Clifford structures on manifolds. (English. Russian original) Zbl 0930.53030

J. Math. Sci., New York 89, No. 3, 1311-1333 (1998); translation from Itogi Nauki Tekh., Ser. Sovrem Mat. Prilozh., Temat. Obz. 30, 220-257 (1996).
This survey paper offers a valuable review of Clifford structures with numerous applications to mathematical physics and differential geometry. In addition, it contains eight theorems (which are probably not entirely new, and derived from the author’s previous publications), which are interesting for their generality. Contents include: an introduction; Clifford differential algebras; Clifford connections; and a bibliography of fifty items.


53C27 Spin and Spin\({}^c\) geometry
15A66 Clifford algebras, spinors
57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
Full Text: DOI


[1] M. P. Burlakov, ”Clifford connections on Riemannian manifolds,”Izv. Vuzov. Mat., No. 7, 3–7 (1990). · Zbl 0711.53015
[2] M. P. Burlakov, ”Grassmann generalized algebras on complex manifolds,”Diff. Geom. Mnogoobr. Figur, No. 22, 15–22 (1991). · Zbl 0895.53052
[3] M. P. Burlakov, ”Clifford structures and integration on smooth manifolds,”Diff. Geom. Mnogoobr. Figur, No. 18, 14–16 (1987). · Zbl 0673.53037
[4] M. P. Burlakov, ”Vector and spinor connections on Riemannian manifolds,” Deposited at VINITI, Grozny (1988). · Zbl 0665.53038
[5] M. P. Burlakov, ”Clifford bundles,” Deposited at VINITI, Grozny (1984).
[6] M. P. Burlakov, ”Clifford differential algebra of space-time,” Deposited at VINITI, Grozny (1984). · Zbl 0557.90035
[7] M. P. Burlakov and A. R. Beglaryan, ”Covariant antiderivation in Clifford bundles,” Deposited at VINITI, Grozny (1988).
[8] M. P. Burlakov, V. V. Pokazeev, and L. E. Freidenzon, ”Integral representations of functions with values in Dirac algebra,” Actual Questions Concerning the Theory of Boundary-Value Problems and Their Applications, Cheboksary (1988), pp. 1–17.
[9] A. M. Vasil’ev,Theory of Geometric-Differential Structures [in Russian], Moscow Univ. Press, Moscow (1987).
[10] H. Weyl,Group Theory and Quantum Mechanics [Russian translation], Nauka, Moscow (1986).
[11] V. V. Vishnevsky, A. P. Shirokov, and B. A. Rosenfeld, ”On the development of the geometry of spaces over algebras,”Izv. Vuzov. Mat., No. 7, 38–44 (1984).
[12] V. V. Vishnevsky, A. P. Shirokov, and V. V. Shurygin,Spaces over Algebras [in Russian], Kazan Univ. Press, Kazan (1985).
[13] E. Cartan,Leçons sur la théorie des spincurs, Paris (1938).
[14] A. P. Kotel’nikov,Theory of Screws and Some of Its Applications in Geometry and Mechanics [in Russian], Kazan (1895).
[15] S. P. Kuznetsov, ”OnB-sets in Clifford algebras,”Issled po Kraevym Zadacham i ikh Pril. Cheboksary, 1–91 (1992).
[16] S. P. Kuznetsov and V. V. Mochalov, ”Internal automorphisms of Clifford algebras and strongly regular functions,” Deposited at VINITI, Cheboksary (1991). · Zbl 0814.30029
[17] B. A. Rosenfeld,Non-Euclidean Geometrics [in Russian], Gostekhizdat (1954).
[18] N. A. Chernikov and N. S. Shavokhina, ”The metric nature of a spinor field” [in Russian], Comb. Inst. Nucl. Invest., Dubna. Preprint (1985).
[19] B. Schutz,Geometrical Methods of Mathematical Physics, Cambridge Univ. Press, Cambridge, U.K. (1980). · Zbl 0462.58001
[20] D. Bambusi, ”Clifford algebra description of non-Abelian gauge fields,”J. Geom. Phys.,7, No. 1, 1–12 (1990). · Zbl 0715.15010
[21] G. P. Barker and J. R. Urani, ”Dirac general covariance and tetrads: 1. Clifford and Lie bundles and torsion,”J. Math. Phys., No. 10, 2407–2410 (1983).
[22] P. Basarab-Horwath and R. W. Tucker, ”Propagateurs spinoriels et formes differentielles,”C. R. Acad. Sci. Ser. 1,299, No. 20, 1029–1032 (1984). · Zbl 0566.58001
[23] I. M. Benn and R. W. Tucker, ”The differential approach to spinors and their symmetries,”Nuovo Cim.,A 88, No. 3, 273–285 (1985).
[24] I. M. Benn, B. P. Dolan, and R. W. Tucker, ”Algebraic spin structures,”Phys. Lett.,B 150, Nos. 1–3, 100–102 (1985).
[25] M. Blau, ”Connections on Clifford bundles and the Dirac operator,”Lett. Math. Phys.,13, No. 1, 83–92 (1987). · Zbl 0644.58028
[26] R. Brauer and H. Weyl, ”Spinors inn dimensions,”Amer. J. Math.,57, 425–449 (1935). · Zbl 0011.24401
[27] P. Budinich and L. Dabrowski, ”On factorization properties of algebraic spinors,”Lett. Math. Phys.,10, No. 1, 7–10 (1985). · Zbl 0585.15004
[28] K. Bugajska, ”Geometrical properties of the algebraic spinors forR 3.1,”J. Math. Phys.,27, No. 1, 143–150 (1986). · Zbl 0601.53011
[29] E. R. Caianiello and A. Giovannini, ”Pure spinors as Pfaffians connecting Clifford and Grassmann algebras,”Lett. Nuovo Cim.,34, No. 10, 301–304 (1982).
[30] J. S. R. Chisholm and R. S. Farwell, ”Clifford approach to metric manifolds. 10th Winter Sch. Geom. and Phys.,”Suppl. Rend. Circ. Mat. Palermo, No. 26, 123–133 (1991). · Zbl 0752.53014
[31] L. Dabrowski, ”Group actions on spinors,”Napoli Bibli.,VI, 1–160 (1988). · Zbl 0703.53001
[32] L. Dabrowski and A. Trautman, ”Spinor structures on spheres and projective spaces,”J. Math. Phys.,27, No. 8, 2022–2028 (1986). · Zbl 0599.53030
[33] R. Deheuvels, ”Groupes conforme et algebres de Clifford,”Rend. Sem. Vat. Univ. Politechn. Torino,40, No. 2, 205–226 (1985). · Zbl 0599.53012
[34] R. Delanghe and F. Brackx, ”Duality in hypercomplex function theory,”J. Funct. Anal.,37, No. 2, 164–181 (1980). · Zbl 0446.46029
[35] R. Delanghe, F. Brackx and F. Sommen,Clifford Analysis, Pitman Publ. Inc., Boston-London-Melbourne (1982).
[36] A. Dimakis and F. Muller-Hoissen, ”Clifford calculus with applications to classical field theories,”Class. Quantum Grav.,8, No. 11, 2093–2132 (1991). · Zbl 0744.53039
[37] G. Dixon, ”Fermionic Clifford algebras and supersymmetry,” In:Clifford Algebras and Appl. Math. Phys. Proc. NATO and SERC Workshop, Canterbury, 15–27Sept., 1985, Dordrecht (1986), pp. 393–398.
[38] J. Froelich and N. Salingaros, ”The exponential mapping in Clifford algebras,”J. Math. Phys.,25, No. 8, 2347–2350 (1984). · Zbl 0563.22014
[39] J. Gilbert and Margaret A. M. Murray, ”Clifford algebras and Dirac operators in harmonic analysis,”Cambridge Stud. Adv. Math.,26, 1–334 (1991). · Zbl 0733.43001
[40] G. Harnett, ”The bivector Clifford algebra and the geometry of Hodge dual operators,”J. Phys. A.,25, No. 21, 5649–5662 (1992). · Zbl 0778.53014
[41] Z. Hasiewicz, A. K. Kwasniewski, and P. Morawiec, ”Supersymmetry and Clifford algebras,”,J. Math. Phys.,25, No. 6, 2031–2036 (1984).
[42] P. Lounesto and G. P. Wene, ”Idempotent structure of Clifford algebras,”Acta Appl. Math.,9, No. 3, 165–173 (1987). · Zbl 0658.15030
[43] A. K. Kwasniewski, ”Clifford and Grassmann like algebras,”Rend. Circ. Mat. Palermo,34, Suppl. 9, 141–155 (1985). · Zbl 0575.35088
[44] Adolfo Maia Jr., Erasmo Recami, Valbyr Rodrigues Jr., and Marcio A. F. Rosa, ”Magnetic monopoles without string in the Kaller-Clifford algebra bundle: a geometrical interpretation,”J. Math. Phys. 31, No. 2, 502–505 (1990). · Zbl 0703.53078
[45] M. K. Makhool and A. O. Morris, ”Real projective representations of Clifford algebras, and symmetric groups,”J. London Math. Soc.,43 No. 3, 412–420 (1991). · Zbl 0789.20012
[46] W. Marcinek, ”Clifford structures on real vector spaces and manifolds,”Repts. Math. Phys.,27, No. 3, 1989 (361–375). · Zbl 0744.53012
[47] J. Ryan, ”Applications of Clifford analysis to axially symmetric partial differential equations,”Complex Variables: Theory Appl.,16, Nos. 2–3, 137–151 (1991).
[48] N. Salingaros, ”The relationship between finite groups and Clifford, algebras,”J. Math. Phys.,25, No. 4, 738–742 (1984). · Zbl 0552.20008
[49] N. Salingaros and G. P. Wene, ”The Clifford algebra of differential forms for Minkowski space (II),” (RSR),28, No. 2, 173–179 (1986). · Zbl 0644.53079
[50] N. Salingaros and G. P. Wene, ”The Clifford algebra of differential forms,”Acta Appl. Math. 4, Nos. 2–3, 271–292 (1985). · Zbl 0572.15011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.