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On the complete decomposition of curvature tensors of Riemannian manifolds with symmetric connection. (English) Zbl 0728.53016
Let V be a real vector space of dimension n with inner product g and let \({\mathcal R}(V)\) denote the vector space formed by all the algebraic curvature tensors having the same symmetries as the curvature tensor associated to a symmetric connnection on a Riemannian manifold. The main part of the paper consists in deriving complete decompositions of \({\mathcal R}(V)\) under the action of SO(n). The dimensions of the factors are determined and the projections of an element of \({\mathcal R}(V)\) on these factors are derived. The consideration of the norms of these projections leads to (in)equalities for quadratic invariants which may be used in Riemannian geometry to characterize special spaces. Several examples of applications are given with focus on projective curvature tensors and projective transformations and on locally affine symmetric manifolds.

53B20 Local Riemannian geometry
53B05 Linear and affine connections
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[1] Abbena E., Garbiero S.,Almost Hermitian homogeneous structures, Proc. Eding. Math. Soc.31 (1988), 375–395. · Zbl 0637.53065
[2] Berger M., Gauduchon P. Mazet E.,Le spectre d’une variete riemannienne, Lecture Notes in Math. vol.194 Springer-Verlag, Berlin and New York, 1971. · Zbl 0223.53034
[3] Bokan N.,Curvature tensors of Riemannian manifolds with connection without torsion, Mat. Vesnik37 (1985), 356–364. · Zbl 0592.53011
[4] Borel A.,Semi simple groups and symmetric spaces, mimeographed notes, Tata Institute, 1961.
[5] Catto D., Francaviglia M.,On certain decompositions of tensor fields of order four, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur.114, No 3–4, (1980/81), 243–248. · Zbl 0503.53011
[6] Dieudonne J., Carell J.,Invariant theory old and new, Adv. in Math.4 (1970), 1–80. · Zbl 0196.05802
[7] Gray A.,Some relations between curvature and characteristic classes, Math. Ann.184 (1970), 257–267. · Zbl 0183.50502
[8] Gray A.,Invariants of curvature operators of four-dimensional Riemannian manifolds, Proc. Thirteenth Biennial Semin. Canad. Math. Cong., Halifax, vol.2, (1971), 42–65.
[9] Gray A., Hervella L. M.,The sixteen classes of almost Hermitian manifolds, Ann. Mat. Pura Appl.123 (1980), 35–58. · Zbl 0444.53032
[10] Gray A., Vanhecke L.,Decomposition of the space of covariant derivatives of curvature operators preprint.
[11] Hano J.,On affine transformations of a Riemannian manifold, Nagoya. Math. Jour.9 (1955), 99–109. · Zbl 0067.14601
[12] Hopf E.,Elementare Bemerkungen über die Losungen partieller Differentialgleichunger zweiter Ordung von elliptischen Typus, Sitzungsberichte der Deutschen Akademie der Wissenschaften zu Berlin19 (1927), 147–152.
[13] Ishihara S.,Groups of proiective transformations and groups of conformal transformations, Jour. Math. Soc. of Japan9, No 2. 2, (1957), 195–227. · Zbl 0080.37401
[14] Ishihara S., Obata M.,Affine transformations in a Riemannian manifold, Tôhoku Math. Jour.7 (1975), 146–150. · Zbl 0067.14602
[15] Iwahori M.,Some remarks on tensor invariants of O(n), U(n), Sp(n), J. Math. Soc. Japan10 (1958), 145–160. · Zbl 0082.15601
[16] Johnson D.L.,Sectional curvature and curvature normal forms, Michigan Math. J.27 (1980), 275–294. · Zbl 0439.53035
[17] Kobayashi S.,A theorem on the affine transformation group of a Riemannian manifold, Nagoya Math. Jour.9 (1955), 39–41. · Zbl 0067.14501
[18] Kowalski O.,Partial curvature structures and conformal geometry of submanifolds, J. Diff. Geom.8 (1973), 53–70. · Zbl 0273.53012
[19] Kulkarni R.S.,On the Bianchi identities, Math. Ann.199 (1972), 175–204. · Zbl 0238.53013
[20] Kulkarni R.S.,Index Theorems of Atiyah–Bott–Patodi and curvature invariants, Les Presses de l’Universite de Montreal, Montreal, 1975. · Zbl 0332.58015
[21] Mori H.,On the decompsition of generalized K-curvature tensor fields, Tôhoku Math. J.25, (1973), 225–235. · Zbl 0271.53030
[22] Naveira A.M.,A classication of Riemannian almost-product manifolds, Rendicotni di Matemiatica3, Serie VII, (1983), 577–592. · Zbl 0538.53045
[23] Nomizu K.,Studies on Riemannian homogeneous spaces, Nagoya Math. Jour.9 (1955), 43–56. · Zbl 0067.14502
[24] Nomizu K.,On the decomposition of generalized curvature tensor fields, Differential geometry (in honor of K. Yano), Kinokuniya, Tokyo, (1972) 335–345. · Zbl 0244.53032
[25] Norden A.P.,Prostranstvaa affinnoj swqznosti, Nauka, Moskwa 1976.
[26] Poor W.A.,Differential geometric structures, McGraw-Hill Book Company, New York, 1981. · Zbl 0493.53027
[27] Rasewskij P.K.,Riemanowa geoemtria i tenzornij analiz, Gostehizdat, Moskwa, 1953.
[28] Schouten J.A.,Ricci-Calculus, 2nd ed., Springer Verlag, Berlin 1954.
[29] Simon U.,Connections and conformal structures in affine differential geometry Diff. Geom. and its Applic., Proc. of the Conference, August 24–30 (1986), 315–328, Brno, Czechoslovakia, D. Reidel Publishing Company, Dordrecht.
[30] Singer I.M., Thorpe J.A.,The curvature of 4-dimensional Einstein spaces, Global Analysis (papers inhonor of K. Kodaira), Univ. of Tokyo Press, Tokyo, (1969), 355–365.
[31] Sitaramayya M.,Curvature tensors in Kahler manifolds, Trans. Amer. Math. Soc.183 (1973), 341–351. · Zbl 0266.53023
[32] Strichartz R.S.,Linear algebra of curvature tensors and their covariant derivatives, Can. J. Math.15 (1988), 1105–1143. · Zbl 0652.53012
[33] Tricerri F., Vanhecke L.,Curvature tensors on almost Hermitian manifolds, Trans. of the Amer. Math. Soc.267 (1981), 365–398. · Zbl 0484.53014
[34] Tricerri F., Vanhecke L.,Homogeneous structures on Riemannian manifolds, London Math. Soc., Lect. Notes 83, Cambridge Univ. Press., 1983. · Zbl 0509.53043
[35] Vanhecke L.,Curvature tensors, J. Korean Math. Soc.14 (1977), 143–151. · Zbl 0367.53006
[36] Weyl H.,Zur Infinitesimalgeometrie, Einordnung der projektiven und der konformen Auffassung. Gottinger Nachrichten (1921), 99–112. · JFM 48.0844.04
[37] Weyl H.,Classical groups, their invariants and representations, Princeton Univ. Press, Princenton, N.J., 1946. · Zbl 1024.20502
[38] Yano K., Bochner S.,Curvature and Betti numbers, Annals of Mathematics Studies, No 32, Princeton, Princeton University Press, 1953.
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