Appendix A Geometry and Tensor Calculus A.l Literature There is plenty of introductory literature on differential geometry and tensor cal­ culus. It depends very much on scientific background which of the following references provides the most suitable starting point. The list is merely a sugges­ tion and is not meant to give a complete overview. • Y. Choquet-Bruhat, C. DeWitt-Morette, and M. Dillard-Bleick, Analysis, Man­ ifolds, and Physics. Part I: Basics [36]: overview of many fundamental notions of analysis, differential calculus, the theory of differentiable manifolds, the theory of distributions, and Morse theory. An excellent summary of stan­ dard mathematical techniques of generic applicability. A good reminder for those who have seen some of it already; probably less suited for initiates. • M. Spivak, Calculus on Manifolds [264]: a condensed introduction to basic geometry. A "black piste" that starts off gently at secondary school level. A good introduction to Spivak's more comprehensive work. • M. Spivak, Differential Geometry [265]: a standard reference for mathemati­ cians. Arguably a bit of an overkill; however, the first volume contains virtually all tensor calculus one is likely to run into in the image literature for the next decade or so. • c. W. Misner, K. S. Thome, and J. A. Wheeler, Gravitation: an excellent course in geometry from a physicist's point of view [211]. Intuitive and practical, even if you have not got the slightest interest in gravitation (in which case you may want to skip the second half of the book). • J. J. Koenderink, Solid Shape [158]: an informal explanation of geometric concepts, helpful in developing a pictorial attitude. The emphasis is on 206 Geometry and Tensor Calculus understanding geometry. For this reason a useful companion to any tutorial on the subject. • A P. Lightman, W. H. Press, R. H. Price, and S. A Teukolsky, Problem Book in Relativity and Gravitation [185]: ±500 problems on relativity and gravitation to practice with; a good way to acquire computational skill in geometry. • M. Friedman, Foundations of Space-Time Theories: Relativistic Physics and Phi­ losophy of Science [92]: a philosophical view on differential geometry and its relation to classical, special relativistic, and general relativistic spacetime theories. Technique is de-emphasized in favour of understanding the role of geometry in physics. • Other classics: R. Abraham, J. E. Marsden, T. Ratiu, Manifolds, Tensor Anal­ ysis, and Applications [1]; V. I. Arnold, Mathematical Methods of Classical Me­ chanics [6]; M. P. do Carmo, Riemannian Geometry [31] and Differential Geome­ try of Curves and Surfaces [30]; R. L. Bishop and S. I. Goldberg, Tensor Analysis on Manifolds [18]; H. Flanders, Differential Forms [72]. The next section contains a highly condensed summary of geometric concepts introduced in this book. The reader is referred to the text and to aforementioned literature for detailed explanations. A.2 Geometric Concepts A.2.1 Preliminaries Kronecker symbol or Kronecker tensor: 1 if { a=!3 (A1) 8$ = 0 if a -#!3 Generalised Kronecker or permutation symbols: 8~11 ... 88~:'~k' ) = 8~11:::::kk = det:( r- 8~; Vk +1 ~ (ILl, ... ,ILk) even permuta~ion of (VI, ... ,Vk) , { -1 If (ILl, ... ,ILk) odd permutation of (VI, ... ,Vk) , (A2) o otherwise. Completely antisymmetric symbol in n dimensions: if (ILl, ... ,ILn) is an even permutation of (1, ... ,n) , if (ILl, ... , ILn) is an odd permutation of (1, ... , n), (A3) otherwise. A.2 Geometric Concepts 207 The tensor product of any two lR-valued operators: if X: DomX -t lR x I--t X(X) , (A4) Y : Dom Y -t lR : y I--t Y(y) , (AS) then x 0 Y : DomX x Dom Y -t lR : (x; y) I--t X 0 Y(x; y) (A6) is the operator defined by x 0 Y(x; y) ~f X(x) . Y(y). (A7) Alternation or antisymmetrisation of an operator: if then AltX: V 0 ... 0 V -t lR: (Xl, ... ,Xk) I--t AltX(Xl, ... ,Xk) ~ k def= k!1 '"L..J sgn 7r X( X"'[l] , ... ,X"'[k] ) , (A9) ". in which the sum extends over all permutations 7r of (1, ... , k), and in which sgn 7r denotes the sign of 7r. Symmetrisation of an operator: if (A 10) then (All) in which the sum extends over all permutations 7r of (1, ... , k). Index symmetrisation and antisymmetrisation: (A12) X(JL1 ... JLd = ~! L X JLr(l) ... JLr(k) ' ". 1 -k! '"L..J sgn7r X 1',,(1) "'JLr(k) • (A13) ". Einstein summation convention in n dimensions: n Xa ya ~ L:Xaya . (A14) a=l 208 Geometry and Tensor Calculus Einstein summation convention for antisymmetric sums: if XI-'I ... l-'k and YI-'I"'l-'k are antisymmetric, then (A.1S) 1-'1 < ... <I-'k A.2.2 Vectors A vector lives in a vector (or linear) space: vEV. (A. 16) Pictorial representation: a vector is an arrow with its tail attached to some base point (Figure A.l). Vectors are rates: if I is a lR-valued function, then v[/] = v(df) = dl (v) = va oa/, (A. 1 7) i.e. the rate of change of I along v (for definition of dl, see Sections A.2.3 and A.2.7). This also shows you how to define polynomials and power series of a vector, e.g. exp v [I] = f ~Val Oal ... van Oan I . (A.18) n=O n. This is just Taylor's expansion at the (implicit) base point to which v is assumed to be attached (see "tangent space"). Leibniz's product rule: v[/g] = gv[J] + I v[g]. (A.19) Decomposition relative to basis vectors: .... a .... V = v ea. (A.20) Standard coordinate basis: (A.2l) Operational definition: a vector is a linear, lR-valued function of covectors (or i-forms), producing the contraction of the vector and the covector: v:.... V* -+ lR :wl-tv- .... (-)w. (A.22) Often, one attaches a vector space V to each base point p of a manifold M (Fig­ ure A.l): (A.23) 1Mp is the tangent space at p, or the "fibre over p" of the manifold's tangent bundle (Figure A.l): TM= U IMp. (A.24) pEM A.2 Geometric Concepts 209 ..., ...... ,P ?k= TMp vector y at base point p ! proJectIoa map : " : hallie point Figure A.l: Vector, tangent space, tangent bundle. The "vector vat base point p": (A.2S) One can find out to which base point a vector is attached by means of the projec­ tion map: 7r : TM ~ M : vp I-t p. (A.26) The tangent space TMp projects, by definition, to p: 7r-1 : M ~ TM : p I-t TMp. (A.27) A vector field is a smooth section of the tangent bundle: s : M ~ TM: p I-t s(p) = vp. (A.28) Any vector field projects to the identity map of the base manifold: (A.29) A.2.3 Covectors A covector (or 11orm) lives in a covector space: wE V*. (A.30) Pictorial representation: a covector is an oriented stack of planes "attached" to some base point (a "planar wave disregarding phase"; see Figure A.2). Covectors are gradients. Decomposition relative to basis covectors: w=w",e- -'" . (A.31) Standard coordinate basis: e'" =dx"'. (A.32) 210 Geometry and Tensor Calculus cuudul'" (I) at baR: point p ~ Figure A.2: Covector. Operational definition: a covector is a linear, IR-valued function of vectors, pro­ ducing the contraction of the covector and the vector: w: Y ~ IR : iit--+ w(iJ). (A.33) Often, one attaches a covector space Y* to each base point p of a manifold M: Y* "'-'T*Mp. (A.34) T*Mp is the cotangent space at p, or the fibre over p of the manifold's cotangent bundle: T*M= U T*Mp. (A.35) pEM The "covector wat base point p": (A.36) One can find out to which base point a covector is attached by means of the pro­ jection map: 7r' : T*M ~ M : wp H p. (A.37) The cotangent space T*Mp projects, by definition, to p: 7r,-l : M ~ T*M: pH T*Mp. (A.38) A covector field is a smooth section of the cotangent bundle: s' : M ~ T*M: p H s'(p) = wp, (A.39) Any covector field projects to the identity map of the base manifold: 7r' 0 s' = 1M. (A.40) A.2.4 Dual Bases Dual bases (no metric required!): e-0< (-)e{3 = e{3- (-0<)e = U{3.1:0< . (A.41) A.2 Geometric Concepts 211 .. ~ ..... ,....,.." : Figure A.3: The gauge figure. A.2.S Riemannian Metric Riemannian metric tensor: G : Vx V -+ IR : (v, 'Iii) I-t G( V, w) . (A.42) A Riemannian metric is a symmetric, bilinear, positive definite, IR-valued map­ ping of two vectors: linearity: G('x ii + j.£ v, w) 'xG(17,w) + j.£G(v,w) , symmetry: G( ii, v) = G(v, 17) , (A.43) positivity: G( ii, ii) > 0 '</17# o. The scalar product is another name for the metric: V·-- W def= G(-v,w.-) (A.44) Pictorial representation: a Riemannian metric is a quadric II centred" at some base point, the gauge figure: Figure A.3. The sharp operator converts vectors into covectors: u V V* - u_defG(- )def -a W : -+ : v I-t wV = V,. = Va e . (A.4S) The "-operator is invertible, "-1 == " (the flat operator): L V* V - L - def H( - ) def a- V : -+ : V I-t V V = V,. = V ea , (A.46) with H(" V," W) ~f G(V, W) . (A.47) In particular, " converts a basis vector into a corresponding covector: "ea = ga{3 e{3 . (A.48) Similarly, " converts a basis covector into a corresponding vector: "ea = ga{3 e{3 .