Length, Angle, Distance, and Energy 229 For these exceptions the concept of the "energy" of a vector or curve has been found to be a meaningful and effective substitute for length. In other words, the vector field alat is determined uniquely by co and is a coordinate vector field of any contact coordinate system. We begin by defining the idea of a smooth one-parameter family of curves which will be called a C1 rectangle. APPENDIX TO CHAPTER 3 An arrow should be read "leads to." ..., 8,,-,) = p dx11 ...dx',(a1, =0 (see Problem 4.2.1). The formulas (6.4.4) and (6.4.5) are the local expressions for the fact that X satisfies (a') and (b'): a,p,FF o p = p. Willmore, T. J., An introduction to differential geometry, Clarendon Press, Oxford, 1959. It is lengthminimizing if ly,J is a minimum when t = 0 for every variation of y such that y = yo. h,(dim, dim) = ai(m)aif = (aiaif)m I A critical point m off is nondegenerate if Hf is nondegenerate. r,C, of oriented p-cubes C, with real M;,, defined by a (c, a/ap,(n)) = c, dx'(pn) is a linear isomorphism independent of the choice of coordinates x'. A special basis for E is singled out Orientability A pair of coordinate systems x` and y' is consistently oriented if the jacobian determinant det (8x1/ 3y') is positive wherever defined. Proposition 5.5.1. For if 6.6. At a given t c- TM, 185 Differential geometry Auslander, L., and MacKenzie, R., Introduction to differentiable manifolds, McGraw-Hill, New York, 1963. Schouten, J., Ricci calculus, Springer, Berlin, 1954. 270 Show that a connexion D on µ is compatible if the connexion forms of D with respect to such orthonormal bases satisfy the skew-adjointness property: w; = -a,ajw; (no sum). RIEMANNIAN AND SEMI-RIEMANNIAN MANIFOLDS [Ch. Thus rs = y(hs/IyI ), which shows that r is a reparametrization of y. Proble 5.4.1. More generally, we have that Proposition 6.7.1 is valid for an arbitrary contact structure. We do not require, for example, that they arise from Although flat spaces are themselves very special, Theorem 5.6.1, its proof, and the corollaries generalize without essential change to all semiriemannian spaces. Lemma 6.4.2. Dt Y = Dt(f'Xf) = (tf')Xf(n) + (f'n)DtX, _ [a"Z(n)ft9Xt(n) + (f'n)a"Dz"(n)X, = a"[Z"(n)f' + (f'n)(fi"n)]X,(n) Here we have used i and j as summation indices running through 1, ..., d and a as a summation index running through 1, ..., e. Thus D is determined locally by its C- coefficients F. For t e N,t we also define the coefficients of Dt to be the numbers P1, defined y) dy) dx. In fact, if r is any other closed 1-form on R2 - {0} and c = (21T)-1 f81 r, then r - cr1 is exact, where S' is the central counterclockwise-oriented unit circle. The terms now match those of fe(a w) 0. field. 225 coordinates x' on M, that is, p, = Pa, and q' = x' -p. Instead, we may try to find coordinates which simplify the expression for H. In particular, if we can include solutions of {f, H} = 0 (including H itself) among the coordinates, then these coordinates will not appear except in the specification of their initial (and hence perpetual) values. S2 = dpi A dq' = dP, A dQ'; (6.7.3) that is, the expression for the 2-form S2 has the same appearance for any contact coordinate system. The pair (M, b) is then called a riemannian manifold. f For X fixed, Ax is a tensor field of type (1, 1), viewed as a field of linear transformations of tangent spaces. 6.8. It is case (b) which applies to a contact form, with k = r. If M is a contact manifold with contact form w, then the one-dimensional codistribution spanned by to has as its associated distribution the 2r-dimensional distribution D annihilated by w; that is, (b) °M is orientable. p dµp and thus [d0,1 Spain, B., Tensor calculus, Interscience, New York, 1953. ut2 But Lx,O = di(W)O + i(W) dO = d + S2W, so W = - Vd. In other words, the vector field alat is determined uniquely by co and is a coordinate vector field of any contact coordinate system. + (vivfvhvk + w'Wfwhwk - 4v'w'v"w')T(2, 2) + 2(v'vfvhwk - v'wfwhwk)T(l, 3) + v'vf WhwkT(0, 4)] + , /Subtype /Image Thus the contact coordinates are more general than hamiltonian coordinates, and consequently the freedom to operate with contact coordinates gives us greater simplifying power. (The power of dw is the iterated wedge product.) This occurrence does not ruin the analysis since we may insert If the same particles are viewed as being hand, it is obvious that shortest (least-energy) curves are always length (energy)-minimizing, since the derivative of a function of t at an absolute minimum always vanishes. M is parallelizable iff there is a diffeomorphism µ: TM M x R d such that the first factor of µ is 7r: TM ->- M and for each m the second factor of µ restricted to M. is a linear function Mm -± R'. To Volume 1 This work represents our effort to present the basic concepts of vector and tensor analysis. Volume I begins with a brief discussion of algebraic structures followed by a rather detailed discussion of the algebra of vectors and tensors. (3) When d is applied twice the result is 0, written d2 = 0: d(dO) = 0 for every p-form 0. i)X = - The 1-form To each k-dimensional codistribution A there is the associated (d - k)-dimensional distribution D given by D(m) = {t I t e Mm, = 0 for every w e . Integration of Forms RIEMANNIAN AND SEMI-RIEMANNIAN MANIFOLDS [Ch. points ao = a, a,, . base, 214 characteristic, 203 coordinate, 50 energy-critical, 215 integral, 121 length-critical, 215 longitudinal, 214 moving frame along, 129 parallel translation along, 227 shortest, 210, 215 transverse, 214 Cycle, 186 Cylinder, 24 261 i - 2, is X,Yy' = YX,y' + X,Yyi - YX,yi = Y8 + [X,, Y]yi ing of only the cartesian coordinate system, the parallel translation of the associated connexion is the familiar parallel translation of vectors in Rd. Flanders, H., Differential forms, Academic Press, New York, 1963. That this property is enjoyed by the connexion form matrix (P;, dxk) = (Cu) is immediate from (5.11.7). [Ch. R= -Xi®w'0Qj'. This agrees with our previous notion of parallelism since the projection is zero if and only if the turning is in a direction normal to the surface. a;{Xp, -pjri(X)}F, op , 200 Then the 1-chain D = r,(a,, du) has boundary C' theory, 147 r,m,, where r, = 0. This point of view was employed by Painleve in 1894, but with a euclidean metric for the most part. _ an invariant subset of this extension. Problem 4.5.2. Local expressions for 0 and dO will be produced in the process. where T is the torsion of D. Show that D* is actually a connexion on M and that the torsion of D* is -T. If D and E are connexions on µ: N -> M and f is a function on N into R, show that fD + (1 - f )E is a connexion on µ. parametrization of, 29 Trace, 84, 85 Trajectories, 266 Transformation law of tangent, 54 of tensor, 83 sional integral submanifolds. However, an adequate notation for riemannian geometry was still lacking. Apt - pjr{(X) = 0. 160 standard on R. 11 Both products are in common use and we shall continue with our original definition.) = f;X, is a parallelization of M. Conversely, any two parallelizations are related by such a matrix. connexion, 220 coordinates, 216, 219 manifold, 224 structure, 224, 238 Angle, 209 Anticommutativity, 94 Arc connected (arcwise connected), 14, 46 length, reduced, 209 Atlas, 21 If m is a critical point off, we define the hessian off at m to be the bilinear form Hi on M. defined as follows. The interaction of these two items, the force 1-form and the kinetic-energy metric, are a sufficient formulation of the structure of a classical mechanics problem. Now we show that the trajectories are determined by co, the 1-form = -dT + p*F, and the kinetic-energy function T. First we have that i(a/at)s2 = 0, so equation (6.5.4) is still valid with the new trajectory field X given by (6.6.1) on S; that is, i(X)S2 = r. (6.7.4) This implies that r belongs to the codistribution E*. Problem 6.3.3. The Main Aim Of This Book Is To Precisely Explain The Fundamentals Of Tensors And Their Applications To Mechanics, Elasticity, Theory Of Relativity, Electromagnetic, Riemannian Geometry And Many Other Disciplines Of Science And Engineering, ... As a first step we simplify our notation for the problem by writing Xa = X'a8,, a = I, ..., k, so what we are looking for are functions annihilated by the k vector fields X,, ..., Xk; that is, Xaf = 0, 210 Let 0, = µ*: T*M--± T*M, Mn*, -* µ_, op. (s, d) Hamilton-Jacobi equations, 267 Hamiltonian (see Coordinates, Function, tangent bundle, 160 Quadratic form, 101 index of, 106 nullity of, 106 signature of, 106 Range, 5 Rank of 2-form, 111 Rectangle C°°, 214 broken, 215 Related by a map vector fields, 138, 220 tensor fields, 139 1-form, 119 One-parameter group, 125, 126 1-1 (one-to-one), 5 Onto function, 5 Operator alternating, 94, 97, 131 Hodge star, 108, 195 Laplace Beltrami, 198 laplacian, 170, 198 symmetric, 88, 97, 131 Orientable, 29, 162 Orientation, 107 induced, 185 Orthogonal, 35, 103, 108, 113, 162 Pair, ordered, 3 Paracompactness, 17, 208 Parallel translation, 226 along curve, 227 Parallelization, 160 of a map, 224 Parametrization by a chain, 182 of Klein bottle (see Klein bottle) of sphere (see Sphere) of torus (see Torus) Permutation inversions of, 93 sign of, 93 Pfaffian (see Form, System) Phase space, 118, 266 Plane(s) field of h-, 151 hyperbolic, 142 projective, 33, 39 Poincarc duality, 186 lemma, 169 converse of, 175 Poisson Pxn = . Thus This surface S is an integral submanifold of the three-dimensional distribution (b) There are a few important properties of d which are also sufficient to determine d completely, that is, axioms for d: Even then the existence result was in a peculiar direction, reverse to the direction in which individual contravariant tensors map. The velocity distribution is not integrable, since any configuration can be reached from any other by sufficient rolling. In terms of these new coordinates the expression for f has the form r Torsion, 231, 247 forms, 231 (b) If (dw)k+ 1 = 0 in a neighborhood of m and wm A (dwm)k Contact Manifolds Struik, D., Differential geometry, 2nd ed., Addison-Wesley, Reading, Mass., 1961. If we move along these y' curves from a point at which y' = 0, the derivative of Yy', has dimension n + 2 and so annihilates only a space of dimension 2n + 1 - (n + 2) = n - 1, which is a contradiction. Spain, B., Tensor calculus, Interscience, New York, 1953. The Betti numbers r,k) dX' A dxk. I aiplpill bracket, 257, 258 Contact Coordinates ., xd with respect to x1, ..., xd is nonsingular at m, so y1, x2,. 268 Hence if E(y,) is a differentiable function of t, its derivative must vanish when t = 0. Y over µ and every n such that µn c U we may write Y(n) = f'(n)X;(µn) This defines d real-valued functions f on j 'U It is easily seen that the f' are C', for if {w'} is the dual basis of 1-forms on U, then f' = w' o Y, which is a composition of the C'° maps Y: N-> TM and w': TU --* R. Hence an arbitrary = m1 -Pt=F,°p. show the necessity of some restrictive hypothesis on the domains. It is only slightly more restrictive to confine our attention to the case where µ is the identity, that is, to a metric on a manifold. Thus if pt, q' are hamiltonian coordinates for the canonical structure on T*M, the equations of motion are the HamiltonJacobi equations Moreover, a force field is frequently given by the differentiation of a potential field U, which makes invariant sense only if the force is -dU. The definition of a tensor comes only in Chapter 6 – when the reader is ready for it. While this text maintains a consistent level of rigor, it takes great care to avoid formalizing the subject.
$.' The functions = [ij, k] The connexion D of the parallelization {F,} is compatible but it will not generally be the semi-riemannian connexion since its torsion is essentially given by the c;k. Under what conditions on the c;k will the symmetrized connexion SD be the semi-riemannian connexion? Show that axiom (2) unifies the following formulas of vector (2.22): *dx = dy dz, *dy = dz dx, *dz = dx dy, *(dx dy dz) = 1, and for the other cases we can use ** = the identity. The second-order differential equations of motion can be viewed as a vector field on the tangent bundle of the configuration space. Thus we have (b) The sphere S2 has been parametrized in Example (d), Section 4.6, with the 2-cube a defined on U = [0, Tr] x [0, 27r] by equations Torsion, 231, 247 forms, 231 Topology, 8 metric, 10 relative, I1 From this viewpoint names such as "tangential tensor fields" and "cotangential tensor fields" would seem more appropriate. (i/au'(0)) = e,, which shows that q is nonsingular at 0 and hence q is a diffeomorphism on some neighborhood of 0. are continuous functions of the parameters ul, . connexion, 220 coordinates, 216, 219 manifold, 224 structure, 224, 238 Angle, 209 Anticommutativity, 94 Arc connected (arcwise connected), 14, 46 length, reduced, 209 Atlas, 21 For any Cm vector field X on M we define a CW function Px on T*M, as we defined p, for X, in (6.3.1), by It should be clear that equivalence of p-cubes is ., y' are related by jacobian determinant J = det (ax'/8y'), then dx1 dx' = J dy' dy" (cf. . The equations of motion (6.5.2) are still valid but we must bear in mind that F, o p is not defined on T*M but on S = T*M x R. 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