0 so that the exponential map expp : V ! U −1 is a diffeomorphism. But TpM, and thus V , is Euclidian, so the triple fexpp ; U; V g form a local chart of M near p. Usually one takes such U's to be geodesic balls (and thus V to be Euclidean balls). We will fix an orthonormal basis feig of TpM, which gives us linear coordinates for V . We denote the corresponding coordinate functions on U by fxig. Definition 1.1. The local chart fU; x1; ··· ; xmg described above is called normal coordinate system at p. This is a particularly nice coordinate system. For example, the parametric i equation for the geodesic γ(t) = expp(tXp), where Xp = x @i, is given by γ : x(t) = (tx1; tx2; ··· ; txm); Moreover, we have Lemma 1.2. Let fU; x1; ··· ; xmg be a normal coordinate system at p. Then (1) For all 1 ≤ i; j ≤ m, gij(p) = δij. k (2) For all 1 ≤ i; j; k ≤ m, Γij(p) = 0. (3) For all 1 ≤ i; j; k ≤ m, @kgij(p) = 0. Proof. (1) This is obvious, since @ijp = d(expp)0ei = ei. (2) Put the geodesic parametric equation x(t) = (tx1; tx2; ··· ; txm): into the geodesic equation, we get for 1 ≤ k ≤ m, k i j k i j k 0 =x ¨ (t) +x _ (t)_x (t)Γij(γ(t)) = x x Γij(γ(t)): Letting t = 0, we conclude that for any 1 ≤ k ≤ m, i j k x x Γij(p) = 0 for all xi; xj. It follows that for any fixed k, all eigenvalues of the symmetric matrix k k (Γij)1≤i;j≤m are 0. So Γij(p) = 0 for all i; j; k. (3) This is a consequence of (2) and the metric compatibility. 1 2 LECTURE 14: NORMAL COORDINATES From (1) and (3) above, we see that the Taylor's expansion of gij(x) at p using normal coordinates start with the constant term is δij, and has no linear term. What is the next term? Theorem 1.3. With respect to normal coordinates near p, the functions gij's admit the following Taylor expansion at x = 0, 1 (1) g (x) = δ + R (p)xkxl + O(jx3j): ij ij 3 iklj Proof. Let (U; x1; ··· ; xm) be a normal coordinate system near p. Fix xi's and let i γ be the geodesic starting at p in the direction Xp = x @i, γ(t) = (tx1; ··· ; txm); 0 ≤ t ≤ 1: For each 1 ≤ i ≤ m, consider a geodesic variation 1 i m fi(t; s) = (tx ; ··· ; t(x + s); ··· ; tx ): Its variation field Vi = t@i is thus a Jacobi field along γ, which satisfies Vi(0) = 0; rγ_ (0)Vi = @i: In view of the first equation and the Jacobi field equation rγ_ rγ_ V + R(_γ; V )_γ = 0 we also have (2) rγ_ (0)Vi := rγ_ (0)rγ_ Vi = 0: Moreover, if we take the (k − 2)th covariant derivative of the Jacobi field equation, we get k−2 (k) X k − 2 (k−2−l) (l) r V + (r R)(γ; _ r V )_γ = 0; γ_ i l γ_ γ_ i l=0 where we used the facts rW (R(X; Y )Z) = (rW R)(X; Y )Z +R(rW X; Y )Z +R(X; rW Y )Z +R(X; Y )rW Z and rγ_ γ_ = 0. Taking k = 3, we get (3) rγ_ Vi + (rγ_ R)(γ; _ Vi)_γ + R(_γ; rγ_ Vi)_γ = 0: Evaluating at t = 0, and using Vi(0) = 0, we get (3) rγ_ (0)Vi = −R(Xp;@i)Xp = R(@i;Xp)Xp: So if we let 2 1 m h(t) = t gij(tx ; ··· ; tx ) = hVi(t);Vj(t)i; LECTURE 14: NORMAL COORDINATES 3 then h(0) = hVi(0);Vj(0)i = 0; 0 h (0) = hrγ_ (0)Vi;Vj(0)i + hVi; rγ_ (0)Vj(0)i = 0; 00 (2) (2) h (0) = hrγ_ (0)Vi;Vj(0)i + 2hrγ_ (0)Vi; rγ_ (0)Vji + hVi; rγ_ (0)Vj(0)i = 2δij; 3 X 3 (3−l) (l) h000(0) = hr V ; r V i = 0; l γ_ (0) i γ_ (0) j l=0 4 X 4 (4−l) (l) h0000(0) = hr V ; r V i = 8R(@ ;X ;X ;@ ): l γ_ (0) i γ_ (0) j i p p j l=0 As a consequence, we get 1 g (tx1; ··· ; txm) = hV (t);V (t)i ij t2 i j 1 8 = δ t2 + R(@ ;X ;X ;@ )t4 + O(t5) t2 ij 4! i p p j 1 = δ + R(@ ;X ;X ;@ )t2 + O(t3) ij 3 i p p j 1 = δ + R(@ ;@ ;@ ;@ )(txk)(txl) + O(t3): ij 3 i k l j This proves the theorem. (k) Remark. One can continue to calculate rγ_ (0)Vi's and get a full expansion of gij in normal coordinates. For example, the next two terms are 1 1 2 R xkxlxr + ( R + R mR )xkxlxrxs: 6 iklj;r 20 iklj;rs 45 kil rjsm Taking derivative of (1), we get 1 1 @ g = R xl + R xk + O(jxj2): r ij 3 irlj 3 ikrj Taking derivative again and evaluate at p, we get 1 1 @ @ g (0) = R (p) + R (p): s r ij 3 irsj 3 isrj As a consequence, we get Corollary 1.4. With respect to normal coordinates, one has 1 R (p) = (@ @ g + @ @ g − @ @ g − @ @ g )(0): ijkl 2 i l jk j k il i k jl j l ik Proof. The right hand side equals 1 (R + R + R + R − R − R − R − R )(p); 6 jlik jilk ikjl ijkl jkil jikl iljk ijlk 4 LECTURE 14: NORMAL COORDINATES which equals Rijkl(p) by using symmetries of the Riemann curvature tensor. Remark. This is Riemann's original definition of the curvature tensor. 2. Geometric Meaning of Curvatures Now we are ready to give geometric interpretations of sectional, Ricci and scalar curvatures. 0 Theorem 2.1. Let Πp ⊂ TpM be a 2-dimensional plane. Denote by Cr the circle of 0 radius r in Πp centered at p, and Cr = expp(Cr ). Let Lr be the length of Cr. Then 2πr − Lr π lim = K(Πp): r!0 r3 3 Proof. Take an orthonormal basis fe1; ··· ; emg of TpM so that Πp is spanned by e1; e2, and consider the normal coordinate system with respect to feig. Then for r small, the cricel Cr has equation 1 2 k Cr : x (t) = r cos t; x (t) = r sin t; x (t) = 0 (k ≥ 3): It follows that _ 2 i j jCr(t)j = gij(Cr(t))x _ (t)_x (t) 1 1 1 = 1 + R x2x2 x_ 1x_ 1 + 1 + R x1x1 x_ 2x_ 2 + 2 R x1x2x_ 1x_ 2 + O(r5) 3 1221 3 2112 3 1212 r4 = r2 − K(Π ) + O(r5): 3 p So r Z 2π Z 2π r2 _ 3 Lr = Length(Cr) = jCrjdt = r 1 − K(Πp) + O(r )dt 0 0 3 π = 2πr − K(Π )r3 + O(r4): 3 p This implies the theorem. So the sectional curvature measures the deviation of the geodesic circle to the standard circle in Euclidean space. To give a geometric interpretation of the Ricci curvature, we first prove Lemma 2.2. In a normal coordinate system near p, we have 1 det(g ) = 1 − Ric (p)xkxl + O(jxj3): ij 3 kl 1 k l 3 Proof. Let A = ln(gij). Since (gij) = I + 3 Riklj(p)x x + O(jx j) , and B2 Bk ln(I + B) = B + + ··· + + ··· ; 2 k LECTURE 14: NORMAL COORDINATES 5 we see 1 A = R (p)xkxl + O(jx3j) : 3 iklj In particular, 1 1 tr(A) = R (p)xkxl + O(jxj3) = − Ric (p)xkxl + O(jxj3); 3 ikli 3 kl where in the last step we used the fact δij = gij. As a consequence, 1 det(g ) = det(exp(A)) = exp(tr(A)) = 1 − R (p)xkxl + O(jxj3): ij 3 kl As an immediate consequence, we get p Corollary 2.3. The volume element det(gij) in normal coordinates has the ex- pansion q 1 det(g ) = 1 − Ric (p)xkxl + O(jxj3): ij 6 kl In particular, we get Corollary 2.4. Let up 2 SpM be a unit tangent vector at p, and let γ(t) be the geodesic starting at p with γ_ (0) = up. Then q Ric(u ) det(g (γ(t))) = 1 − p t2 + O(t3): ij 6 Proof. Take an orthonormal basis of TpM so that up = e1. Then with respect to the corresponding normal coordinates, the geodesic γ(t) = expp(tup) can be written as γ : x1(t) = t; x2 = ··· = xm = 0: It follows q 1 1 det(g (γ(t))) = 1 − Ric (p)t2 + O(t3) = 1 − Ric(u )t2 + O(t3): ij 6 11 3 p So the Ricci curvature measures the change of the volume element in the given direction. Finally let's study the scalar curvature S. We have Theorem 2.5. For r small enough, S(p) Vol(B (p)) = ! rm 1 − r2 + O(jr3j) ; r m 6(m + 2) m where !m is the volume of the unit ball in R . 6 LECTURE 14: NORMAL COORDINATES Proof. By definition Z q 1 m Vol(Br(p)) = det(gij)dx ··· dx Br(0) Z 1 k l 1 m 3 = 1 − Rickl(p)x x dx ··· dx + O(r ) Br(0) 6 Z m Rickl(p) k l 1 m 3 = !mr − x x dx ··· dx + O(r ): 6 Br(0) An elementary computation shows Z ! xkxldx1 ··· dxm = m rm+2δkl Br(0) m + 2 and the theorem follows.
Load more

Recommended publications
  • Vectors, Matrices and Coordinate Transformations
    S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between them, physical laws can often be written in a simple form. Since we will making extensive use of vectors in Dynamics, we will summarize some of their important properties. Vectors For our purposes we will think of a vector as a mathematical representation of a physical entity which has both magnitude and direction in a 3D space. Examples of physical vectors are forces, moments, and velocities. Geometrically, a vector can be represented as arrows. The length of the arrow represents its magnitude. Unless indicated otherwise, we shall assume that parallel translation does not change a vector, and we shall call the vectors satisfying this property, free vectors. Thus, two vectors are equal if and only if they are parallel, point in the same direction, and have equal length. Vectors are usually typed in boldface and scalar quantities appear in lightface italic type, e.g. the vector quantity A has magnitude, or modulus, A = |A|. In handwritten text, vectors are often expressed using the −→ arrow, or underbar notation, e.g. A , A. Vector Algebra Here, we introduce a few useful operations which are defined for free vectors. Multiplication by a scalar If we multiply a vector A by a scalar α, the result is a vector B = αA, which has magnitude B = |α|A. The vector B, is parallel to A and points in the same direction if α > 0.
    [Show full text]
  • Chapter 5 ANGULAR MOMENTUM and ROTATIONS
    Chapter 5 ANGULAR MOMENTUM AND ROTATIONS In classical mechanics the total angular momentum L~ of an isolated system about any …xed point is conserved. The existence of a conserved vector L~ associated with such a system is itself a consequence of the fact that the associated Hamiltonian (or Lagrangian) is invariant under rotations, i.e., if the coordinates and momenta of the entire system are rotated “rigidly” about some point, the energy of the system is unchanged and, more importantly, is the same function of the dynamical variables as it was before the rotation. Such a circumstance would not apply, e.g., to a system lying in an externally imposed gravitational …eld pointing in some speci…c direction. Thus, the invariance of an isolated system under rotations ultimately arises from the fact that, in the absence of external …elds of this sort, space is isotropic; it behaves the same way in all directions. Not surprisingly, therefore, in quantum mechanics the individual Cartesian com- ponents Li of the total angular momentum operator L~ of an isolated system are also constants of the motion. The di¤erent components of L~ are not, however, compatible quantum observables. Indeed, as we will see the operators representing the components of angular momentum along di¤erent directions do not generally commute with one an- other. Thus, the vector operator L~ is not, strictly speaking, an observable, since it does not have a complete basis of eigenstates (which would have to be simultaneous eigenstates of all of its non-commuting components). This lack of commutivity often seems, at …rst encounter, as somewhat of a nuisance but, in fact, it intimately re‡ects the underlying structure of the three dimensional space in which we are immersed, and has its source in the fact that rotations in three dimensions about di¤erent axes do not commute with one another.
    [Show full text]
  • Solving the Geodesic Equation
    Solving the Geodesic Equation Jeremy Atkins December 12, 2018 Abstract We find the general form of the geodesic equation and discuss the closed form relation to find Christoffel symbols. We then show how to use metric independence to find Killing vector fields, which allow us to solve the geodesic equation when there are helpful symmetries. We also discuss a more general way to find Killing vector fields, and some of their properties as a Lie algebra. 1 The Variational Method We will exploit the following variational principle to characterize motion in general relativity: The world line of a free test particle between two timelike separated points extremizes the proper time between them. where a test particle is one that is not a significant source of spacetime cur- vature, and a free particles is one that is only under the influence of curved spacetime. Similarly to classical Lagrangian mechanics, we can use this to de- duce the equations of motion for a metric. The proper time along a timeline worldline between point A and point B for the metric gµν is given by Z B Z B µ ν 1=2 τAB = dτ = (−gµν (x)dx dx ) (1) A A using the Einstein summation notation, and µ, ν = 0; 1; 2; 3. We can parame- terize the four coordinates with the parameter σ where σ = 0 at A and σ = 1 at B. This gives us the following equation for the proper time: Z 1 dxµ dxν 1=2 τAB = dσ −gµν (x) (2) 0 dσ dσ We can treat the integrand as a Lagrangian, dxµ dxν 1=2 L = −gµν (x) (3) dσ dσ and it's clear that the world lines extremizing proper time are those that satisfy the Euler-Lagrange equation: @L d @L − = 0 (4) @xµ dσ @(dxµ/dσ) 1 These four equations together give the equation for the worldline extremizing the proper time.
    [Show full text]
  • Geodetic Position Computations
    GEODETIC POSITION COMPUTATIONS E. J. KRAKIWSKY D. B. THOMSON February 1974 TECHNICALLECTURE NOTES REPORT NO.NO. 21739 PREFACE In order to make our extensive series of lecture notes more readily available, we have scanned the old master copies and produced electronic versions in Portable Document Format. The quality of the images varies depending on the quality of the originals. The images have not been converted to searchable text. GEODETIC POSITION COMPUTATIONS E.J. Krakiwsky D.B. Thomson Department of Geodesy and Geomatics Engineering University of New Brunswick P.O. Box 4400 Fredericton. N .B. Canada E3B5A3 February 197 4 Latest Reprinting December 1995 PREFACE The purpose of these notes is to give the theory and use of some methods of computing the geodetic positions of points on a reference ellipsoid and on the terrain. Justification for the first three sections o{ these lecture notes, which are concerned with the classical problem of "cCDputation of geodetic positions on the surface of an ellipsoid" is not easy to come by. It can onl.y be stated that the attempt has been to produce a self contained package , cont8.i.ning the complete development of same representative methods that exist in the literature. The last section is an introduction to three dimensional computation methods , and is offered as an alternative to the classical approach. Several problems, and their respective solutions, are presented. The approach t~en herein is to perform complete derivations, thus stqing awrq f'rcm the practice of giving a list of for11111lae to use in the solution of' a problem.
    [Show full text]
  • Rotation Matrix - Wikipedia, the Free Encyclopedia Page 1 of 22
    Rotation matrix - Wikipedia, the free encyclopedia Page 1 of 22 Rotation matrix From Wikipedia, the free encyclopedia In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. For example the matrix rotates points in the xy -Cartesian plane counterclockwise through an angle θ about the origin of the Cartesian coordinate system. To perform the rotation, the position of each point must be represented by a column vector v, containing the coordinates of the point. A rotated vector is obtained by using the matrix multiplication Rv (see below for details). In two and three dimensions, rotation matrices are among the simplest algebraic descriptions of rotations, and are used extensively for computations in geometry, physics, and computer graphics. Though most applications involve rotations in two or three dimensions, rotation matrices can be defined for n-dimensional space. Rotation matrices are always square, with real entries. Algebraically, a rotation matrix in n-dimensions is a n × n special orthogonal matrix, i.e. an orthogonal matrix whose determinant is 1: . The set of all rotation matrices forms a group, known as the rotation group or the special orthogonal group. It is a subset of the orthogonal group, which includes reflections and consists of all orthogonal matrices with determinant 1 or -1, and of the special linear group, which includes all volume-preserving transformations and consists of matrices with determinant 1. Contents 1 Rotations in two dimensions 1.1 Non-standard orientation
    [Show full text]
  • Coordinate Transformation
    Coordinate Transformation Coordinate Transformations In this chapter, we explore mappings – where a mapping is a function that "maps" one set to another, usually in a way that preserves at least some of the underlyign geometry of the sets. For example, a 2-dimensional coordinate transformation is a mapping of the form T (u; v) = x (u; v) ; y (u; v) h i The functions x (u; v) and y (u; v) are called the components of the transforma- tion. Moreover, the transformation T maps a set S in the uv-plane to a set T (S) in the xy-plane: If S is a region, then we use the components x = f (u; v) and y = g (u; v) to …nd the image of S under T (u; v) : EXAMPLE 1 Find T (S) when T (u; v) = uv; u2 v2 and S is the unit square in the uv-plane (i.e., S = [0; 1] [0; 1]). Solution: To do so, let’s determine the boundary of T (S) in the xy-plane. We use x = uv and y = u2 v2 to …nd the image of the lines bounding the unit square: Side of Square Result of T (u; v) Image in xy-plane v = 0; u in [0; 1] x = 0; y = u2; u in [0; 1] y-axis for 0 y 1 u = 1; v in [0; 1] x = v; y = 1 v2; v in [0; 1] y = 1 x2; x in[0; 1] v = 1; u in [0; 1] x = u; y = u2 1; u in [0; 1] y = x2 1; x in [0; 1] u = 0; u in [0; 1] x = 0; y = v2; v in [0; 1] y-axis for 1 y 0 1 As a result, T (S) is the region in the xy-plane bounded by x = 0; y = x2 1; and y = 1 x2: Linear transformations are coordinate transformations of the form T (u; v) = au + bv; cu + dv h i where a; b; c; and d are constants.
    [Show full text]
  • Lectures on Differential Geometry Math 240C
    Lectures on Differential Geometry Math 240C John Douglas Moore Department of Mathematics University of California Santa Barbara, CA, USA 93106 e-mail: [email protected] June 6, 2011 Preface This is a set of lecture notes for the course Math 240C given during the Spring of 2011. The notes will evolve as the course progresses. The starred sections are less central to the course, and may be omitted by some readers. i Contents 1 Riemannian geometry 1 1.1 Review of tangent and cotangent spaces . 1 1.2 Riemannian metrics . 4 1.3 Geodesics . 8 1.3.1 Smooth paths . 8 1.3.2 Piecewise smooth paths . 12 1.4 Hamilton's principle* . 13 1.5 The Levi-Civita connection . 19 1.6 First variation of J: intrinsic version . 25 1.7 Lorentz manifolds . 28 1.8 The Riemann-Christoffel curvature tensor . 31 1.9 Curvature symmetries; sectional curvature . 39 1.10 Gaussian curvature of surfaces . 42 1.11 Matrix Lie groups . 48 1.12 Lie groups with biinvariant metrics . 52 1.13 Projective spaces; Grassmann manifolds . 57 2 Normal coordinates 64 2.1 Definition of normal coordinates . 64 2.2 The Gauss Lemma . 68 2.3 Curvature in normal coordinates . 70 2.4 Tensor analysis . 75 2.5 Riemannian manifolds as metric spaces . 84 2.6 Completeness . 86 2.7 Smooth closed geodesics . 88 3 Curvature and topology 94 3.1 Overview . 94 3.2 Parallel transport along curves . 96 3.3 Geodesics and curvature . 97 3.4 The Hadamard-Cartan Theorem . 101 3.5 The fundamental group* .
    [Show full text]
  • 3. Introducing Riemannian Geometry
    3. Introducing Riemannian Geometry We have yet to meet the star of the show. There is one object that we can place on a manifold whose importance dwarfs all others, at least when it comes to understanding gravity. This is the metric. The existence of a metric brings a whole host of new concepts to the table which, collectively, are called Riemannian geometry.Infact,strictlyspeakingwewillneeda slightly di↵erent kind of metric for our study of gravity, one which, like the Minkowski metric, has some strange minus signs. This is referred to as Lorentzian Geometry and a slightly better name for this section would be “Introducing Riemannian and Lorentzian Geometry”. However, for our immediate purposes the di↵erences are minor. The novelties of Lorentzian geometry will become more pronounced later in the course when we explore some of the physical consequences such as horizons. 3.1 The Metric In Section 1, we informally introduced the metric as a way to measure distances between points. It does, indeed, provide this service but it is not its initial purpose. Instead, the metric is an inner product on each vector space Tp(M). Definition:Ametric g is a (0, 2) tensor field that is: Symmetric: g(X, Y )=g(Y,X). • Non-Degenerate: If, for any p M, g(X, Y ) =0forallY T (M)thenX =0. • 2 p 2 p p With a choice of coordinates, we can write the metric as g = g (x) dxµ dx⌫ µ⌫ ⌦ The object g is often written as a line element ds2 and this expression is abbreviated as 2 µ ⌫ ds = gµ⌫(x) dx dx This is the form that we saw previously in (1.4).
    [Show full text]
  • Unit 5: Change of Coordinates
    LINEAR ALGEBRA AND VECTOR ANALYSIS MATH 22B Unit 5: Change of Coordinates Lecture 5.1. Given a basis B in a linear space X, we can write an element v in X in a unique 3 way as a sum of basis elements. For example, if v = is a vector in X = 2 and 4 R 1 1 2 B = fv = ; v = g, then v = 2v + v . We say that are the B 1 −1 2 6 1 2 1 B 3 coordinates of v. The standard coordinates are v = are assumed if no other 4 basis is specified. This means v = 3e1 + 4e2. n 5.2. If B = fv1; v2; ··· ; vng is a basis of R , then the matrix S which contains the vectors vk as column vectors is called the coordinate change matrix. Theorem: If S is the matrix of B, then S−1v are the B coordinates of v. 1 1 6 −1 5.3. In the above example, S = has the inverse S−1 = =7. We −1 6 1 1 compute S−1[3; 4]T = [2; 1]T . Proof. If [v]B = [a1; : : : ; an] are the new coordinates of v, this means v = a1v1 + ··· + −1 anvn. But that means v = S[v]B. Since B is a basis, S is invertible and [v]B = S v. Theorem: If T (x) = Ax is a linear map and S is the matrix from a basis change, then B = S−1AS is the matrix of T in the new basis B. Proof. Let y = Ax. The statement [y]B = B[x]B can be written using the last theorem as S−1y = BS−1x so that y = SBS−1x.
    [Show full text]
  • Hamilton's Ricci Flow
    The University of Melbourne, Department of Mathematics and Statistics Hamilton's Ricci Flow Nick Sheridan Supervisor: Associate Professor Craig Hodgson Second Reader: Professor Hyam Rubinstein Honours Thesis, November 2006. Abstract The aim of this project is to introduce the basics of Hamilton's Ricci Flow. The Ricci flow is a pde for evolving the metric tensor in a Riemannian manifold to make it \rounder", in the hope that one may draw topological conclusions from the existence of such \round" metrics. Indeed, the Ricci flow has recently been used to prove two very deep theorems in topology, namely the Geometrization and Poincar´eConjectures. We begin with a brief survey of the differential geometry that is needed in the Ricci flow, then proceed to introduce its basic properties and the basic techniques used to understand it, for example, proving existence and uniqueness and bounds on derivatives of curvature under the Ricci flow using the maximum principle. We use these results to prove the \original" Ricci flow theorem { the 1982 theorem of Richard Hamilton that closed 3-manifolds which admit metrics of strictly positive Ricci curvature are diffeomorphic to quotients of the round 3-sphere by finite groups of isometries acting freely. We conclude with a qualitative discussion of the ideas behind the proof of the Geometrization Conjecture using the Ricci flow. Most of the project is based on the book by Chow and Knopf [6], the notes by Peter Topping [28] (which have recently been made into a book, see [29]), the papers of Richard Hamilton (in particular [9]) and the lecture course on Geometric Evolution Equations presented by Ben Andrews at the 2006 ICE-EM Graduate School held at the University of Queensland.
    [Show full text]
  • COORDINATE TRANSFORMATIONS Members of a Structural System Are Typically Oriented in Differing Directions, E.G., Fig
    COORDINATE TRANSFORMATIONS Members of a structural system are typically oriented in differing directions, e.g., Fig. 17.1. In order to perform an analysis, the ele- ment stiffness equations need to be expressed in a common coor- dinate system – typically the global coordinate system. Once the element equations are expressed in a common coordinate system, the equations for each element comprising the structure can be Figure 17.1 – Frame Structure 1 (Kassimali, 1999) 2 assembled. Coordinate Transformations: Frame Elements Consider frame element m of Fig. 17.7. Y e x y m b X (a) Frame Q , u 6 6 Q , u m 4 4 Q3, u3 Q , u Q5, u5 1 1 Figure 17.7: Local – Global Q2, u2 Coordinate Relationships (b) Local Coordinate End Forces and Displacements 3 4 1 Figure 17.7 shows that the local – QcosFsinFxXY global coordinate transformations QsinFcosFyXY (17.9) can be expressed as QFzZ x = cos X + sin Y y = -sin X + cos Y where x, X = 1 or 4; y, Y = 2 or 5; and z, Z = 3 or 6. and since z and Z are parallel, this coordinate transformation is Utilizing (17.9) for all six member expressed as force components and expressing z = Z the resulting transformations in Using the above coordinate matrix form gives transformations, the end force and QFb [t] [0] b displacement transformations can QFee[0] [t] be expressed as or 5 {Q} = [T] {F} (17.11) T {Q} Q Q Q T where b 123 = member; {Q} = <<Q>b <Q>e> = beginning node local coordinate element local coordinate force T T force vector; {Q}e = <Q4 Q5 Q6> = vector; {F} = <<F>b <F>e> = end node local coordinate force element global coordinate force T vector; {F}b = <F1 F2 F3> = [t] [0] beginning node global coordinate vector; and [T] = = T [0] [t] force vector; {F}e = <F4 F5 F6> = end node global coordinate force element local to global coordinate cos sin 0 transformation matrix.
    [Show full text]
  • 1 Vectors & Tensors
    1 Vectors & Tensors The mathematical modeling of the physical world requires knowledge of quite a few different mathematics subjects, such as Calculus, Differential Equations and Linear Algebra. These topics are usually encountered in fundamental mathematics courses. However, in a more thorough and in-depth treatment of mechanics, it is essential to describe the physical world using the concept of the tensor, and so we begin this book with a comprehensive chapter on the tensor. The chapter is divided into three parts. The first part covers vectors (§1.1-1.7). The second part is concerned with second, and higher-order, tensors (§1.8-1.15). The second part covers much of the same ground as done in the first part, mainly generalizing the vector concepts and expressions to tensors. The final part (§1.16-1.19) (not required in the vast majority of applications) is concerned with generalizing the earlier work to curvilinear coordinate systems. The first part comprises basic vector algebra, such as the dot product and the cross product; the mathematics of how the components of a vector transform between different coordinate systems; the symbolic, index and matrix notations for vectors; the differentiation of vectors, including the gradient, the divergence and the curl; the integration of vectors, including line, double, surface and volume integrals, and the integral theorems. The second part comprises the definition of the tensor (and a re-definition of the vector); dyads and dyadics; the manipulation of tensors; properties of tensors, such as the trace, transpose, norm, determinant and principal values; special tensors, such as the spherical, identity and orthogonal tensors; the transformation of tensor components between different coordinate systems; the calculus of tensors, including the gradient of vectors and higher order tensors and the divergence of higher order tensors and special fourth order tensors.
    [Show full text]