an overview of various occurrences of general expressions for the coefficients of lovelock...

8/11/2019 An Overview of Various Occurrences of General Expressions for the Coefficients of Lovelock Lagrangians and for L

1/54

AN OVERVIEW OF VARIOUS OCCURRENCES OF GENERAL EXPRESSIONS FOR THE COEFFICIENTS OF LOVELOCK LAGRANGIANSAND FOR LOVELOCK TENSORS FROM THE 0thTO THE 5thORDER IN CURVATURE

C. C. BriggsCenter for Academic Computing, Penn State University, University Park, PA 16802

Wednesday, March 22, 2000Abstract. An overview is given of variousoccurrences of general expressions for the coefficients of Lovelock Lagrangians and for Lovelock tensors from the 0thto the 5thorder icurvature in terms of the Riemann-Christoffel and Ricci curvature tensors and the Riemann curvature scalar for n-dimensional differentiable manifolds having a general lineaconnection.PACS numbers: 02.40.-k, 04.20.Cv, 04.20.Fy

The quintic Lovelock tensor has only recently been evaluated. Davis, S., Symmetric Variations of the Metric and Extremaof the Action for Pure Gravity, Gen. Rel. Grav., 30(1998) 345.

INTRODUCTION

This paper presents an overview of various occurrences of general

expressions for the coefficientsL(p)of Lovelock Lagrangians1-3(also called

Euler densities,4-5dimensionally continued Euler forms,6-8 sectional

curvatures,9-13Gauss curvature forms,14-16Gauss-Bonnet forms,17-19

Gauss-Kronecker curvatures,20-52Lipschitz-Killing curvatures,53-74each

of the latter two as distinct from the other,75-76etc.) and for Lovelock tensors77

G(p)ab(also called generalized Einstein curvature tensors,78-79etc.) from the

0thto the 5thorder in curvature in terms of the Riemann-Christoffel and Ricci

curvature tensors and Riemann curvature scalar for n-dimensi

differentiable manifolds having a general linear connection, where

Riemann-Christoffel curvature tensorRabcdis given by80

Rabcd

2 (

[a

b]

d

c+

[a

d

|e|

b]

e

c+

a

e

b

e

d

c),theRicci curvature tensorRa

bby

RabRca

bc=Racbc=Rca

cb+2 ([cQa]bc+Sca

dQdbc),

and the Riemann curvature scalarRby

RRaa=Rba

ab=Rabab=Rab

ba=Rbaba,

where ais the Pfaffian derivative, ab

cthe connection coefficient, ab

object of anholonomity, Qabcthe non-metricity tensor, and Sab

cthe to

tensor.

1 Mena Marugn, G. A., Dynamically generated four-dimensional models in Lovelock cosmology, Phys. Rev. D, 46(1992) 4340.2 Demaret, J., Y. De Rop, P. Tombal, and A. Moussiaux, Qualitative Analysis of Ten-Dimensional Lovelock Cosmological Models, Gen. Rel. Grav., 24(1992) 1169.3 Verwimp, T., Unified prescription for the generation of electroweak and gravitational gauge field Lagrangian on a principal fiber bundle,Journ. Math. Phys., 31(1990) 3047.4 Euler, L., Recherches sur la courbure des surfaces, Mmoires de lacadmie [sic] des sciences de Berlin, 1767, 16(1760) 119.5 Mardones, A., and J. Zanelli, Lovelock-Cartan theory of gravity, Class. Quantum Grav., 8(1991) 1545.6 Mller-Hoissen, F., Dimensionally continued Euler forms: Kaluza-Klein cosmology and dimensional reduction, Class. Quantum Grav., 3(1986) 665.7 Arik, M., E. Hizel, and A. Mostafazadeh, The Schwarzschild solution in non-Abelian Kaluza-Klein theory, Class. Quantum Grav., 7(1990) 1425.8 Dereli, T., and G. oluk, Direct-curvature Yang-Mills field couplings induced by the Kaluza-Klein reduction of Euler form actions in seven dimensions, Class. Quantum Gr

(1990) 533.9 Thorpe, J. A., Sectional Curvatures and Characteristic Classes, Annals of Mathematics , 80(1964) 429; On the Curvatures of Riemannian Manifolds, Illinois Journ. Mat

(1966) 412; Some Remarks on the Gauss-Bonnet Integral,Journ. Math. Mech., 18(1969) 779.10 Gray, A., Some Relations Between Curvature and Characteristic Classes,Mathematische Annalen, 184(1970) 257.11 Bishop, R. L., and S. I. Goldberg, Some Implications of the Generalized Gauss-Bonnet Theorem, Trans. Amer. Math. Soc., 112(1964) 508.12 Cheung, Y. K., and C. C. Hsiung, Curvature and Characteristic Classes of Compact Riemannian Manifolds,Journ. Diff. Geom., 1(1967) 89.13 Kulkarni, R. S., Curvature and Metric, Annals of Mathematics , 91(1970) 311; Curvature Structures and Conformal Transformations, Journ. Di ff. Geom., 4(1970) 425;

Theorem of F. Schur, ibid., 4(1970) 453.14 Gauss, C. F., Disquisitiones Generales circa Superficies Curvas, Commentationes societatis regiae scientiarum Gottingensis recentiores [Nouveaux Mmoires de Gottingue], vol

a. 1823-1827), Gottingae, Germany, 1828, Commentationes classis mathematicae, p. 99.15 Eells, J., A generalization of the Gauss-Bonnet Theorem, Trans. Amer. Math. Soc., 92(1959) 142.16 Thorpe, J. A., (1964), op. cit.17 Bonnet, P. O., Mmoire sur la thorie gnrale des surfaces,Journal de lcole Polytechnique, 19/Cahier 32 (1848) 1; cf. Sur quelques proprits gnrales des surfaces et des

traces sur les surfaces, Comptes rendus hebdomadaires des sances de lAcademie des Sciences (Paris) , 19(1844) 980.18 Mller-Hoissen, F., Spontaneous Compactification with Quadratic and Cubic Curvature Terms, Phys. Lett., 163B(1985) 106; Gravity Actions, Boundary Terms and Second-

Field Equations,Nucl. Phys. , B337(1990) 709; From Chern-Simons to Gauss-Bonnet, ibid., B346(1990) 235.19 Mardones, A., and J. Zanelli, op. cit.20 Kronecker, L., ber Systeme von Functionen [sic] mehrer Variabeln [Zweite Abhandlung] [sic],Monatsberichte der kniglich preussischen Akademie der Wissenschaften zu B

Gesammtsitzung [sic] vom 5. August 1869, (1869) 688.21 Chen, B.-Y., Some Integral Formulas of the Gauss-Kronecker Curvature, Kdai Math. Sem. Rep., 20(1968) 410.22 Cheng, Q.-M., Complete Maximal Spacelike Hypersurfaces of H41(C),Manuscripta Math. , 82(1994) 149.23 de Almeida, S. C., and F. G. B. Brito, Minimal Hypersurfaces of S4with Constant Gauss-Kronecker Curvature,Mathematische Zeitschrift, 195(1987) 99.24 Fu, J. H. G., Curvature Measures and Generalized Morse Theory,Journ. Diff. Geom., 30(1989) 619.25 Griffiths, P. A., Complex Differential and Integral Geometry and Curvature Integrals Associated to Singularities of Complex Analytic Varieties, Duke Mathematical Journ

(1978) 427.26 Guan, B., and J. Spruck, Boundary-value problems on Snfor surfaces of constant Gauss curvature,Annals of Mathematics, 138(1993) 601.27 Hsiung, C.-C., and S. S. Mittra, Isometries of Compact Hypersurfaces with Boundary in a Riemannian Space, in Kobayashi, S., M. Obata, and T. Takahashi (eds.), Differ

Geometry, in Honor of Kentaro Yano , Kinokuniya Book-Store Co., Ltd., Tokyo, Japan (1972), p. 145.28 Huang, W.-H., Superharmonicity of Curvatures for Surfaces of Constant Mean Curvature, Pacific Journal of Mathematics, 152(1992) 291.29 Hug, D., Curvature Relations and Affine Surface Area for a General Convex Body and its [sic] Polar,Results in Mathematics, 29(1996) 233; Contributions to Affine Surface A

Manuscripta Mathematica, 91(1996) 283.30 Kojima, M., q-Conformally flat hypersurfaces,Differential Geometry and its Applications, 5(1995) 51.31 Kowalski, O., On the Gauss-Kronecker Curvature Tensors,Mathematische Annalen, 203(1973) 335.32 Kulkarni, R. S., On the Bianchi identities,Mathematische Annalen,199(1972) 175.33 Langevin, R., and H. Rosenberg, Fenchel type theorems for submanifolds of Sn, Comment. Math. Helvetici, 71(1996) 594.34 Li, A.-M., Spacelike hypersurfaces with constant Gauss-Kronecker curvature in the Minkowski space,Arch. Math. (Basel), 64(1995) 534.35 Li, Y. Y., Group invariant convex hypersurfaces with prescribed Gauss-Kronecker curvature, in Multidimensional Complex Analysis and Partial Differential Equations; A Coll

of Papers in Honor of Franois Treves, Contemporary Mathematics, 205(1997) 203.36 Martnez, A., and F. Miln, On the Affine Bernstein Problem, Geometriae Dedicata, 37(1991) 295.37 Miln, F., Pick Invariant and Affine Gauss-Kronecker Curvature, Geometriae Dedicata, 45(1993) 41.38 Nelli, B., and H. Rosenberg, Some Remarks on Positive Scalar and Gauss-Kronecker Curvature Hypersurfaces ofRn+ 1andHn+ 1,Ann. Inst. Fourier, Grenoble, 47(1997) 120939 Qingming, C., Hypersurfaces With [sic] Constant Quasi-Gauss-Kronecker Curvature in S4(1),Advances in Mathematics (China), 22(1993) 125.40 Ramanathan, J., Minimal hypersurfaces in S4with vanishing Gauss-Kronecker curvature,Mathematische Zeitschrift, 205(1990) 645.41 Ros, A., Compact Hypersurfaces with Constant Scalar Curvature and a Congruence Theorem,Journ. Diff. Geom., 27(1988) 215.42 Rosenberg, H., and J. Spruck, On the Existence of Convex Hypersurfaces of Constant Gauss Curvature in Hyperbolic Space, Journ. Diff. Geom., 40(1994) 379.43 Schmuckenschlaeger, M., A Simple Proof of an Approximation Theorem of H. Minkowski, Geometriae Dedicata, 48(1993) 319.44 Schneider, R., Polyhedral Approximation of Smooth Convex Bodies,Journal of Mathematical Analysis and Applications, 128(1987) 470.


2/54

2 Wednesday, March 22, 2000

In accordance with various general definitions given by Mller-Hoissen81

and Verwimp,82L(p)and G(p)abare given, using anholonomic coordinates, by

L(p)=

1, ifp= 0

(2p)!2p

R [i1i2i1i2R i3i4

i3i4 R i2p 1i2p]i2p 1i2p, ifp> 0

(4)

and

G(p)ab =

ba, ifp= 0

(2p + 1)!2p+ 1p

b[aR i1i2

i1i2R i3i4i3i4 R i2p 1i2p]

i2p 1i2p, ifp> 0

respectively, whereRabcd= gceRabe

dand bais the Kronecker delta.

Some numerical properties ofL(p)and G(p)abfor 0p 5 appear in Ta

1 and 2 (see below). Complete expressions and the overview follow.

TABLE 1. SOME NUMERICAL PROPERTIES OFL(p)FOR 0 p5. TABLE 2. SOME NUMERICAL PROPERTIES OF G(p)abFOR 0 p5

ORDER QUANTITY NUMBER OFTERMS

NUMBER OF PERMUTA-TIONS COMPREHENDED

SUM OF NUMERICALFACTORS

ORDER QUANTITY NUMBER OFTERMS

NUMBER OF PERMUTA-TIONS COMPREHENDED

SUM OF NUMERICAFACTORS

0 L(0) 1 1 1 0 G(0)ab

1 1 1

1 L(1) 1 2 1 1 G(1)ab

2 6 1 122 L(2) 3 24 6 2 G(2)a

b7 120 7 12

3 L(3) 8 720 90 3 G(3)ab

2 6 5040 105

4 L(4) 2 5 40,320 2520 4 G(4)ab

115 362,880 2835

5 L(5) 8 5 3,628,800 113,400 5 G(5)ab

596 39,916,800 124,740

45 Schneider, R., and J. A. Wieacker, Random Polytopes in a Convex Body,Zeitschrift fr Wahrscheinlichkeitstheorie und verwandte Gebiete, 52(1980) 69.46 Schtt, C., On the Affine Surface Area, Proc. Amer. Math. Soc., 118(1993) 1213; Random Polytopes and Affine Surface Area,Mathematische Nachrichten, 170(1994) 22747 Tso, K., Deforming a Hypersurface by Its Gauss-Kronecker Curvature, Comm. on Pure and Appl. Math., 38(1985) 867; Convex Hypersurfaces with Prescribed Gauss-Kron

Curvature, Journ. Dif f. Geom. , 34(1991) 389; A direct method approach for the existence of convex hypersurfaces with prescribed Gauss-Kronecker curvature, MathemaZeitschri ft, 209(1992) 339.

48 Wang, X.-J., Existence of Convex Hypersurfaces with Prescribed Gauss-Kronecker Curvature, Trans. Amer. Math. Soc., 348(1996) 4501.49 Hopf, H., Die Curvatura integra Clifford-Kleinscher Raumformen,Nachrichten von der Gesellschaft der Wissenschaften zu Gttingen, Mathematisch-physikalische Klasse, 2(

131.50 Chern, S. S., and R. K. Lashof, On the total curvature of immersed manifolds, I, Amer. Journ. Math. , 79(1957) 306; On the total curvature of immersed manifolds, II, Mic

Math. Journ. , 5(1958) 5.51 Chern, S. S., Integral Formulas for Hypersurfaces in Euclidean Space and Their Applications to Uniqueness Theorems,Journ. Math. Mech., 8(1959) 947.52 Kulkarni, R. S., (1972), op. cit.53 Lipschitz, R., Entwicklung einiger Eigenschaften der quadratischen Formen von nDifferentialen (Erste Mittheilung.),Journal fr die reine und angewandte Mathematik, 71(

274; Entwicklung einiger Eigenschaften der quadratischen Formen von nDifferentialen (Zweite Mittheilung.), ibid., 288; Fortgesetzte Untersuchen in Betreff der ganzen homFunctionen von nDifferentialen, ibid., 72(1870) 1; Ausdehnung der Theorie der Minimalflchen, ibid., 78(1874) 1; Beitrag zu der Theorie der Krmmung, ibid., 81(1876Ausdehnung der Theorie der Minimalflchen [Einleitung],Monatsberichte der kniglich preussischen Akademie der Wissenschaften zu Berlin, Sitzung der physikalisch-mathematKlasse vom 27. Mai 1872, (1872) 361.

54 Killing, W.,Die nicht-euklidischen Raumformen in analytischer Behandlung, Verlag von B. G. Teubner, Leipzig, Germany (1885), pp. 210-211, 234-235, 236, 243-257, and 263-55 Cheung, Y. K., and C. C. Hsiung, op. cit.56 Brito, F. G. B., Une obstruction gomtrique lexistence de feuilletages de codimension 1 totalement godsiques,Journ. Diff. Geom., 16(1981) 675.57 Brito, F. G. B., R. Langevin, and H. Rosenberg, Intgrales de courbure sur des varits feuilletes,Journ. Diff. Geom., 16(1981) 19.58 Budach, L., Lipschitz-Killing Curvatures of Angular Partially Ordered Sets, Advances in Mathematics , 78(1989) 140; On the Combinatorial Foundations of Regge-Calc

Annalen der Physik (7), 46(1989) 1.59 Cheeger, J., W. Mller, and R. Schrader, On the Curvature of Piecewise Flat Spaces, Commun. Math. Phys., 92(1984) 405; Kinematic and Tube Formulas for Piecewise L

Spaces,Indiana Univ. Math. Journ., 35(1986) 737.60 Enomoto, K., Compactification of Submanifolds in Euclidean Space by the Inversion,Advanced Studies in Pure Mathematics; Progress in Differential Geometry, 22(1993) 1.61 Houh, C.-S., Surfaces with Maximal Lipschitz-Killing Curvature in the Direction of Mean Curvature Vector, Proc. Amer. Math. Soc., 35(1972) 537.62 Khnel, W., Total Curvature of Manifolds with Boundary in E,Journ. London Math. Soc. (2), 15(1977) 173; (n 2)-Tightness and Curvature of Submanifolds with Boun

Internat. Journ. Math. and Math. Sci. , 1(1978) 421.63 Langevin, R., and L. D. Trng, Courbure au voisinage dune singularit, Comptes rendus hebdomadaires des sances de lAcademie des Sciences (Paris), 290(1980) A95.64 Montesinos, A., Some Integral Invariants of Plane Fields on Riemannian Manifolds, in Dold, A., and B. Eckmann (eds.), Differential Geometry. Proceedings of the Interna

Symposium held at Pescola, Spain, October 3-10, 1982 , Springer-Verlag, Berlin, Germany (1984),Lecture Notes in Mathematics , 1045(1984) 134.65 Rochowski, M., A Scheme of Generating Canonical Frames in Normal Bundles of Immersed Manifolds in Euclidean Spaces, Demonstratio Mathematica, 23(1990) 271; Pa

Cross Sections and Canonical Frames in Normal Bundles of Immersed Manifolds in Euclidean Space, ibid., 557.66 Teufel, E., Eine differentialtopologische Berechnung der totalen Krmmung und totalen Absolutkrmmung in der sphrischen Differentialgeometrie,Manuscripta Mathematic

(1980) 119; Anwendungen der differentialtopologischen Berechnung der totalen Krmmung und totalen Absolutkrmmung in der sphrischen Differentialgeometrie, op. cit., 32(239; On the Total Absolute Curvature of Immersions into Hyperbolic Spaces, in Szenthe, J. and L. Tamssy (eds.), Colloquia Mathematica Societatis Jnos Bolyai, 46, Top

Differential Geometry, Debrecen, Hungary, North-Holland Publishing House, Amsterdam, The Netherlands (1984), vol. 2, p. 1201.67 Vranceanu, G., Une interprtation gomtrique de la courbure Lipschitz-Killing pour les varits diffrentiables sans torsion,Revue Roumaine de Mathmatiques Pures et Appli

21(1976) 1461.68 Wintgen, P., On the Total Curvature of Surfaces in E4, Colloquium Mathematicum, 39(1978) 289.69 Zhle, M., Absolute Curvature Measures, Mathematische Nachrichten , 140(1989) 83; Approximation and Characterization of Generalised Lipschitz-Killing Curvatures,

Global Anal. Geom., 8(1990) 249.70 Chern, S. S., and R. K. Lashof, op. cit.71 Thorpe, J. A., (1964), op. cit.; (1966), op. cit.; (1969), op. cit.72 Thas, C., Properties of Ruled Surfaces in the Euclidean Space En,Bulletin of the Institute of Mathematics Academia Sinica, 6(1978) 133.73 Lafontaine, J., Mesures de courbure des varits lisses et des polydres, Sminaire Bourbaki, Volume 1985/86, Astrisque 145-146 (1987), xpose 664 (1987), p. 241.74 Cheeger, J., Spectral Geometry of Singular Riemannian Spaces, Journ. Diff. Geom. , 18(1983) 575.75 Willmore, T. J., and B. A. Saleemi, The Total Absolute Curvature of Immersed Manifolds,Journ. London Math. Soc., 41(1966) 153.76 Gray, A., op. cit.77 Arik, M., E. Hizel, and A. Mostafazadeh, op. cit.78 Lovelock, D., Divergence-Free Tensorial Concomitants,Aequationes Mathematicae, 4(1970) 127; The Einstein tensor and its generalizations,Journ. Math. Phys., 12(1971

The four-dimensionality of space and the Einstein tensor, ibid., 13(1972) 874.79 Farhoudi, M., Lovelock Tensor as Generalized Einstein Tensor, University of London preprint no. QMW-PH-95-37, Los Alamos preprint no. gr-qc/9510060; Classical

Anomaly, University of London preprint no. QMW-PH-95-41, Los Alamos preprint no. gr-qc/9511047.80 Schouten, J. A.,Ricci Calculus, Springer-Verlag, Berlin, Germany (1954), p. 172.81 Mller-Hoissen, F., (1985), op. cit.; From Chern-Simons to Gauss-Bonnet, (1990), op. cit.82 Verwimp, T., On higher dimensional gravity: the Lagrangian, its dimensional reduction and a cosmological model, Class. Quantum Grav., 6(1989) 1655.


3/54


4/54


5/54


6/54


7/54


8/54


1920RaecdRcf

beRdhfgRjk

hiRgijk+960Rae

cdRcfbeRhi

fgRjkhiRdg

jk3840RaecdRcf

beRhjfgRdk

hiRgijk120Rae

cdRfgbeRcd

fgRjkhiRhi

jk

+960RaecdRfg

beRchfgRjk

hiRdijk240Rae

cdRfgbeRhi

fgRjkhiRcd

jk+960RaecdRfg

beRhjfgRck

hiRdijk+960Rae

cdRfhbeRcd

fgRjkhiRgi

jk+

+1920RaecdRfh

beRcifgRjk

hiRdgjk1920Rae

cdRfhbeRcj

fgRdkhiRgi

jk+3840RaecdRfh

beRijfgRck

hiRdgjk+960Rae

cdRfhbeRij

fgRgkhiRcd

j

960RafcdRcd

beRehfgRjk

hiRgijk+480Raf

cdRcdbeRhi

fgRjkhiReg

jk1920RafcdRcd

beRhjfgRek

hiRgijk+480Raf

cdRcgbeRde

fgRjkhiRhi

jk

1920RafcdRcg

beRdhfgRjk

hiReijk+1920Raf

cdRcgbeReh

fgRjkhiRdi

jk+960RafcdRcg

beRhifgRjk

hiRdejk+3840Raf

cdRcgbeRhj

fgRekhiRdi

j

+960RafcdRgh

beRcdfgRjk

hiReijk1920Raf

cdRghbeRce

fgRjkhiRdi

jk+1920RafcdRgh

beRcifgRjk

hiRdejk3840Raf

cdRghbeRcj

fgRdkhiRei

j

+960RafcdRgh

beReifgRjk

hiRcdjk3840Raf

cdRghbeRej

fgRckhiRdi

jk+3840RafcdRgh

beRijfgRck

hiRdejk+1920Rah

cdRcfbeRde

fgRjkhiRgi

j

+1920RahcdRcf

beRdifgRjk

hiRegjk3840Rah

cdRcfbeRdj

fgRekhiRgi

jk+3840RahcdRcf

beRdjfgRgk

hiReijk1920Rah

cdRcfbeRei

fgRjkhiRd

+3840RahcdRcf

beRejfgRdk

hiRgijk3840Rah

cdRcfbeRej

fgRgkhiRdi

jk+3840RahcdRcf

beRijfgRdk

hiRegjk3840Rah

cdRcfbeRij

fgRekhiRd

+3840RahcdRcf

beRijfgRgk

hiRdejk+1920Rah

cdRcfbeRjk

fgRdehiRgi

jk1920RahcdRcf

beRjkfgRdg

hiReijk+1920Rah

cdRcfbeRjk

fgReghiRd

+480RahcdRfg

beRcdfgRjk

hiReijk960Rah

cdRfgbeRce

fgRjkhiRdi

jk+960RahcdRfg

beRcifgRjk

hiRdejk+480Rah

cdRfgbeRei

fgRjkhiRcd

jk+

+960RahcdRfg

beRijfgRek

hiRcdjk960Rah

cdRfibeRcd

fgRjkhiReg

jk+1920RahcdRfi

beRcefgRjk

hiRdgjk+3840Rah

cdRfibeRej

fgRckhiRdg

jk

+1920RahcdRfi

beRejfgRgk

hiRcdjk+1920Raj

cdRfgbeRch

fgRdkhiRei

jk1920RajcdRfg

beRchfgRek

hiRdijk+1920Raj

cdRfgbeReh

fgRckhiRd

+960RajcdRfg

beRhifgRck

hiRdejk+480Raj

cdRfgbeRhi

fgRekhiRcd

jk+1920RajcdRfg

beRhkfgRce

hiRdijk+3840Raj

cdRfhbeRce

fgRdkhiRgi

jk

3840RajcdRfh

beRcefgRgk

hiRdijk+3840Raj

cdRfhbeRci

fgRdkhiReg

jk+3840RajcdRfh

beRcifgRgk

hiRdejk+3840Raj

cdRfhbeRei

fgRckhiRd

+1920RajcdRfhbeReifgRgkhiRcdjk1920RajcdRfhbeRekfgRcdhiRgijk+3840RajcdRfhbeRekfgRcghiRdijk1920RajcdRfhbeRikfgRcdhiRe

For a check, notefor 1 p5that (1) G(p)aa =

n 2p2p

L(p)and that (2) the magnitudes of the numerical coefficients of G(p)abadd up to

(2p + 1)!2p+ 1p

.

TABLE 3. OVERVIEW OF AGREEMENT UP TO AN OVERALL NUMERICAL FACTOR (DISREGARDING SIGNS) WITH VARIOUS REFERENCES

REF-ER-

ENCEL(p) L(0) L(1) L(2) L(3) L(4) L(5) G(p)a

b G(0)ab G(1)a

b G(2)ab G(3)a

b G(4)ab G(5)

[83] COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE COMPL[84] COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE[85] COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE[86] COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE Partial Partial COMPLETE COMPLETE COMPLETE COMPLETE Partial Partial Parti[87] COMPLETE COMPLETE COMPLETE COMPLETE Partial[88] COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE[89] COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE

[90] COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE[91] COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE[92] COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE[93] COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE Partial[94] COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE[95] COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE[96] COMPLETE COMPLETE COMPLETE COMPLETE[97] COMPLETE COMPLETE COMPLETE COMPLETE[98] COMPLETE COMPLETE COMPLETE COMPLETE[99] COMPLETE COMPLETE COMPLETE Partial Partial COMPLETE COMPLETE COMPLETE COMPLETE Partial

[100] COMPLETE COMPLETE COMPLETE Partial[101] COMPLETE COMPLETE COMPLETE[102] COMPLETE COMPLETE COMPLETE[103] COMPLETE COMPLETE COMPLETE[104] COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE[105] COMPLETE COMPLETE COMPLETE COMPLETE[106] COMPLETE COMPLETE COMPLETE Partial[107] COMPLETE COMPLETE COMPLETE Partial Partial[108] COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE[109] COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE[110] COMPLETE COMPLETE COMPLETE COMPLETE[111] COMPLETE COMPLETE COMPLETE Partial Partial[112] COMPLETE COMPLETE COMPLETE COMPLETE[113] COMPLETE COMPLETE COMPLETE[114] COMPLETE COMPLETE Partial Partial Partial[115] COMPLETE COMPLETE Partial COMPLETE[116] COMPLETE COMPLETE Partial COMPLETE[117] COMPLETE COMPLETE Partial Partial[118] COMPLETE COMPLETE COMPLETE COMPLETE[119] COMPLETE COMPLETE COMPLETE COMPLETE

......

......

......

......

......

......

......

...

(Continued on Next Page)


9/54

Z-26 Wednesday, March 22, 2000

TABLE 3. OVERVIEW OF AGREEMENT UP TO AN OVERALL NUMERICAL FACTOR (DISREGARDING SIGNS) WITH VARIOUS REFERENCESContinued

REF-ER-


b G(0)ab G(1)a

b G(2)ab G(3)a

b G(4)ab G(5)

......

......

......

......

......

......

......

...[120] COMPLETE COMPLETE COMPLETE[121] COMPLETE COMPLETE[122] COMPLETE COMPLETE[123] COMPLETE COMPLETE[124] COMPLETE COMPLETE[125] COMPLETE COMPLETE[126] COMPLETE COMPLETE[127] COMPLETE COMPLETE[128] COMPLETE COMPLETE[129] COMPLETE COMPLETE COMPLETE COMPLETE[130] COMPLETE COMPLETE COMPLETE[131] COMPLETE COMPLETE COMPLETE[132] COMPLETE COMPLETE Partial[133] COMPLETE COMPLETE COMPLETE[134] COMPLETE COMPLETE[135] COMPLETE Partial Partial Partial[136] COMPLETE Partial[137] COMPLETE COMPLETE COMPLETE COMPL[138] COMPLETE COMPLETE COMPLETE[139] COMPLETE COMPLETE Partial[140] COMPLETE COMPLETE[141] COMPLETE COMPLETE

[142] COMPLETE Partial Partial[143] COMPLETE Partial[144] COMPLETE Partial[145] COMPLETE[146] COMPLETE[147] COMPLETE[148] COMPLETE[149] COMPLETE[150] COMPLETE[151] COMPLETE[152] COMPLETE[153] COMPLETE[154] COMPLETE[155] COMPLETE[156] COMPLETE[157] COMPLETE

[158] COMPLETE[159] COMPLETE[160] COMPLETE[161] COMPLETE[162] COMPLETE[163] COMPLETE[164] COMPLETE[165] COMPLETE[166] COMPLETE[167] COMPLETE[168] COMPLETE[169] COMPLETE[170] COMPLETE[171] Partial [172] Partial[173] COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE Partial[174] COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE[175] COMPLETE COMPLETE COMPLETE Partial COMPLETE COMPLETE[176] COMPLETE COMPLETE COMPLETE Partial Partial[177] COMPLETE COMPLETE COMPLETE Partial[178] COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE[179] COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE[180] COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE[181] COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE[182] COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE[183] COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE[184] COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE Partial[185] COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE Partial[186] COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE Partial[187] COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE Partial

......

......

......

......

......

......

......

...



10/54



REF-ER-


b G(0)ab G(1)a

b G(2)ab G(3)a

b G(4)ab G(5)

......

......

......

......

......

......

......

...[188] COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE Partial[189] COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE Partial[190] COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE[191] COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE[192] COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE[193] COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE[194] COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE[195] COMPLETE COMPLETE COMPLETE Partial COMPLETE[196] COMPLETE COMPLETE COMPLETE COMPLETE COMPLETE[197] COMPLETE COMPLETE COMPLETE COMPLETE Partial[198] COMPLETE COMPLETE COMPLETE COMPLETE[199] COMPLETE COMPLETE COMPLETE COMPLETE[200] COMPLETE COMPLETE COMPLETE Partial[201] COMPLETE COMPLETE COMPLETE[202] COMPLETE COMPLETE COMPLETE[203] COMPLETE COMPLETE COMPLETE[204] COMPLETE COMPLETE COMPLETE[205] COMPLETE COMPLETE COMPLETE[206] COMPLETE COMPLETE COMPLETE[207] COMPLETE COMPLETE COMPLETE[208] COMPLETE COMPLETE COMPLETE[209] COMPLETE COMPLETE COMPLETE

[210] COMPLETE COMPLETE COMPLETE[211] COMPLETE COMPLETE COMPLETE[212] COMPLETE COMPLETE COMPLETE[213] COMPLETE COMPLETE COMPLETE[214] COMPLETE COMPLETE COMPLETE[215] COMPLETE COMPLETE COMPLETE[216] COMPLETE COMPLETE Partial Partial Partial Partial COMPLETE COMPLETE Parti[217] COMPLETE COMPLETE Partial Partial Partial COMPLETE COMPLETE[218] COMPLETE COMPLETE Partial Partial Partial COMPLETE COMPLETE[219] COMPLETE COMPLETE Partial Partial Partial COMPLETE COMPLETE[220] COMPLETE COMPLETE Partial Partial Partial COMPLETE Partial[221] COMPLETE COMPLETE Partial Partial COMPLETE[222] COMPLETE COMPLETE Partial Partial Partial[223] COMPLETE COMPLETE Partial COMPLETE COMPLETE Partial[224] COMPLETE COMPLETE Partial COMPLETE COMPLETE Partial[225] COMPLETE COMPLETE Partial COMPLETE COMPLETE Partial

[226] COMPLETE COMPLETE Partial COMPLETE COMPLETE Partial[227] COMPLETE COMPLETE Partial COMPLETE COMPLETE Partial[228] COMPLETE COMPLETE Partial COMPLETE COMPLETE Partial[229] COMPLETE COMPLETE Partial COMPLETE COMPLETE Partial[230] COMPLETE COMPLETE Partial COMPLETE COMPLETE Partial[231] COMPLETE COMPLETE Partial COMPLETE COMPLETE Partial[232] COMPLETE COMPLETE Partial COMPLETE COMPLETE Partial[233] COMPLETE COMPLETE Partial COMPLETE COMPLETE[234] COMPLETE COMPLETE Partial COMPLETE COMPLETE[235] COMPLETE COMPLETE Partial COMPLETE COMPLETE[236] COMPLETE COMPLETE Partial COMPLETE COMPLETE[237] COMPLETE COMPLETE Partial COMPLETE COMPLETE[238] COMPLETE COMPLETE Partial COMPLETE COMPLETE[239] COMPLETE COMPLETE Partial COMPLETE COMPLETE[240] COMPLETE COMPLETE Partial COMPLETE COMPLETE[241] COMPLETE COMPLETE Partial COMPLETE Partial[242] COMPLETE COMPLETE Partial Partial COMPLETE[243] COMPLETE COMPLETE Partial COMPLETE Partial[244] COMPLETE COMPLETE Partial COMPLETE Partial[245] COMPLETE COMPLETE Partial Partial Partial[246] COMPLETE COMPLETE Partial Partial[247] COMPLETE COMPLETE Partial Partial[248] COMPLETE COMPLETE Partial Partial[249] COMPLETE COMPLETE Partial[250] COMPLETE COMPLETE Partial[251] COMPLETE COMPLETE Partial[252] COMPLETE COMPLETE Partial[253] COMPLETE COMPLETE Partial[254] COMPLETE COMPLETE COMPLETE COMPLETE Partial[255] COMPLETE COMPLETE COMPLETE COMPLETE

......

......

......

......

......

......

......

...



11/54



REF-ER-


b G(0)ab G(1)a

b G(2)ab G(3)a

b G(4)ab G(5)

......

......

......

......

......

......

......

...[256] COMPLETE COMPLETE COMPLETE COMPLETE[257] COMPLETE COMPLETE COMPLETE COMPLETE[258] COMPLETE COMPLETE COMPLETE COMPLETE[259] COMPLETE COMPLETE COMPLETE COMPLETE[260] COMPLETE COMPLETE COMPLETE COMPLETE[261] COMPLETE COMPLETE COMPLETE Partial Partial[262] COMPLETE COMPLETE COMPLETE Partial[263] COMPLETE COMPLETE COMPLETE[264] COMPLETE COMPLETE Partial Partial[265] COMPLETE COMPLETE Partial Partial[266] COMPLETE COMPLETE[267] COMPLETE COMPLETE[268] COMPLETE COMPLETE COMPLETE COMPLETE Partial[269] COMPLETE Partial COMPLETE COMPLETE[270] COMPLETE Partial COMPLETE[271] COMPLETE Partial COMPLETE[272] COMPLETE COMPLETE Partial[273] COMPLETE Partial Partial[274] COMPLETE Partial[275] COMPLETE[276] COMPLETE[277] COMPLETE COMPLETE COMPLETE Partial COMPLETE COMPLETE

[278] COMPLETE COMPLETE COMPLETE Partial[279] COMPLETE COMPLETE Partial Partial COMPLETE[280] COMPLETE COMPLETE Partial Partial[281] COMPLETE COMPLETE Partial Partial[282] COMPLETE COMPLETE Partial Partial COMPLETE Partial Partial[283] COMPLETE COMPLETE Partial Partial Partial[284] COMPLETE COMPLETE Partial Partial Partial[285] COMPLETE COMPLETE Partial COMPLETE COMPLETE Partial[286] COMPLETE COMPLETE Partial Partial[287] COMPLETE COMPLETE Partial Partial[288] COMPLETE COMPLETE Partial[289] COMPLETE COMPLETE Partial[290] COMPLETE COMPLETE Partial[291] COMPLETE COMPLETE Partial[292] COMPLETE COMPLETE Partial[293] COMPLETE COMPLETE Partial

[294] COMPLETE COMPLETE Partial[295] COMPLETE COMPLETE Partial[296] COMPLETE COMPLETE Partial Partial[297] COMPLETE COMPLETE Partial Partial[298] COMPLETE COMPLETE COMPLETE COMPLETE[299] COMPLETE COMPLETE COMPLETE COMPLETE Partial[300] COMPLETE COMPLETE COMPLETE Partial Partial[301] COMPLETE COMPLETE Partial COMPLETE[302] COMPLETE COMPLETE Partial Partial[303] COMPLETE COMPLETE Partial[304] COMPLETE COMPLETE COMPLETE COMPLETE[305] COMPLETE COMPLETE COMPLETE COMPLETE[306] COMPLETE COMPLETE COMPLETE COMPLETE[307] COMPLETE COMPLETE COMPLETE COMPLETE[308] COMPLETE COMPLETE COMPLETE COMPLETE[309] COMPLETE COMPLETE COMPLETE COMPLETE[310] COMPLETE COMPLETE COMPLETE COMPLETE[311] COMPLETE COMPLETE COMPLETE COMPLETE[312] COMPLETE COMPLETE COMPLETE COMPLETE[313] COMPLETE COMPLETE COMPLETE COMPLETE[314] COMPLETE COMPLETE COMPLETE Partial[315] COMPLETE COMPLETE COMPLETE Partial[316] COMPLETE COMPLETE COMPLETE Partial[317] COMPLETE COMPLETE COMPLETE Partial[318] COMPLETE COMPLETE COMPLETE Partial[319] COMPLETE COMPLETE COMPLETE[320] COMPLETE COMPLETE COMPLETE[321] COMPLETE COMPLETE COMPLETE[322] COMPLETE COMPLETE COMPLETE[323] COMPLETE COMPLETE COMPLETE

......

......

......

......

......

......

......

...



12/54



REF-ER-


b G(0)ab G(1)a

b G(2)ab G(3)a

b G(4)ab G(5)

......

......

......

......

......

......

......

...[324] COMPLETE COMPLETE COMPLETE[325] COMPLETE COMPLETE Partial COMPLETE[326] COMPLETE COMPLETE Partial Partial[327] COMPLETE COMPLETE Partial Partial[328] COMPLETE COMPLETE Partial Partial[329] COMPLETE COMPLETE Partial[330] COMPLETE COMPLETE Partial[331] COMPLETE COMPLETE Partial[332] COMPLETE COMPLETE Partial[333] COMPLETE COMPLETE COMPLETE[334] COMPLETE COMPLETE COMPLETE[335] COMPLETE COMPLETE COMPLETE[336] COMPLETE COMPLETE COMPLETE[337] COMPLETE COMPLETE Partial[338] COMPLETE COMPLETE Partial[339] COMPLETE COMPLETE Partial[340] COMPLETE COMPLETE[341] COMPLETE COMPLETE[342] COMPLETE COMPLETE[343] COMPLETE COMPLETE[344] COMPLETE COMPLETE[345] COMPLETE COMPLETE

[346] COMPLETE COMPLETE[347] COMPLETE COMPLETE[348] COMPLETE COMPLETE[349] COMPLETE COMPLETE[350] COMPLETE COMPLETE[351] COMPLETE COMPLETE[352] COMPLETE COMPLETE[353] COMPLETE COMPLETE[354] COMPLETE COMPLETE[355] COMPLETE COMPLETE[356] COMPLETE COMPLETE[357] COMPLETE COMPLETE[358] COMPLETE COMPLETE[359] COMPLETE COMPLETE[360] COMPLETE COMPLETE[361] COMPLETE COMPLETE

[362] COMPLETE COMPLETE[363] COMPLETE COMPLETE[364] COMPLETE COMPLETE[365] COMPLETE COMPLETE[366] COMPLETE COMPLETE[367] COMPLETE COMPLETE[368] COMPLETE COMPLETE[369] COMPLETE Partial Partial Partial Partial COMPLETE Partial Partial[370] COMPLETE Partial Partial Partial Partial Partial Partial Partial[371] COMPLETE Partial Partial Partial Partial COMPLETE[372] COMPLETE Partial Partial Partial Partial COMPLETE[373] COMPLETE Partial Partial Partial Partial Partial[374] COMPLETE Partial Partial Partial COMPLETE Partial Partial[375] COMPLETE Partial Partial Partial Partial Partial[376] COMPLETE Partial Partial Partial COMPLETE Partial Partial[377] COMPLETE Partial Partial Partial COMPLETE Partial[378] COMPLETE Partial Partial Partial Partial Partial[379] COMPLETE Partial Partial Partial Partial Partial[380] COMPLETE Partial Partial Partial Partial[381] COMPLETE Partial Partial Partial[382] COMPLETE Partial Partial Partial COMPLETE Partial Partial[383] COMPLETE Partial Partial Partial[384] COMPLETE Partial Partial COMPLETE Partial Partial[385] COMPLETE Partial Partial COMPLETE[386] COMPLETE Partial Partial COMPLETE[387] COMPLETE Partial Partial COMPLETE[388] COMPLETE Partial Partial Partial[389] COMPLETE Partial Partial Partial[390] COMPLETE Partial Partial Partial[391] COMPLETE Partial Partial Partial Partial

......

......

......

......

......

......

......

...



13/54



REF-ER-


b G(0)ab G(1)a

b G(2)ab G(3)a

b G(4)ab G(5)

......

......

......

......

......

......

......

...[392] COMPLETE Partial Partial[393] COMPLETE Partial Partial[394] COMPLETE Partial Partial[395] COMPLETE Partial Partial[396] COMPLETE Partial COMPLETE COMPLETE Partial[397] COMPLETE Partial COMPLETE COMPLETE[398] COMPLETE Partial COMPLETE Partial[399] COMPLETE Partial COMPLETE[400] COMPLETE Partial Partial COMPLETE Partial[401] COMPLETE Partial Partial COMPLETE Partial[402] COMPLETE Partial Partial COMPLETE[403] COMPLETE Partial Partial COMPLETE[404] COMPLETE Partial Partial Partial Partial[405] COMPLETE Partial Partial Partial[406] COMPLETE Partial Partial Partial[407] COMPLETE Partial Partial[408] COMPLETE Partial COMPLETE Partial[409] COMPLETE Partial COMPLETE Partial[410] COMPLETE Partial COMPLETE Partial[411] COMPLETE Partial COMPLETE Partial[412] COMPLETE Partial COMPLETE Partial[413] COMPLETE Partial COMPLETE Partial

[414] COMPLETE Partial COMPLETE Partial[415] COMPLETE Partial COMPLETE Partial[416] COMPLETE Partial COMPLETE Partial[417] COMPLETE Partial COMPLETE Partial[418] COMPLETE Partial COMPLETE Partial[419] COMPLETE Partial COMPLETE Partial[420] COMPLETE Partial COMPLETE Partial[421] COMPLETE Partial COMPLETE Partial[422] COMPLETE Partial COMPLETE Partial[423] COMPLETE Partial COMPLETE Partial[424] COMPLETE Partial COMPLETE Partial[425] COMPLETE Partial COMPLETE Partial[426] COMPLETE Partial COMPLETE Partial[427] COMPLETE Partial COMPLETE Partial[428] COMPLETE Partial COMPLETE Partial[429] COMPLETE Partial COMPLETE Partial

[430] COMPLETE Partial COMPLETE[431] COMPLETE Partial COMPLETE[432] COMPLETE Partial COMPLETE[433] COMPLETE Partial COMPLETE[434] COMPLETE Partial COMPLETE[435] COMPLETE Partial COMPLETE[436] COMPLETE Partial COMPLETE[437] COMPLETE Partial COMPLETE[438] COMPLETE Partial COMPLETE[439] COMPLETE Partial COMPLETE[440] COMPLETE Partial COMPLETE[441] COMPLETE Partial COMPLETE[442] COMPLETE Partial Partial Partial[443] COMPLETE Partial Partial Partial[444] COMPLETE Partial Partial Partial[445] COMPLETE Partial Partial Partial[446] COMPLETE Partial Partial Partial[447] COMPLETE Partial Partial Partial[448] COMPLETE Partial Partial Partial[449] COMPLETE Partial Partial Partial[450] COMPLETE Partial Partial Partial[451] COMPLETE Partial Partial Partial[452] COMPLETE Partial Partial[453] COMPLETE Partial Partial[454] COMPLETE Partial Partial[455] COMPLETE Partial Partial[456] COMPLETE Partial Partial[457] COMPLETE Partial Partial[458] COMPLETE Partial Partial[459] COMPLETE Partial Partial

......

......

......

......

......

......

......

...



14/54



REF-ER-


b G(0)ab G(1)a

b G(2)ab G(3)a

b G(4)ab G(5)

......

......

......

......

......

......

......

...[460] COMPLETE Partial Partial[461] COMPLETE Partial Partial[462] COMPLETE Partial Partial[463] COMPLETE Partial Partial[464] COMPLETE Partial Partial[465] COMPLETE Partial Partial[466] COMPLETE Partial[467] COMPLETE Partial[468] COMPLETE Partial[469] COMPLETE Partial[470] COMPLETE Partial[471] COMPLETE Partial[472] COMPLETE Partial[473] COMPLETE Partial[474] COMPLETE Partial[475] COMPLETE Partial[476] COMPLETE Partial[477] COMPLETE Partial[478] COMPLETE Partial[479] COMPLETE Partial[480] COMPLETE Partial[481] COMPLETE Partial

[482] COMPLETE Partial[483] COMPLETE Partial[484] COMPLETE Partial[485] COMPLETE Partial[486] COMPLETE Partial[487] COMPLETE Partial[488] COMPLETE Partial[489] COMPLETE Partial[490] COMPLETE Partial[491] COMPLETE Partial[492] COMPLETE Partial[493] COMPLETE Partial[494] COMPLETE Partial Partial Partial Partial[495] COMPLETE Partial Partial Partial Partial Partial[496] COMPLETE Partial Partial Partial Partial[497] COMPLETE Partial Partial

[498] COMPLETE Partial Partial Partial[499] COMPLETE Partial Partial Partial[500] COMPLETE Partial Partial[501] COMPLETE Partial Partial Partial[502] COMPLETE Partial Partial[503] COMPLETE Partial[504] COMPLETE COMPLETE COMPLETE[505] COMPLETE COMPLETE COMPLETE[506] COMPLETE COMPLETE COMPLETE[507] COMPLETE COMPLETE COMPLETE[508] COMPLETE COMPLETE Partial[509] COMPLETE COMPLETE Partial[510] COMPLETE Partial COMPLETE[511] COMPLETE Partial COMPLETE[512] COMPLETE Partial COMPLETE[513] COMPLETE Partial COMPLETE[514] COMPLETE Partial COMPLETE[515] COMPLETE Partial COMPLETE[516] COMPLETE Partial COMPLETE[517] COMPLETE Partial COMPLETE[518] COMPLETE Partial COMPLETE[519] COMPLETE Partial COMPLETE[520] COMPLETE Partial COMPLETE[521] COMPLETE Partial COMPLETE[522] COMPLETE Partial COMPLETE[523] COMPLETE Partial COMPLETE[524] COMPLETE Partial Partial Partial[525] COMPLETE Partial Partial Partial[526] COMPLETE Partial Partial[527] COMPLETE Partial Partial

......

......

......

......

......

......

......

...



15/54



REF-ER-


b G(0)ab G(1)a

b G(2)ab G(3)a

b G(4)ab G(5)

......

......

......

......

......

......

......

...[528] COMPLETE Partial Partial[529] COMPLETE Partial Partial[530] COMPLETE Partial Partial[531] COMPLETE Partial Partial[532] COMPLETE Partial[533] COMPLETE Partial[534] COMPLETE Partial[535] COMPLETE Partial[536] COMPLETE Partial[537] COMPLETE Partial[538] COMPLETE Partial[539] COMPLETE COMPLETE Partial[540] COMPLETE COMPLETE Partial[541] COMPLETE COMPLETE[542] COMPLETE COMPLETE[543] COMPLETE COMPLETE[544] COMPLETE COMPLETE[545] COMPLETE COMPLETE[546] COMPLETE COMPLETE[547] COMPLETE COMPLETE[548] COMPLETE COMPLETE[549] COMPLETE COMPLETE

[550] COMPLETE COMPLETE[551] COMPLETE COMPLETE[552] COMPLETE COMPLETE[553] COMPLETE COMPLETE[554] COMPLETE COMPLETE[555] COMPLETE COMPLETE[556] COMPLETE Partial Partial[557] COMPLETE Partial Partial[558] COMPLETE Partial Partial[559] COMPLETE Partial[560] COMPLETE Partial[561] COMPLETE Partial[562] COMPLETE Partial[563] COMPLETE Partial[564] COMPLETE[565] COMPLETE

[566] COMPLETE[567] COMPLETE[568] COMPLETE[569] COMPLETE[570] COMPLETE[571] COMPLETE[572] COMPLETE[573] COMPLETE[574] COMPLETE[575] COMPLETE[576] COMPLETE[577] COMPLETE[578] COMPLETE[579] COMPLETE[580] COMPLETE Partial Partial COMPLETE COMPLETE[581] Partial COMPLETE[582] Partial Partial Partial[583] Partial Partial[584] Partial Partial[585] COMPLETE Partial COMPLETE Partial[586] COMPLETE Partial[587] COMPLETE Partial[588] COMPLETE Partial[589] COMPLETE Partial[590] COMPLETE Partial[591] COMPLETE COMPLETE COMPLETE[592] COMPLETE COMPLETE COMPLETE[593] COMPLETE COMPLETE Partial Partial[594] COMPLETE Partial Partial Partial[595] COMPLETE Partial

......

......

......

......

......

......

......

...



16/54



REF-ER-


b G(0)ab G(1)a

b G(2)ab G(3)a

b G(4)ab G(5)

......

......

......

......

......

......

......

...[596] COMPLETE COMPLETE COMPLETE[597] COMPLETE COMPLETE COMPLETE[598] COMPLETE Partial[599] COMPLETE COMPLETE[600] COMPLETE COMPLETE[601] COMPLETE Partial[602] COMPLETE Partial[603] COMPLETE[604] COMPLETE[605] COMPLETE[606] COMPLETE[607] COMPLETE[608] COMPLETE[609] COMPLETE[610] COMPLETE[611] COMPLETE[612] COMPLETE[613] COMPLETE[614] COMPLETE[615] COMPLETE[616] COMPLETE[617] COMPLETE

[618] Partial Partial Partial Partial Partial Partial Partial[619] Partial Partial COMPLETE Partial Parti[620] Partial Partial COMPLETE[621] Partial Partial[622] Partial COMPLETE COMPLETE[623] Partial COMPLETE Partial Partial[624] Partial COMPLETE Partial Partial[625] Partial COMPLETE Partial Partial[626] Partial COMPLETE Partial[627] Partial COMPLETE Partial[628] Partial COMPLETE Partial[629] Partial Partial Partial[630] Partial Partial Partial[631] Partial Partial Partial[632] Partial Partial Partial[633] Partial Partial

[634] Partial Partial[635] Partial Partial[636] Partial COMPLETE[637] Partial COMPLETE[638] Partial COMPLETE[639] Partial COMPLETE[640] Partial COMPLETE[641] Partial COMPLETE[642] Partial COMPLETE[643] Partial Partial[644] Partial Partial[645] Partial Partial[646] Partial Partial[647] Partial Partial[648] Partial[649] Partial[650] Partial[651] Partial[652] Partial[653] Partial[654] Partial[655] Partial[656] Partial[657] Partial[658] Partial[659] Partial[660] Partial[661] Partial[662] Partial Partial Partial[663] Partial Partial

......

......

......

......

......

......

......

...



17/54



REF-ER-


b G(0)ab G(1)a

b G(2)ab G(3)a

b G(4)ab G(5)

......

......

......

......

......

......

......

...[664] Partial[665] Partial[666] Partial[667] Partial[668] Partial[669] COMPLETE COMPLETE COMPLETE[670] COMPLETE[671] Partial COMPLETE COMPLETE COMPLETE[672] Partial COMPLETE COMPLETE[673] COMPLETE COMPLETE[674] COMPLETE COMPLETE[675] COMPLETE COMPLETE[676] COMPLETE Partial[677] COMPLETE Partial[678] COMPLETE Partial[679] COMPLETE Partial[680] COMPLETE Partial[681] Partial COMPLETE[682] Partial COMPLETE[683] Partial COMPLETE[684] Partial Partial Partial[685] Partial Partial

[686] Partial Partial[687] Partial Partial[688] Partial Partial[689] Partial Partial[690] Partial Partial[691] Partial Partial[692] Partial Partial[693] Partial Partial[694] Partial Partial[695] Partial Partial[696] Partial Partial[697] Partial Partial[698] Partial Partial[699] Partial Partial[700] Partial Partial[701] Partial Partial

[702] Partial Partial[703] Partial Partial[704] Partial Partial[705] Partial Partial[706] Partial Partial[707] Partial[708] Partial[709] Partial[710] Partial[711] Partial[712] Partial[713] Partial[714] Partial[715] Partial[716] Partial[717] Partial[718] Partial[719] Partial[720] Partial[721] Partial[722] COMPLETE COMPLETE Partial[723] COMPLETE Partial[724] COMPLETE Partial[725] COMPLETE Partial[726] COMPLETE Partial[727] COMPLETE[728] COMPLETE[729] COMPLETE[730] COMPLETE[731] COMPLETE

......

......

......

......

......

......

......

...



18/54



REF-ER-


b G(0)ab G(1)a

b G(2)ab G(3)a

b G(4)ab G(5)

......

......

......

......

......

......

......

...[732] COMPLETE[733] Partial Partial[734] Partial Partial[735] Partial[736] Partial[737] Partial[738] Partial[739] Partial[740] Partial[741] Partial[742] Partial[743] COMPLETE[744] COMPLETE[745] COMPLETE[746] Partial[747] Partial[748][749][750][751][752][753]

[754][755][756][757][758][759][760][761][762][763][764][765][766][767][768][769]

[770]

TABLE NOTES83 [This paper].84 Briggs, C. C., A General Expression for the Quartic Lovelock Tensor, Los Alamos

preprint no. gr-qc/9703074.85 Lovelock, D., Intrinsic Expressions for Curvatures of Even Order of Hypersurfaces in

a Euclidean Space, Tensor, New Series, 22(1971) 274.86 Briggs, C. C., Some Possible Features of General Expressions for Lovelock Tensors

and for the Coefficients of Lovelock Lagrangians Up to the 15thOrder in Curvature (andBeyond), Los Alamos preprint no. gr-qc/9808050.

In Briggs, L(4) partially agrees, insofar as the concomitants belonging to the 1st

through 10thterms ofL(4)per Eq. (16) occur withmoreoverthe magnitudes of theirnumerical coefficients in a ratio of 1:24:6:64:96:96:8:32:48:96, which completelyagrees; L(5)partially agrees, insofar as the concomitants belonging to the 1

st through10 th terms of L(5) per Eq. (17) occur withmoreoverthe magnitudes of their

numerical coefficients in a ratio of 1:40:10:160:240:240:20:80:240:480, whichcompletely agrees; G(3)abpartially agrees, insofar as the concomitants belonging to the

1stthrough 10thterms of G(3)abper Eq. (27) occur withmoreoverthe magnitudes of

their numerical coefficients in a ratio of 1:12:3:16:24:24:2:8:6:24, which completelyagrees; G(4)a

bpartially agrees, insofar as the concomitants belonging to the 1stthrough10 thterms of G(4) a

b per Eq. (28) occur withmoreoverthe magnitudes of theirnumerical coefficients in a ratio of 1:24:6:64:96:96:8:32:48:96, which completelyagrees; and G(5)a

b partially agrees, insofar as the concomitants belonging to the 1st

through 10thterms of G(5)abper Eq. (29) occur withmoreoverthe magnitudes of

their numerical coefficients in a ratio of 1:40:10:160:240:240:20:80:240:480, whichcompletely agrees.

87 Gray, A., Tubes, Addison-Wesley Pub. Co., Redwood City, CA (1990), pp. 63-64, 72,and 78.

In Gray,L(3)partially agrees, insofar as the concomitants belonging to the 1stthrough

8thterms ofL(3)per Eq. (15) occur.88 Rund, H., and D. Lovelock, Variational Principles in the General Theory of

Relativity,Jahresbericht der Deutschen Mathematische-Vereinigung, 74(1972) 1.

(Continued in Next Column)

TABLE NOTESContinued89 Lovelock, D., and H. Rund, Tensors, Differential Forms, and Variational Princ

John Wiley & Sons, New York, NY (1975), pp. 293, 296, and 322.90 Wurmser, D., Vacuum state and Schwarzschild solution in ten-dimensional gra

Phys. Rev. D, 36 (1987) 2970.91 Lovelock, D., and H. Rund, Variational principles in the general theory of relati

in Vanstone, J. R. (ed.), ProceedingsoftheThirteenthBie nni al SeminarCanadianMathematicalCongressonDifferentialTopology;DifferentialGeomet

Applications; Volume1, Canadian Mathematical Congress, Montreal, Canada (1p. 51.

92 Deruelle, N., and J. Madore, On the Vanishing of the Cosmological Constant,Lett ., 114A (1986) 185.

93 Faria-Busto, L., Some new cosmological results of quadratic Lagrangians,Rev. D, 38 (1988) 1741.

In Faria-Busto, G(2)ab

partially agrees, insofar as (A) the concomitants belongthe 1stand 4thterms of G(2)abper Eq. (26) occur withmoreoverthe magnitud

their numerical coefficients in a ratio of 1:4, which completely agrees, as (Bconcomitants belonging to the 2ndand 5thterms of G(2)a

bper Eq. (26) occur, buthe magnitudes of their numerical coefficients in a ratio of 1:4 instead of 1:2, and the concomitants belonging to the 3rdand 7thterms of G(2)a

bper Eq. (26) occur wmoreoverthe magnitudes of their numerical coefficients in a ratio of 1:4, completely agrees.

94 Rund, H., Invariant theory of variational problems on subspaces of a Riemamanifold, Hamburger Mathematische Einzelschriften (Neue Folge) , no. 5 ([Herausgegeben von Mathematischen Seminar der Universitt Hamburg, Vandenand Ruprecht, Gttingen, Germany (1971)], pp. 12, 41, 42, 53, and 54.

95 Wiltshire, D. L., Spherically Symmetric Solutions of Einstein-Maxwell Theorya Gauss-Bonnet Term, Phys. Lett., 169B(1986) 36.

96 Giorgini, B., and R. Kerner, Cosmology in ten dimensions with the genergravitational Lagrangian, Class. Quantum Grav., 5(1988) 339.

97 Gnther, P., and R. Schimming, Curvature and Spectrum of Compact Riema



19/54


TABLE NOTESContinuedManifolds,Journ. Diff. Geom., 12(1977) 599.

98 Kerner, R., Discussion (Papers IV.4.-IV.5.), discussion on papers nos. iv.4 and iv.5,i.e. Deruelle, N., and J. Madore, Kaluza-Klein Cosmology with the LovelockLagrangian, and Kerner, R., Kaluza-Klein Cosmology with DoubleCompactification, respectively, in Origin an dEarlyHistoryoftheUniverse;Proceedingsofthe26thLigeInternationalAstrophysicalColloquium,July1-4,1986,Universite de Lie ge [sic],Ins tit utdAstrophysique, Liege [sic],Belgiq ue , Liege,Belgium (1987), pp. 271 and 277, respectively.

99 Davis, S., Symmetric Variations of the Metric and Extrema of the Action for PureGravity, Univ. of Cambridge preprint DAMTP-R/96/7, Los Alamos preprint no.gr-qc/9608025.

In Davis,L(2)partially agrees, insofar as an analogue of the concomitant belonging to

the 3rdterm ofL(2)per Eq. (14) occurs;L(4)partially agrees, insofar as (A) analogues ofthe concomitants belonging to the 20 ththrough 25thterms of L(4)per Eq. (16) occur,but with the magnitudes of their numerical coefficients in a ratio of 1:8:1:1:8:1 insteadof 1:16:2:32:16:32 and with the analogue of the concomitant belonging to the 23rd

term of L(4) per Eq. (16) having an unnatural arrangement of indices, and as (B)analogues of the concomitants belonging to the 23rdand 25thterms ofL(4)per Eq. (16)occur, but with the magnitudes of their numerical coefficients in a ratio of 1:2 insteadof 1:1 and with the analogue of the concomitant belonging to the 23rdterm ofL(4)perEq. (16) having an unnatural arrangement of indices; and G(4)a

bpartially agrees, insofaras (A) the 1stfour terms in Daviss Eq. (73) are sums of analogues of the 107 thand111 thterms of G(4)a

bper Eq. (28) and as (B) the last term in Daviss Eq. (73) is ananalogue of the 109thterm of G(4)a

bper Eq. (28).00 Rund, H., Integral Formulae on Hypersurfaces in Riemannian Manifolds, Annali di

Matematica Pura ed Applicata (4), 88(1971) 99.In Rund, G(1)a

bpartially agrees, insofar as the concomitant belonging to the 2 ndtermof G(1)a

bper Eq. (25) occurs.01 Jacobson, T., and R. C. Myers, Black Hole Entropy and Higher-Curvature

Interactions, Phys. Rev. Lett., 70(1993) 3684.02 Jacobson, T., and R. C. Myers, Entropy of Lovelock Blackholes, Phys.Rev.Lett. ,70(1993) 3684.

03 Baados, M., C. Teitelboim, and J. Zanelli, Dimensionally Continued Black Holes,Los Alamos preprint no. gr-qc/9307033; Phys. Rev. D, 49(1994) 975.

04 Choquet-Bruhat, Y., High Frequency Gauss-Bonnet Gravity, in Ferrarese, G. (ed.),ClassicalMec han ics andRelativit y: Relationship andConsistency , Bibliopolis,edizioni di filosofia e scienze, Napoli, Italy (1991), p. 57.

05 Lovelock, D., The Einstein Tensor and Its Generalizations, Journ. Math. Phys. , 12(1971) 498.

06 Goroff, M. H., and A. Sagnotti, The Ultraviolet Behavior of Einstein Gravity, Nucl.Phys., B266 (1986) 709.

In Goroff and Sagnotti,L(3)partially agrees, insofar as the concomitants belonging tothe 1stthrough 8thterms ofL(3)per Eq. (15) occur.

07 Davis, S., Higher-Derivative Quantum Cosmology, Los Alamos preprint no.gr-qc/9911021.

In Davis, G(1)abpartially agrees, insofar as the concomitant belonging to the 2ndterm

of G(1)abper Eq. (25) occurs; and L(5)partially agrees, insofar as it is mentioned byname.

08 Deruelle, N., and J. Madore, The Friedmann Universe as an Attractor of aKaluza-Klein Cosmology, Mod. Phys. Let t. A, 1(1986) 237.

09 Farhoudi, M., Lovelock Tensor as Generalized Einstein Tensor, University of Londonpreprint no. QMW-PH-95-37, Los Alamos preprint no. gr-qc/9510060.

10 Bertrand, C., R. Kerner, and S. Mignemi, Generalization of Mantons Construction ofthe Weinberg-Salam Model with a Gauss-Bonnet Term, Int. Journ. of Mod. Phys. A,7(1992) 7741.

11 Kleidis, K., and D. B. Papadopoulos, Particle creation, renormalizability conditionsand the mass-energy spectrum in gravity theories of quadratic Lagrangians, LosAlamos preprint no. gr-qc/9905041.

In Kleidis and Papadopoulos, G (1)ab partially agrees, insofar as the concomitant

belonging to the 2ndterm of G (1)ab per Eq. (25) is mentioned; and G (2)a

bpartiallyagrees, insofar as the concomitants belonging to the 5 ththrough 7thterms of G(2)a

bperEq. (26) occur, but with the magnitudes of their numerical coefficients in a ratio of2:1:1 instead of 2:2:1.

12 Madore, J., Cosmological applications of the Lanczos Lagrangian, Class. QuantumGrav., 3(1986) 361.

13 Kerner, R., Cosmology and Kaluza-Klein Theories, in Garca, P. L., and A.Prez-Rendn (eds.), Dif fere ntialGeometric Met hodsin Mat hemati cal Physics;Proceedingsofthe14thInternationalConferenceheld[sic] inSalamanca,Spain,June24-29, 1985, No. 1251 of Dold, A., and B. Eckmann (eds.), LectureNo te s in

Mathematics, Springer-Verlag, Berlin, Germany (1987), p. 150.14 Fradkin, E. S., and A. A. Tseytlin, Quantum Properties of Higher Dimensional and

Dimensionally Reduced Supersymmetric Theories,Nucl. Phys., B227(1983) 252.In Fradkin and Tseytlin, L(3)partially agrees, insofar asaside from typos, i.e. in

Fradkin and Tseytlins Eq. (2.9), one of the contravariant indices M should becovariant and one of the covariant indices P should be contravariant in the analogue ofthe concomitant belonging to the 8thterm of L(3) per Eq. (15), and in Fradkin andTseytlins Eq. (A.5), one of the contravariant indices M should be covariant and oneof the covariant indices P should be contravariant in Fradkin and Tseytlins scalarI2,which corresponds to the 8thterm ofL(3)per Eq. (15), and one of the covariant indicesK should be contravariant and one of the covariant indices S should be contravariant


TABLE NOTESContinuedin Fradkin and Tseytlins scalarI4, which also has an unnatural arrangement of iand which corresponds either to both the 7 thand the 8thterms ofL(3)per Eq. (else to neitheranalogues of the concomitants belonging to the 7 thand 8thter

L(3)per Eq. (15) occur, but with the magnitudes of their numerical coefficientratio of 17:28 instead of 1:4; and G(1)a

bpartially agrees, insofar as the concombelonging to the 2ndterm of G(1)a

bper Eq. (25) occurs.115 Fabris, J. C., and R. Kerner, Gnralisation de la supergravit 11 dimensions

les invariants dEuler,Helvetica Physica Acta, 62(1989) 427.In Fabris and Kerner, L(2) partially agrees, insofar asaside from a typo, i.

symbol T appears instead of R in the 1st covariant Ricci curvature tensor appearing in the 2ndterm in the expression for L(2)in Fabris and Kerners Eq. which otherwise contains the concomitant belonging to the 2 ndterm of L(2)p

(14)the concomitants belonging to the 1st through 3rdterms of L(2)per Eqoccur withmoreoverthe magnitudes of their numerical coefficients in a ra1:4:1, which completely agrees.

116 Jacobson, T., G. Kang, and R. C. Myers, On Black Hole Entropy, Los Apreprint no. gr-qc/9312023, Phys.Rev.D, 49(1994) 6587.

In Jacobson etal.,L(2)partially agrees, insofar as the concomitants belonging 1st through 3rd terms of L (2) per Eq. (14) occur, but with arbitrary numcoefficients.

117 Cai, R-G., and K.-S. Soh, Topological black holes in the dimensionally contgravity, Los Alamos preprint no. gr-qc/9808067.

In Cai and Soh,L(2)partially agrees, insofar as the concomitants belonging to tand 3rdterms of L(2)per Eq. (14) occur; and G (1)a

b partially agrees, insofar concomitant belonging to the 2ndterm of G(1)a

bper Eq. (25) occurs.118 Lovelock, D., On the Non-Decomposability of Certain Degenerate Lag

Densities, Tensor, New Series, 23(1972) 362.119 Vasilic, M., Euler Forms in Kaluza-Klein Theories, Il Nuovo Cimento,

(1994) 1083.

120 It, K., (ed.), Encyclopedic Dictionary of Mathematics, Second Edition, bMathematical Society of Japan, Volume II, O-Z, Appendices and Indexes[sicMIT Press, Cambridge, MA (1993), pp. 1333, 1350, and 1353.

121 Kleidis, K., A. Kuiroukidis, D. B. Papadopoulos, and H. VarvoHigher-dimensional models in gravitational theories of quartic Lagrangians,Alamos preprint no. gr-qc/9905042.

122 Horndeski, G. W., Dimensionally dependent divergences, Proceedings oCambridge Philosophical Society, 72(1972) 77.

123 Thorpe, J. A., Some Remarks on the Gauss-Bonnet Integral, Journ. Math. M18(1969) 779.

124 Baados, M., C. Teitelboim, and J. Zanelli, Black Hole Entropy and the Gauss-BTheorem, Phys. Rev. Lett., 72(1994) 957.

125 Teitelboim, C., and J. Zanelli, Dimensionally continued topological gravitheory in Hamiltonian form, Class.QuantumGrav., 4(1987) L125.

126 Baados, M., C. Teitelboim, and J. Zanelli, Black Hole Entropy and the DimenContinuation of the Gauss-Bonnet Theorem, Los Alamos preprint no. gr-qc/930Phys.Rev. Lett. , 72 (1994) 957.

127 Teitelboim, C., Topological Roots of Black Hole Entropy, in Brown, J. D., Chu, D. C. Ellison, and R. J. Plemmons (eds.), Proceedings oftheCorneliusLa

InternationalCentenaryConference,Raleigh,NorthCarolina,December12-17,Society for Industrial and Applied Mathematics, Philadelphia, PA (1993), p. 223.

128 Pavelle, R., A decomposition identity for a class of Riemannian invariaProceedings of the Cambridge Philosophical Society, 72(1972) 459.

129 Buchdahl, H. A., On functionally constant invariants of the Riemann tenProceedings of the Cambridge Philosophical Society , 68 (1970) 179 [G(p)a

bgiven correctly in Eq. (4.6) of Buchdahl as opposed to Eq. (4.7) of Buchdahl, whictwo typos, i.e. a subscript is missing and the Kronecker delta is too small].

130 Fabris, J. C., On the Stability of Homogeneous Solutions of the Gravity TheoryDimensions with Euler Invariants, Phys. Lett., 223B(1989) 144.

131 Dubois-Violette, M., and J. Madore, Conservation Laws and Integrability Condfor Gravitational and Yang-Mills Field Equations, Commun. Math. Phys(1987) 213.

132 Levine, J., and J. D. Zund, The Euler-Poincar and Pontrjagin Characteristic Cof Psuedo-Riemannian Manifolds, Tensor, New Series, 21(1970) 250

In Levine and Zund, G(1)abpartially agrees, insofar as analogues of the concombelonging to the 1stand 2ndterms of G(1)a

bper Eq. (25) occur, but with the magnof their numerical coefficients in a ratio of 1:4 instead of 1:2.

133 Myers, R. C., Higher-derivative gravity, surface terms, and string theory, PhysD, 36 (1987) 392.

134 van Nieuwenhuizen, P., and C. C. Wu, On integral relations for invariants constfrom three Riemann tensors and their applications in quantum gravity, Journ. Phys., 18 (1977) 182.

135 Avez, A., Formule de Gauss-Bonnet-Chern en mtrique de signature quelconComptes rendus hebdomadaires des sances de lAcadmie des Sciences (Paris)(1962) 2049.

In Avez,L(2)partially agrees, insofar as the concomitant belonging to the 3rdte

L(2)per Eq. (14) occurs; G(0)abpartially agrees, insofar as the concomitant belong

the 1stterm of G(0)abper Eq. (24) occurs; and G(1)a

b partially agrees, insofar concomitant belonging to the 2ndterm of G(1)a

bper Eq. (25) occurs.136 Nishino, H., and S. J. Gates, Jr., Euler Characteristics in New D= 10,

Superspace Supergravity, Nucl. Phys. , B282(1987) 1.



20/54


TABLE NOTESContinuedIn Nishino and Gates, L(2)partially agrees, insofar as the concomitant belonging to

the 3rdterm ofL(2)per Eq. (14) is mentioned.37 Briggs, C. C., A General Expression for the Quintic Lovelock Tensor, Los Alamos

preprint no. gr-qc/9607033.38 Farhoudi, M., Classical Trace Anomaly, University of London preprint no.

QMW-PH-95-41, Los Alamos preprint no. gr-qc/9511047.39 Deser, S., and Z. Yang, Energy and stability in Einstein-Gauss-Bonnet models,

Class.QuantumGrav., 6(1989) L83.In Deser and Yang, G(1)a

bpartially agrees, insofar as the concomitant belonging to the2ndterm of G(1)a

bper Eq. (25) occurs.40 Arik, M., The Analogue Potential of Kaluza-Klein Cosmology with Euler Forms,

Nucl. Phys., B328 (1989) 308.

41 Ishikawa, K., Mass Spectrum of the M4SD Solution in Euler Invariant TypeHigher Derivative Gravity, Phys. Lett., 188B(1987) 186.

42 Allendoerfer, C. B., Global Theorems Riemannian Geometry, Bull. Amer. Math.Soc., 54 (1948) 249.

In Allendoerfer, G(0)abpartially agrees, insofar as the concomitant belonging to the 1st

term of G (0 )ab per Eq. (24) occurs; and G (1) a

b partially agrees, insofar as theconcomitant belonging to the 2ndterm of G(1)a

bper Eq. (25) occurs.43 Allendoerfer, C. B., Characteristic Cohomology Classes in a Riemann Manifold,

Annals of Mathematics (2), 51(1950) 551.In Allendoerfer, G(0)a

bpartially agrees, insofar as the concomitant belonging to the 1st

term of G(0)abper Eq. (24) occurs.

44 Allendoerfer, C. B., and A. Weil, The Gauss-Bonnet Theorem for RiemannianPolyhedra, Trans. Amer. Math. Soc., 53(1943) 101.

In Allendoerfer and Weil, G(0)abpartially agrees, insofar as the concomitant belonging

to the 1stterm of G(0)abper Eq. (24) occurs.

45 Chern, S.-S., The Geometry of G-Structures, Bull. Amer. Math. Soc., 72 (1966)167.

46 Chern, S.-S., Differential Geometry; Its Past and Its Future, Actes, Congrs intern.math., 1(1970) 41.

47 Arik, M., E. Hizel, and A. Mostafazadeh, The Schwarzschild solution in non-AbelianKaluza-Klein theory, Class. Quantum Grav., 7(1990) 1425.

48 Nijenhuis, A., On Cherns Kinematic Formula in Integral Geometry, Journ. Diff.Geom., 6(1974) 475.

49 Weinstein, A., Remarks on Curvature and the Euler Integrand,Journ. Diff. Geom., 6(1971) 259.

50 Nasu, T., On Conformal Invariants of Higher Order, Hiroshima Math. Journ., 5(1975) 43.

51 Tomonaga, Y., Euler-Poincar Characteristic and Sectional Curvature, in Kobayashi,S., M. Obata, and T. Takahashi (eds.), Differentia l Geometry, in Honor of K. Yano,Kinokuniya Book-Store Co., Ltd., Tokyo, Japan (1972), p. 501.

52 Dowker, J. S., and J. P. Schofield, Conformal transformations and the effective actionin the presence of boundaries,Journ. Math. Phys., 31(1990) 808.

53 Mller-Hoissen, F., Gravity Actions, Boundary Terms and Second-Order FieldEquations, Nucl. Phys., B337(1990) 709.

54 de Roo, M., H. Suelmann, and A. Wiedemann, Supersymmetric R4-actions in tendimensions, Phys. Lett., 280B(1992) 39.

55 fl, .. , ,Izvestija Akademii Nauk SSSR, Serija Matematicheskaja(fl , flfl), 13(1949) 125.

56 Pontrjagin, L. S., On Some Topologic Invariants of Riemannian Manifolds,Comptes Rendus (Doklady) de lAcadmie des Sciences de lURSS, 35(1944) 91.

57 Pinl, M., and H. W. Trapp, Stationre Krmmungsdichten auf Hyperflchen deseuklidischenRn+ 1,Mathematische Annalen, 176(1968) 257.

58 Weyl, H., On the Volume of Tubes, Amer. Journ. of Math., 61(1939) 461.59 Jacobowitz, H., Curvature Operators on the Exterior Algebra, Lin. Mult. Alg., 7

(1979) 93.60 Thorpe, J. A., Sectional Curvatures and Characteristic Classes, Anna ls of

Mathematics, 80(1964) 429.61 Cheung, Y. K., and C. C. Hsiung, Curvature and Characteristic Classes of Compact

Riemannian Manifolds,Journ. Diff. Geom. , 1(1967) 89.62 Patodi, V. K., Curvature and the Eigenforms of the Laplace Operator, Journ. Diff.

Geom., 8(1971) 233.63 Alvarez-Gaum, L., Supersymmetry and the Atiya-Singer Index Theorem, Commun.

Math. Phys., 90 (1983) 161.64 Willmore, T. J., Les plans parallles dans les espaces riemanniens globaux, Comptes

rendus hebdomadaires des sances de lAcademie des Sciences (Paris), 232(1951) 298.65 Allendoerfer, C. B., The Euler number of a Riemann manifold,Amer. Journ. Math.,

62(1940) 243.66 Arik, M., and T. Dereli, Euler-Form Actions and the Vanishing of the Cosmological

Constant, Phys. Rev. Lett., 62(1988) 5.67 Dorey, N., V. V. Khoze, and M. P. Mattis, Multi-Instantons, Three-Dimensional

Gauge Theory, and the Gauss-Bonnet-Chern Theorem,Nucl.Phys., B502(1997) 94.68 Myers, R. C., and J. Z. Simon, Black Hole Evaporation and Higher-Derivative

Gravity, Gen.Rel.Grav., 21(1989) 761.69 Stehney, A., Courbure dordre pet les classes de Pontrjagin, Journ. Dif f. Geom., 8

(1973) 125.70 Stehney, A., Extremal Sets of p-th Sectional Curvature, Journ. Diff . Geom. , 8

(1973) 383.


TABLE NOTESContinued171 Mena Marugn, G. A., Perturbative formalism of Lovelock gravity, Phys. R

46(1992) 4320.In Mena Marugn,L(p)partially agrees, insofar as it agrees withL(p)per Eq. (4)

from a typo, i.e. the last contravariant index of Mena Marugns expression foshould read j2m instead of i2 m.

172 Arik, M., and T. Dereli, Euler-Poincar Lagrangians and Kaluza-Klein Theory,Lett ., 189B (1987) 96.

In Arik and Dereli,L(p)partially agrees, insofar as it agrees withL(p)per Eq. (4)from a typo, i.e. the generalized Kronecker delta in the expression for L(p)in ArDerelis Eq. (1) has a covariant summation index that should read dN instead of

173 Mller-Hoissen, F., Spontaneous Compactification with Quadratic and CCurvature Terms, Phys. Lett., 163B(1985) 106.

In Mller-Hoissen, G(3)abpartially agrees, insofar as the expression for G(3)ab4thof Mller-Hoissens Eqs. (9) agrees aside from a typo, i.e. the contravariant h should be a covariant index in the concomitants belonging to the 9ththrougterms of G(3)a

bper Eq. (27), which are the 18 terms of Mller-Hoissens expressiG(3)a

bthat are not identically parallel to the Kronecker delta.174 Mller-Hoissen, F., and R. Stckl, Coset spaces and ten-dimensional u

theories, Class. Quantum Grav., 5(1988) 27.175 Perry, M., Quantum Gravity, in Hall, G. S., and J. R. Pulham (eds.), Ge

Relativity; Proceedings of the Forty Sixth [sic] Scottish Universities Summer Sin Physics, Aberdeen, July 1995, SUSSP Publications, Edinburgh, ScotlandInstitute of Physics Publishing, London, England (1996), p. 377.

In Perry,L(3)partially agrees, insofar as the concomitant belonging to the 7thte

L(3)per Eq. (15) occurs.176 Lemos, J. P. S., A Profusion of Black Holes from Two to Ten Dimensions,

Alamos preprint no. hep-th/9701121.In Lemos,L(3)partially agrees, insofar as the concomitant belonging to the 1

s

of L(3)per Eq. (15) occurs; and G (1)ab partially agrees, insofar as the concom

belonging to the 2ndterm of G(1)abper Eq. (25) occurs.177 Duff, M. J., Ultraviolet Divergences in Extended Supergravity, in Ferrara, S., G. Taylor (eds.), Supergravity 81, Cambridge University Press, Cambridge, En(1982), p. 197.

In Duff,L(3)partially agrees, insofar as the concomitant belonging to the 7thte

L(3)per Eq. (15) occurs.178 Lovelock, D., Vector-tensor field theories and the Einstein-Maxwell field equat

Proc. Roy. Soc. London (A), 341(1974) 285.179 DeWitt, B. S., Dynamical Theory of Groups and Fields, in DeWitt, C., and

DeWitt (eds.), Rela tiv ity, Group s and Topology , Gordon and Breach ScPublishers, Inc., New York, NY (1964), p. 585.

180 Lovelock, D., Divergence-Free Tensorial Concomitants,Aequationes Mathema4(1970) 127.

181 Boulware, D. G., and S. Deser, String-Generated Gravity Models, Phys. Rev.55(1985) 2656.

182 Paul, B. C., and S. Mukherjee, Higher-dimensional cosmology with Gauss-Bterms and the cosmological-constant problem, Phys. Rev. D, 42(1990) 2595.

183 Ray, J. R., A variational derivation of the Bach-Lanczos identity, Journ. Phys., 19 (1978) 100.

184 Ruzmaikina, T. V., and A. A. Ruzmaikin, Quadratic Corrections to the LagraDensity of the Gravitational Field and the Singularity, Soviet Physics JET(1970) 372.

In Ruzmaikina and Ruzmaikin, G(2)abpartially agrees, insofar as (A) the concom

belonging to the 1stand 4thterms of G(2)abper Eq. (26) occur withmoreover

magnitudes of their numerical coefficients in a ratio of 1:4, which completely aand as (B)aside from a typo, i.e. the concomitant belonging to the 5thterm ofper Eq. (26) has, in Ruzmaikina and Ruzmaikins expression, two indices contrthat should be freethe concomitants belonging to the 2ndand 5thterms of G(2)Eq. (26) occur, but with the magnitudes of their numerical coefficients in a ratio instead of 1:2.

185 Barth, N. H., and S. M. Christensen, Quantizing fourth-order gravity theoriesfunctional integral, Phys. Rev. D, 28(1983) 1876.

In Barth and Christensen, G(2)ab partially agrees, insofar as (A) the concom

belonging to the 1stand 4thterms of G(2)abper Eq. (26) occur withmoreover

magnitudes of their numerical coefficients in a ratio of 1:4, which completely aand as (B) the concomitants belonging to the 2ndand 6thterms of G(2)a

bper Eqoccur, but with the magnitudes of their numerical coefficients in a ratio of 1:4 inof 1:2.

186 Birrell, N. D., and P. C. W. Davies, Quantum Fields in Curved Space, CambUniversity Press, Cambridge, England (1982), pp. 154, 161, and 162.

In Birrell and Davies, G(2)abpartially agrees, insofar as it agrees with G(2)a

bp(26) aside from a typo, i.e. the concomitant belonging to the 6thterm of G(2)a

bp(26) in the last term of the 2ndline of Birrell and Daviess Eq. (6.54) has the wRiemann-Christoffel curvature tensor indices (viz. those with respect to whicRiemann-Christoffel curvature tensor is identically antisymmetric) contractedindices of the Ricci curvature tensor (viz. thosein Birrell and Davieswith respwhich the Ricci curvature tensor is, for differentiable manifolds having a metrical connection, identically symmetric).

187 Miji c, M. B., M. S. Morris, and W.-M. Suen, TheR2cosmology: Inflation withphase transition, Phys. Rev. D, 34(1986) 2934.

In Mijic et al., G(2)abpartially agrees, insofar as the concomitants belonging



21/54


TABLE NOTESContinued1stand 4thterms of G(2)a

bper Eq. (26) occur withmoreoverthe magnitudes of theirnumerical coefficients in a ratio of 1:4, which completely agrees.

88 Coley, A. A., Higher Dimensional Vacuum Cosmologies, TheAst rop hys icalJournal, 427(1994) 585.

In Coley, G(2)ab partially agrees, insofar as the concomitants belonging to the 1st

through 5thand the 7thterms of G(2)abper Eq. (26) occur, but with the magnitudes of

their numerical coefficients in a ratio of 1:4:1:4:16:0:4 instead of 1:4:1:4:8:8:4.89 Jttner, F., Beitrge zur Theorie der Materie, Mathematische Annalen , 87 (1922)

270.In Jttner, G(2)a

bpartially agrees, insofar as the concomitants belonging to the 1stand4th terms of G (2)a

b per Eq. (26) occur withmoreoverthe magnitudes of theirnumerical coefficients in a ratio of 1:4, which completely agrees.

90 Kalmykov, M. Yu., K. A. Kazakov, P.I. Pronin, and K. V. Stepanyantz, Detailedanalysis of the dependence of the one-loop counterterms on the gauge andparametrization in the Einstein gravity [sic] with the cosmological constant, LosAlamos preprint no. hep-th/98091679.

91 Verwimp, T., On higher dimensional gravity: the Lagrangian, its dimensionalreduction and a cosmological model, Class. Quantum Grav., 6(1989) 1655.

92 Whitt, B., Fourth-Order Gravity as General Relativity plus Matter, Phys. Lett.,145B (1984) 176.

93 Utiyama, R., and B. S. DeWitt, Renormalization of a Classical Gravitational FieldInteracting with Quantized Matter Fields,Journ. Math. Phys., 3(1962) 608.

94 Demianski, M., Z. Golda, and W. Puszkarz, Dynamics of the D -DimensionalFRW-cosmological Models within the Superstring-generated Gravity Model, Gen.

Rel.Grav., 23(1991) 917.95 Barvinsky, A. O., and G. A. Vilkovisky, Covariant perturbation theory (II). Second

order in the curvature. General algorithms,Nucl. Phys., B333(1990) 471.In Barvinsky and Vilkovisky, G (0)a

b partially agrees, insofar as the concomitantbelonging to the 1stterm of G(0)a

bper Eq. (24) occurs.

96 Azreg-Ainou, M., and G. Clment, Kaluza-Klein and Gauss-Bonnet cosmic strings,Class. Quantum Grav., 13(1996) 2635.

97 Zhenjiu, Z., H. Huanran, B. Gong, and H. Changbai, Quantum Mechanics of BlackHoles in Curved Space-Time, in Audretsch, J., and V. de Sabbata (eds.), Quantum

MechanicsinCurvedSpace-Time, Plenum Press, New York, NY (1990), p. 111.In Zhenjiu etal., G(2)a

bpartially agrees, insofar as (A) the concomitants belonging tothe 1st, 2nd, 4th, and 5thterms of G(2)a

bper Eq. (26) occur, but with the magnitudes oftheir numerical coefficients in a ratio 3:6:8:12 instead of 1:4:4:8, and as (B) theconcomitants belonging to the 1stand 4thterms of G(2)a

bper Eq. (26) occur withmoreoverthe magnitudes of their numerical coefficients in a ratio of 1:4, whichcompletely agrees.

98 Madsen, M. S., and J. D. Barrow, De Sitter ground states and boundary terms ingeneralized gravity,Nucl. Phys., B323(1989) 242.

99 Horiguchi, T., Dimensional reduction and renormalizability of the Wheeler-DeWittequation: next-leading-order contribution,IlNuovoCimento, 111B(1996) 165.

200 Shiraishi, K., The Friedmann Universe and Compact Internal Spaces inHigher-Dimensional Gravity Theories, Prog.Theor.Phys., 76(1986) 321.

In Shiraishi, G(1)abpartially agrees, insofar as the concomitant belonging to the 2 ndterm of G(1)a

bper Eq. (25) occurs.201 Lovelock, D., The Uniqueness of the Einstein field equations in a four-dimensional

space,Arch. Rational Mech. Anal. , 33(1969) 54.202 Deruelle, N., and L. Faria-Busto, Lovelock gravitational field equations in

cosmology, Phys. Rev. D, 41(1990) 3696.203 Deruelle, N., Cosmologies primordiales; Leurs varit, leurs constraintes, Journ. of

Geometry and Physics, 4(1987) 133.204 Deruelle, N., On the Approach to the Cosmological Singularity in Quadratic Theories

of Gravity: The Kasner Regimes, Nucl. Phys. , B327(1989) 253.205 Brunini, S. A., and M. Gomes, The Gauss-Bonnet Identity in Fourth Order Gravity,

Mod. Phys. Let t. A, 8 (1993) 1977.206 Duff, M. J., and P. van Nieuwenhuizen, Quantum Inequivalence of Different Field

Representations, Phys. Lett., 94B(1980) 179.207 Cremmer, E., Dimensional Reduction in Field Theory and Hidden Symmetries in

Extended Supergravity, in Ferrara, S., and J. G. Taylor (eds.), Supergravity 81,Cambridge University Press, Cambridge, England (1982), p. 313.

208 Buchbinder, I. L., and J. J. Wolfengaut, Renormalization group equations and effectiveaction in curved spacetime, Class. Quantum Grav., 5(1988) 1127.

209 Paul, B. C., A. Beesham, and S. Mukherjee, Wormholes in higher dimensions withGauss-Bonnet terms, PramanaJournal of Physics, 44(1995) 133.

210 Barrow, J. D., and A. C. Ottewill, The stability of general relativistic cosmologicaltheory,Journ. Phys. A: Math. Gen., 16(1983) 2757.

211 Reuter, M., and C. Wetterich, Spectrum Degeneracy and New Symmetries forGeneralized Euler Form Actions,Nucl. Phys., B304(1988) 653.

212 Ilha, A., and J. P. S. Lemos, Dimensionally Continued Oppenheimer-SnyderGravitational Collapse. ISolutions in Even Dimensions, Phys.Rev.D, 55 (1997)1788.

213 Boulware, D. G., and S. Deser, Effective Gravity Theories with Dilatons, Phys.Lett. , 175B (1986) 409.

214 Kerner, R., Kaluza-Klein Cosmology with Double Compactification, paper no. iv.4in OriginandEarlyHistoryoftheUniverse;Proceedingsofthe26thLigeInternational

As tr op hysi ca lColloquium,Ju ly 1-4, 1986,Universite de Li eg e [sic], Inst itutdAstrophysique,Liege[sic],Belgique, Liege, Belgium (1987), p. 271.


TABLE NOTESContinued215 Julve, J., and M. Tonin, Quantum Gravity with Higher Derivative Terms, IlN

Cimento, 46B (1978) 137.216 Petrov, A. Z., Einstein Spaces, Kelleher, R. F. (tr.), and J. Woodrow (ed.), Perg

Press, Ltd., Oxford, England (1969); translation of the Russian [Prostranstva Einshteina], published by Fizmatlit, Moscow, U(1961), containing amendments and revisions supplied by the author, pp130-131, and 306.

In Petrov,L(2)partially agrees, insofar as (A) the concomitants belonging to tthrough 3rd terms of L (2 ) per Eq. (14) occur on p. 77 of Petrov, but witmagnitudes of the numerical coefficients of the concomitants belonging to the 12nd terms of L (2) per Eq. (14) being in a ratio of 1:n, where n is an approdimensionality, instead of 1:4, and as (B) analogues of the concomitants belong

the 2ndand 3rdterms of L(2)per Eq. (14) occur in the expressions for Petrovs invariants S2andB2, respectively, on p. 130 of Petrov;L(3)partially agrees, insoanalogues of the concomitants belonging to the 4 th, 5th, and 7thterms ofL(3)p(15) occur in the expressions for Petrovs scalar invariants S3, T1, andB3, respecton p. 130 of Petrov; L(4)partially agrees, insofar as an analogue of the concombelonging to the 10thterm ofL(4)per Eq. (16) occurs in the expression for Pescalar invariant S4and as analogues of the concomitants belonging to the 13

thanterms ofL(4)per Eq. (16) occur in the expression for Petrovs scalar invariant T2130 of Petrov; L (5) partially agrees, insofar as analogues of the concom

RbaRc

bRecRf

dRadef and Rb

aRebRd

cRfdRac

ef belonging to the 29th and 30th3840Rb

aRcbRe

cRfdRad

efand 1920RbaRe

bRdcRf

dRacef, respectively, of L(5)p

(17) occurwith typos having been corrected, i.e. with the covariant indices having been appended to the initial Riemann-Christoffel curvature tensor factothe expression for Petrovs scalar invariant T3 on p. 130 of Petrov, but witmagnitudes of their numerical coefficients in a ratio of 4:3 instead of 2:1; andpartially agrees, insofar as an analogue of the concomitant R a

cRdbRe

dRbelonging to the 116th term 3840Ra

cRdbRe

dRfeRc

fof G(5)ab per Eq. (29) occ

with the aforementioned typos left uncorrectedin the expression for Petrovs invariant T3on p. 130 of Petrov.

See below for correspondences with analogues of Petrovs invariants B1,B2,BS3, S4, T1, T2, and T3as expressed in Eqs. (), (), and () on pp. 130-131 of Pand concomitants belonging to terms of L(1),L(2),L(3),L(4),L(5), and G(5)a

bpe(13) through (17) and Eq. (29), respectively.

PETROVS CORRESPONDING CORRESPONDING TERM(S)INVARIANT ANALOGUE CONTAININGSAME CONCOMITA

B1 Raba b = R , 1stofL(1) per Eq. (13);

B2 RcdabRab

cd, 3rdofL(2) per Eq.(14);B3 Rcd

ab RefcdRab

ef, 7th ofL(3) per Eq. (15);S2 Rb

aRab, 2ndofL(2) per Eq. (14);

S3 RbaRc

bRac, 4th ofL(3) per Eq. (15);

S4 RbaRc

bRdcRa

d, 10th ofL(4) per Eq. (16);T1 Rc

aRdbRab

cd, 5th ofL(3) per Eq. (15);T2

12 (Rb

aRcbRef

cdRadef 13th and14 th ofL(4) per

RcaRd

bRefcdRab

ef), Eq. (16), respectively;

T3 [with typos corrected]4Rb

aRcbRe

cRfdRad

ef+ 29thand 30thofL(5)per+ 3Rb

aRebRd

cRfdRac

ef, Eq. (17), respectively;T3 [with typos left uncorrected]

7RacRd

bRedRf

eRcf, 116thof G(5)a

bper Eq. (217 Donoghue, J. F., Introduction to the Effective Field Theory Description of Gra

preprint no. UMHEP-424, Los Alamos preprint no. gr-qc/9512024.In Donoghue,L(2)partially agrees, insofar as (A) the concomitants belonging

1stand 2ndterms ofL(2)per Eq. (14) occur and as (B) the concomitant belonging 3rdterm of L(2) per Eq. (14) is mentioned; L(3) partially agrees, insofar as (Aconcomitant belonging to the 1stterm ofL(3)per Eq. (15) is mentioned and as (Bconcomitant belonging to the 7thterm of L(3)per Eq. (15) occurs; and L(4) paagrees, insofar as the concomitant belonging to the 1 stterm of L(4)per Eq. (mentioned.

218 , . ., , - , , (1961), pp. 100, 16344.

In ,L(2)partially agrees, insofar as (A) the concomitants belonging to and 2ndterms ofL(2)per Eq. (14) occur on p. 100 of , but with the magnof their numerical coefficients in a ratio of 1: n , where n is an approdimensionality, instead of 1:4, and as (B) analogues of the concomitants belongthe 2ndand 3rdterms ofL(2)per Eq. (14) appear in the expressions for s invariants S2andB2, respectively, on p. 163 of ;L(3)partially agrees, inas analogues of the concomitants belonging to the 4th, 5th, and 7thterms ofL(3)p(15) appear in the expressions for s scalar invariants S3, T1, anrespectively, on p. 163 of ; andL(4)partially agrees, insofar as an analogthe concomitant belonging to the 10th term of L (4) per Eq. (16) appears iexpression for s scalar invariant S4 and as analogues of the concombelonging to the 13thand 14thterms ofL(4)per Eq. (16) appear in the expressios scalar invariant T2on p. 163 of . An expression for scalar invariant T3, which would comprise analogues of the concom

RbaRc

bRecRf

dRadef and Rb

aRebRd

cRfdRac

ef belonging to the 29th and 30th3840Rb

aRcbRe

cRfdRad

efand 1920RbaRe

bRdcRf

dRacef, respectively, of L(5)p

(17), but with the magnitudes of their numerical coefficients in a ratio of 4:3 inst(Continued in Next Column)


22/54


TABLE NOTESContinued2:1, seems to be missing.

See below for correspondences between analogues of s invariantsB1,B2,B3,S2, S3, S4, T1, and T2 as expressed in Eqs. (), (), and () on p. 163 of Petrov andconcomitants belonging to terms of L(1),L(2),L(3),L(4), andL(5), per Eqs. (13) through(17), respectively.

S CORRESPONDING CORRESPONDING TERM(S)INVARIANT ANALOGUE CONTAINING SAME CONCOMITANT(S)

B1 Raba b = R , 1stofL(1)per Eq. (13);

B2 RcdabRab

cd, 3rdofL(2)per Eq. (14);B3 Rcd

ab RefcdRab

ef, 7thofL(3)per Eq. (15);S2 Rb

aRab, 2ndofL(2)per Eq. (14);

S3 RbaRc

bRac, 4thofL(3)per Eq. (15);

S4 RbaRcbRdcRad, 10thofL(4)per Eq. (16);T1 Rc

aRdbRab

cd, 5thofL(3)per Eq. (15);T2

12 (Rb

aRcbRef

cdRadef 13thand 14thofL(4)per

RcaRd

bRefcdRab

ef), Eq. (16), respectively.219 Petrow, A. S., Einstein-Rume, Mathematische Lehrbcher und Monographien, II.

Abt. [Abtei lung], Mathematische Monographien , Bd. XVI, Koch, H. (tr.), and H.-J.Treder (ed.), Akademie-Verlag GmbH., Berlin, Germany (1964), pp. 72, 123, and 279.

In Petrow,L(2)partially agrees, insofar as (A) the concomitants belonging to the 1st

and 2ndterms ofL(2)per Eq. (14) occur on p. 72 of Petrow, but with the magnitudes oftheir numerical coefficients in a ratio of 1:n, where nis an appropriate dimensionality,instead of 1:4, and as (B) analogues of the concomitants belonging to the 2 ndand 3rd

terms ofL(2)per Eq. (14) appear in the expressions for Petrows scalar invariants S2andB2, respectively, on p. 123 of Petrow;L(3)partially agrees, insofar as analogues of theconcomitants belonging to the 4th, 5th, and 7thterms ofL(3)per Eq. (15) appear in theexpressions for Petrows scalar invariants S3, T1, andB3, respectively, on p. 123 ofPetrow; and L(4) partially agrees

an overview of various occurrences of general expressions for the coefficients of lovelock...

Documents