ERRATA TO

ANALYSIS, MANIFOLDS AND PHYSICS. PART I

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22 66 3

*Refers to the revised edition.

Front Matter III. Differentiable Manifolds,

Chapter I

Example 1" Let P be the set N of all N has no maximal element. (p. 17) is open iff it is a neighborhood x" (N (x) - {x }) 71 a --/= in A. The set A is dense For each x 6 X and U 6 @" with x 6 U there exists B 6 ~ such that x 6 B and B C U. is a filter 5 ( x ) . Uj such that Vi C Uj. is simply connected when n > 2. Exercise 2, p. 68. vector space (p. 27) is bounded d l (P, 0) < a, d2 (P, 0) < a, d3 (P, 0) < a continuous (p. 21) into the neighborhood [[olx- p. 56 Haar measure (p. 180). a a-field of subsets, ser Im(a)] < [ml(a) +lx2lp)l/p Let X and Y be two metric spaces, operator T on L2 ( y ) by v 2 - (vlv) add C p in the margin Let P - C(V 3 (3))

531

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67 69

-8 - 1 2

3-10

Vv, w E Vn(s) So we have for k odd

Answer: We shall construct a covering of the product space x~ ,NX~ which has no countable subcovering, let alone a finite one. Let A C N and VA -- x a eN Ua be an element of the topology on the product space defined as follows:

- -J[0,2/3) ifc~6A G / (1/3,1] i fo t~A " Then {VA; A C N} is an open covering of the product space; it has no countable subcovering. Indeed let {VA~}i be any countable subset of {VA; A C N}. One can always find a point x = {x l, x2 . . . . } in the product space such that x ~ {VA~ }i. For example, let

1/6 if i r Ai then X 6 { VAi }i. X i - - 5/6 if i ~_ Ai

Contributed by J. Labelle.

72 9 - 4

77 85

11 5 6

Chapter H

mapping: D/Ix0 = l, D Idlx0 = Id, Vx0. of f . A point where the rank is not maximal is called critical. If n = p the determinant of D f is called the Jacobian for the function q A sufficient condition theorem A sufficient condition for f

Chapter HI

111 127 129

131

135

- 1 2 - 1 5

8 12

- 5 footnote

- 6

x ~ U c X . admissible atlas (see p. 543) replace "identical" by "isomorphic" bundle in particular (p. 376) El a subspace of E or [Osborn II 4] if follows that f * o v = v o f* .

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210 211 216 218

footnote 17

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footnote 2O - 1

12, 13 - 7 - 3

7 7

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-11 9,12 - 1 5

-8

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footnote - 2

5 9

11 16

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det[ x 0] only identity transformations Svg The tensor Svg is the strain tensor generated by the vector field v ag o ah = agh (left action of G on X) or Og ocr h --- Crhg (right action of G on X) the group which defines isometries effectively, transitively, and freely on G. [v~, v~](g) and the n-toms 72 n = R n / Z n (p. 209) (orah Oag)

L and R of v~,,) as well as the generators v(~) v(c,) inserted in (5) 158 and p. 353

- [D,u , D,/~]~ (e)x k is defined by matrix multiplication, antisymmetrized for every v ~ C1 X - 1 V ~ e~e[3 i.e., all rotations be nonsingular, k~/~ defines a metric called Cartan-Killing. jacobian matrix of the embedding mapping (p. 240) (i~) - ( -- ) bl ((g 1)2 _t_ (Z2)2) -2 (Z2) 2 (zl) 2 --2zlz 2

--2zlz 2 (zl) 2 --(Z2) 2 (b 2) Chapter IV

Then if oe and fl are 1-forms By property 2 are vectors and W a pseudo-vector div f W = grad f . W + f div W Cx : ( X ) p

ffsurface

e3

534 ERRATA TO PART I

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20

224 1, 2

225 6 226 - 5 250 23, 24 253 17

254 11 271 7 276 - 9

285 14 287 - 3

302 16 306 5 313 18 314 - 5

- 4 316 - 8

- 4 317 6 331 - 1 2

- 1 3 -11

354

- 8 - 7

12-20

group [See Analysis, Manifolds and Physics, Part II] Move the sentence " H p is often called the de Rham group" and the marginal note "de Rham cohomology" back to line 12. Delete and replace by: "For the cases of 0-chains see for instance [Patterson]." Insert the definition of the Euler-Poincar6 characteristic, which can be found on p. 293. co depends only on x 1 x p �9 �9 . ~ ~ forC Cz Hp, o) cz H p. delete "A differential.., system." vector field. By the theorem on p. 248, C is completely integrable. result in a set {0(k)} �9 equations (see example p. 263) = -dy~

Chapter V

(p. 134) manifold. The metric is called iorentzian. It is

d u i d C i bl ~ C t - -

dt dt ' In a moving frame the (Vvu ) l l - (v,,,v' a lorentzian metric it is a maximal is a minimal hypersurface; in qs* (D f ) - D(qS*/) pp. 482-486. d(*~o) (Vl . . . . . Vn/2, J Vl . . . . . J vn/2) ~CFvr/- 2q:,r/. (1) such that q~ - constant is the group of isometries and dilations - r/)~ Oo~ v )" + r/xc~ 3/~ v )~ - 2,;15 r/~/~ where q~ is obtained by contraction 2q~ -- �89 3a v a Replace lines 12 to 20 by" This transformation is defined in the case of an euclidean metric if one adds to the space a point at infinity, whose image in an inversion is the origin of this inversion. In the

ERRATA TO PART I 535

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359 15 360 16 362 - 1 4 363 7 365 2

3

9,10

14 369 8 371 10

- 13 372 - 13 373 - 1 4 375 4 377 4 379 14 380 - 8 , - 7

- 6 381 - 13

382 15 20 - 4

case of the Lorentz metric the inversion with origin point x0, for instance x0 = 0, is defined only when x 2 - O~fx~x ~ # O, that is when x is not a null (i.e. lightlike) vector. The conformal group is defined in Se- gal cosmos (S 3 x R, ref p. 356) universal cover of the compactified Minkowski spacetime (cf. problem V.8). For more on the compactification of Minkowski space see for instance R. Penrose "Conformal Treatment of Infinity" pp. 563-584 in Relativity, Groups, and Topol- ogy (Les Houches 1963) Eds C. DeWitt and B. DeWitt (Gordon and Breach New York 1964) in empty space.

Chapter V bis subspace of Tp (P). Due to Add: We also write fi -- v• (p). a 1-form on U with values - -Ad(g -1) - ( 4 ) .71 .o9 kg.j i (x) ( v ) ( Ox/Og ) ( v )

110 OLgij(x)g/Ox OLgi.j(x)g/Og w

OMc(g~'i(x)v) delete the composition sign then (p. 367), corresponding to ~ and V u ~P of a covariant vector. Add" "h -= hor" _ . . . . + 1 . . .

~veru0 differential 2-form ~2 on X Delete: "It i s . . . p. 359" connection as defined on p. 359 of the connection, as we have already seen on an the injection f in P of G1; thus if gi -- gij (x )g j

536 ERRATA TO PART I

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- 4 - 3

384 - 1 2

385 17, 18

385 - 9 , - 1

the realization o of G Tg2g 1 - - P X G F action of G1, where (P, X, Jr, G) is a principal fibre bundle Delete and insert: 2) Suppose G is reducible to G, and let P1 denote a reduced bundle with injection f" P1 -+ P. Let/z be the projection P -+ P / G 1. The mapping/x o f Delete and insert the following Theorem. Every fibre bundle (E, X, Jr, F, G) such that the base manifold X is paracompact and the fi- bre F is diffeomorphic to R n admits infinitely many cross sections.

386 21 - 1 5

Proof: It is easy to show that when E is a vector bun- dle, i.e. when Fx is a vector space, it admits infinitely many cross sections. Indeed, let {Oi} be a partition of unity on X subordinates to a locally finite coveting by open sets Vi such that Vi C Uj some open set of an atlas of X over which E is trivializable. Let O" i be an arbitrary cross section over Vi. Then the element of Fx given by the finite sum

ff (X ) - - Z Oi (X )O" i (X ) i

is a cross section over X. The theorem is stronger be- cause it does not require a canonical identification of a point of Fx with the origin of R m, nor the group G to be linear. For the proof I one uses the property that a differentiable function defined on a closed set o f I~ m can be extended to the whole of It{ m together with Zorn's lemma, to show that every cross section defined over a closed set Y C X can be extended to a cross section over X.

I cf. R. Godement, Theorie des Faisceaux (Hermann, Paris, 1958) p. 151 or Kobayashi and Namizu, loc. cit. Vol. I, p. 58.

and px 1 the linear product in R n and px I the linear

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406

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4 5 7

l l 8

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5 11

17,18

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l l ,13 l0

figure 5

l l

--13

in part A (pp. 380-381) delete i.e . . . . and insert: [~0(p ' ) = g~'~o(p)g-1]. p. 388 and by the subspace horizontal field is horizontal (p. 374), and that - (u , v)

1 " ' ' - - " ' ' ( V e t ( I ) , V e t ( Z ) ) , . . . , (2k)! Tx X, x -- H (p) and be unique Add" df(X2)(Vl , . . . , V n ) - rr* df(Q)(Vl . . . . . Vn)

, . . . JT ! -- df(S2) (rr' hor Vl , hor Vn) - df (~ ) (hor v l , . . . , hor Vn) -- Df(X2)(Vl . . . . . Vn) +�89

This, together with the fact q~ being invariant on a vertical vector projects to a form q5 on X,

[ f ( V ) ] 2 curvature 2-form complexification (p. 224) via a scalar product in the fibres, possibly deduced to the scalar product in Decomposition theorem. Proof: It follows formally the same lines as the proof given above for the finite dimensional space E p,x. Its validity in the new context rests therefore Zip is an elliptic ))Ap* (A is exterior product) {Dp, Ep} dx v -- ~ i , Replace 7r i by :rl D~u (v) - -

+ A ~i,xO

538 ERRATA TO PART I

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411 415 419 420

421

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435 441

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468 472 477 478 490 494

Line

7 - 1 2

- 1 0 , - 11 - 5 2O

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- 1 6

6,7,14,17

7 - 3 , - 4

References

13 - 9 - 8 - 7

- 1 0

16 - 2 10 5

- 6 13

Errata (~)(Si (X) ) ) -1

when the connections A is flat. bold face dq~ p: g (1 ) --+ L(C, C) , )/1 - - --(2:rri) -1 tr 'f . . . . . n d ~ C l �9 2zr

S l

i A + = iA_in d~. define the following electromagnetic field define the following Do F+ and F_ Atiyah

change signs before 1/4 change signs before 1/8

if there is no torsion. x R x R T. Regge, "The group manifold approach to unified gravity", in Rela- tivitY, Groupes et Topologie II, eds. B.S. DeWitt and R. Stora (North- Holland, Amsterdam, 1984) pp. 933-1006.

C h a p t e r VI

exists C ~ on U, but cannot in general vanish for it vanishes if ~0' (a) - 0. Remark. ~06~ =

1 d T,, = i dx Proof: X is solution in sO' since, D X - B has at mos t one . . . i f D* has an - ( i y ) ~. p. 272 �9 " = I1(1 + I x 1 2 ) m / 2 5 f l 1 2 �9 .. Flu -- ( - - - )u -- 0

ERRATA TO PART I 539

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499 - 2

512 7

513 13 522 - 9 523 6

532 8 from 538 - 2

to 539 23

f 3q9 3u dx Y~,__. Ox----i Ox---- 7

D O 0 ~

1 = 4----~ 6c.

1 = - d x l A d x Z A d x 3

t 0/1 __ sin 0 cos 4~, 0/2 - - sin 0 sin 4~, or3 -- cos 0 For a system with an infinite �9 .. = (T, L* U) when U is C ~ with compact sup- port Y - ( x ) exp(ax) Replace by"

Answer: a) The elementary kernels E (t, s) are solutions of

( - d 2 / d t 2 - p2)E(t , s) -- ~(t, s). (4)

The Fourier transform of this equation is

~ ( ( - d Z / d t 2 - p2)6 �9 E)(r/) -- l,

(r] 2 -- p 2 ) 3 E - 1,

1 p v ( 1 1 ) = ~ + K l g ( r / - p) + K2a(r/-+- p). (5) ( S E ) (tl) 2p ~ - P 0 -k- p

We can choose K1 and K2 such that the elementary kernel is in the convo- lution algebra ~ ' + or ~ ' - . Recall (Problem VI 7) that

(SY+) ( r / ) -- q : i (Pv 1 :k iJr6o~ q::i(ri T i0) -1 (6) k 77 /

Hence

E + �9 ~ ' + if K + -- - K + -- irr/2p and 5 E + = l ( 1 __ _ 1 )

2p r l - p - i 0 r / + p - i 0 ' E - E ~ ' - if K1- -- - K 2 -- izr/2p and l ( , , ) ~ E - - - 2 p r / - p + i 0 r l + p - i 0

(7a)

(7b)

540 ERRATA TO PART I

Page* Line Errata To compute E + from (5) using (6) one can translate (see p. 458)

( 1 ) Pv + i6 ( r / - p)

r / - p

by p and the other two terms by -p"

E+(t, s) -- qzY+(t - s ) p - 1 sin(pt - ps). (8)

The propagator E - E + - E - is

E(t, s) - _ p - I sin(pt - ps). We could, of course have obtained (8) by solving (4) according to the method developed on p. 469, which says

E+ (t, s) -- Y• (t - s)h + (t - s) where the C ~ functions h + satisfy the homogeneous equation and the fol- lowing boundary conditions:

h + (0) -- 0, h +'(0) - - 1 and h - (0) - 0, h - ' (0 ) - 1.

Equation (7) suggests the following integral representation for the ele-

541 - 3 equation (9)

549 6 571 footnote 590 - 7

593 - 4

Chapter VII

Hilbert space of norm The Morse index is the negative of the index defined on p. 287

----- - d ( . . . + I m2u 2) dx)(U, V).

B

f exp(-~t2/2)h(t) dt A

_ - 1 ( h ( t ) h ( t ) - - )~t )~t 2

h'( t ) ) B - - - - t - - ~ exp(--)~t2/2)lA

ERRATA TO PART I 541

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603

606

608

B if + ~ exp(-'kt2/2) A

h(t) 3 7 - - 3 h'(t)t + h"(t)) dt

References CHERN, S.S., Selected Papers (Springer-Verlag, New York, 1978). ITZYKSON, C. and J.B. ZUBER, Quantum Field Theory (McGraw-Hill, New York, 1980). SCHUTZ, B., Geometrical methods of mathematical physics (Cambridge University Press, Cambridge, 1980).

New and~or Corrected Entries for Index

Page Entries Page Entries

617

618

619

602

620 621

622 623 623

Adjoint, 59 Adjoint representation, 166, 184 Automorphism, 157, 174 Cartan-Killing metric, 188 624 Chain, 217 Compactified Minkowski space, 354 Critical point, 72, 119,546, 547, 567 624 Critical value, 546, 547, 567 Decomposition theorem, 399 625 Degenerate, 285 626 Dense, 13 Dual, star, 295 Duality, 226 627 Dirichlet problem, 502 Einstein equation, 342, 356 628 Energy function, 344 Energy integral, 324, 513 Faithful, 163 Gaussian curvature, 396 629 Haar measure, 39, 180, 189 Hodge decomposition, 399

Hodge theorem, 400 Index of a vector field, 396 Inductive limit topology, 427 Laplacian, 318, 398, 421,449, 495, 527 Leray form, 449 Lorentzian metric, 287, 292 Lusin theorem, 41 Maxwell equations, 271,336 Normal (hypersurface), 292, 315 O(n), 175 Poincar6-Hopf theorem, 396 Principal value, 439, 532 Reflexive, 59 Shock wave, 535 Simplex, 216 Singular field, 339 Spin bundle, 417 Symplectic form, 268, 282, 552 Trace, 430, 487 Trajectory, 144, 274

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