Corollary 6.5: Let M be the Minkowski space [with dim(M);' 3]. For a map h: M- M, the ... Time," Adams Prize Essay, Cambridge (1966). 16S. Lie, Math. Ann. 5, 145 ...
Erratum: New Jacobian ϑ functions and the evaluation of lattice sums I. J. Zucker Citation: Journal of Mathematical Physics 17, 853 (1976); doi: 10.1063/1.522985 View online: http://dx.doi.org/10.1063/1.522985 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/17/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A renormalization method for the evaluation of lattice sums J. Math. Phys. 35, 6036 (1994); 10.1063/1.530726 Comment on ’’New Jacobian theta functions and the evaluation of lattice sums’’ by I.J. Zucker [J. Math. Phys. 16, 2189 (1975)] J. Math. Phys. 18, 187 (1977); 10.1063/1.523129 New Jacobian ϑ functions and the evaluation of lattice sums J. Math. Phys. 16, 2189 (1975); 10.1063/1.522465 Erratum: The evaluation of lattice sums. II. Number theoretic approach J. Math. Phys. 15, 520 (1974); 10.1063/1.1666676 The evaluation of lattice sums. III. Phase modulated sums J. Math. Phys. 15, 188 (1974); 10.1063/1.1666619
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Corollary 6.5: Let M be the Minkowski space [with dim(M);' 3]. For a map h: M- M, the following are equivalent. (1)
h is a
13
\lJ ! - homeomorphism on
M for one i, or a
r-homeomorphism on M for one i,
or a.8 ihomeomorphism on M for one i, or a.8thomeomorphism on M for one i, for i E {O, 1, 2, ... , 00,
w,g},
(2)
h is a 13 I-homeomorphism, \lJ [-homeomorphism, .Bchomeomorphism and.8t-homeomorphism on M for eachi=O,1,2, ... ,co,w,g.
(3)
h is a Lorentz transformation times a (linear) dilation with a constant > 0.
Remarks added in proof: It might be interesting to note, that the analog of Proposition 6. 1 in the case of analytic Riemannian manifolds will be proved by LelongFerrand 19 (Corollaire to Theoreme A): Conformal C2 maps are necessarily analytic. A "global proof" including Euclidean and hyperbolic metrics seems to be unknown.
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pology for Curved SPace-Time which Incorporates the Causal, Differential, and Conformal Structures," to appear in J. Math. Phys. 4R. Penrose, Techniques of Differential Topology in Relativity (Soc. Ind. APpl. Math., Philadelphia, 1972). 5A. Einstein, Uni versidad Nacional de Tucuman Revista A 2, 11 (1941).
6J. Winicour, Springer Lecture Notes in Physics 14, 145 (1972).
7S. Nanda, J. Math. Phys. 12, 394 (1971); 13, 12 (1972). SR. Geroch, Math. Rev. 43, 808 (1972) . 9S. W. Hawking and G. F. R. Ellis, The Large Scale Structure 0/ Space-Time (Cambridge U. P., Cambridge, 1973). IOJ. Liouville, J. Math. Pures Appl. 12, 265 (1847). 11 N. J. Hicks, Notes on Differential Geometry (Van Nostrand, London, 1965). 12R. Narasimhan, Analysis on Real and Complex Manifolds (North-Holland, Amsterdam, 1968). 13R. Beez, Math. Phys. (Leipz) 20, 253 (1875). 14R. Gobel, Die volle kausale Gruppe der Raum-Zeit, Physikal Tei! IT del' Habi!itationsschrift, V{tirzburg, 1973. 15S. W. Hawking, Singularities fInd the Geometry 0/ Space Time," Adams Prize Essay, Cambridge (1966). 16S. Lie, Math. Ann. 5, 145 (1872), cf. p. 186. 17S. Ferrara, R. Gatto, and A. F. Grillo, Springer Tracts Mod. Phys. 67, xxx (1973). I8C. Caratheodory, Sitzungsber. der Preussischen Akad. d. Wiss., phys.-math. Klasse Sec. 25,12 (1924). 19J. Lelong-Ferrand, "Interpretations geometriques de la courbure scalaire et regularite des homoeomorphismes conformes," to appear in the jubilee book, dedicated to A. Lichnerowicz (Reidel, Brussels, to be published).
ERRATA
Erratum: New Jacobian (J functions and the evaluation of lattice sums [J. Math. Phys. 16, 2189 (1975)J I. J. Zucker Department of Physics. University of Surrey. Guilford. Surrey, England (Received 2 January 1976)
On p. 2190, bottom left-hand column: The line reading "=2 s [L sa (s) + LSb(s)]" should read" =2 s [L sa (s) XLSb(s)],"
On p. 2190, top right-hand column: The line reading "= [IT +2ln(1 +..j2)]/J2" should read" =IT..j2ln(1 +";2)0"
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Journal of Mathematical Physics. Vol. 17, No.5. May 1976
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