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Curvature tensors on almost Hermitian manifolds. (English) Zbl 0484.53014


MSC:

53B20 Local Riemannian geometry
53B25 Local submanifolds
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)

Citations:

Zbl 0199.254
Full Text: DOI

References:

[1] M. F. Atiyah, N. J. Hitchin, and I. M. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A 362 (1978), no. 1711, 425 – 461. · Zbl 0389.53011 · doi:10.1098/rspa.1978.0143
[2] Marcel Berger, Paul Gauduchon, and Edmond Mazet, Le spectre d’une variété riemannienne, Lecture Notes in Mathematics, Vol. 194, Springer-Verlag, Berlin-New York, 1971 (French). · Zbl 0223.53034
[3] Jean A. Dieudonné and James B. Carrell, Invariant theory, old and new, Advances in Math. 4 (1970), 1 – 80 (1970). · Zbl 0196.05802 · doi:10.1016/0001-8708(70)90015-0
[4] Harold Donnelly, Topology and Einstein Kaehler metrics, J. Differential Geometry 11 (1976), no. 2, 259 – 264. · Zbl 0332.32004
[5] Harold Donnelly, Invariance theory of Hermitian manifolds, Proc. Amer. Math. Soc. 58 (1976), 229 – 233. · Zbl 0344.53043
[6] Peter B. Gilkey, Spectral geometry and the Kaehler condition for complex manifolds, Invent. Math. 26 (1974), 231 – 258. · Zbl 0303.53057 · doi:10.1007/BF01418951
[7] Alfred Gray, Some examples of almost Hermitian manifolds, Illinois J. Math. 10 (1966), 353 – 366. · Zbl 0183.50803
[8] Alfred Gray, Invariants of curvature operators of four-dimensional Riemannian manifolds, Proceedings of the Thirteenth Biennial Seminar of the Canadian Mathematical Congress (Dalhousie Univ., Halifax, N.S., 1971) Canad. Math. Congr., Montreal, Que., 1972, pp. 42 – 65. · Zbl 0278.53034
[9] Alfred Gray, The structure of nearly Kähler manifolds, Math. Ann. 223 (1976), no. 3, 233 – 248. · Zbl 0345.53019 · doi:10.1007/BF01360955
[10] Alfred Gray, Curvature identities for Hermitian and almost Hermitian manifolds, Tôhoku Math. J. (2) 28 (1976), no. 4, 601 – 612. · Zbl 0351.53040 · doi:10.2748/tmj/1178240746
[11] Alfred Gray and Luis M. Hervella, The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl. (4) 123 (1980), 35 – 58. · Zbl 0444.53032 · doi:10.1007/BF01796539
[12] Alfred Gray and L. Vanhecke, Almost Hermitian manifolds with constant holomorphic sectional curvature, Časopis Pěst. Mat. 104 (1979), no. 2, 170 – 179. · Zbl 0413.53011
[13] -, Decomposition of the space of covariant derivatives of curvatures operators (to appear).
[14] Nigel Hitchin, Compact four-dimensional Einstein manifolds, J. Differential Geometry 9 (1974), 435 – 441. · Zbl 0281.53039
[15] Nagayoshi Iwahori, Some remarks on tensor invariants of \?(\?),\?(\?),\?\?(\?), J. Math. Soc. Japan 10 (1958), 145 – 160. · Zbl 0082.15601 · doi:10.2969/jmsj/01020145
[16] D. L. Johnson, A normal form for curvature, Ph. D. Thesis, M.I.T., 1977 (to appear under the title Critical behaviour of sectional curvature and curvature normal forms for Kähler manifolds).
[17] Hiroshi Mori, On the decomposition of generalized \?-curvature tensor fields, Tôhoku Math. J. (2) 25 (1973), 225 – 235. Collection of articles dedicated to Shigeo Sasaki on his sixtieth birthday. · Zbl 0271.53030 · doi:10.2748/tmj/1178241382
[18] Katsumi Nomizu, On the decomposition of generalized curvature tensor fields. Codazzi, Ricci, Bianchi and Weyl revisited, Differential geometry (in honor of Kentaro Yano), Kinokuniya, Tokyo, 1972, pp. 335 – 345. · Zbl 0244.53032
[19] Sumio Sawaki and Kouei Sekigawa, Almost Hermitian manifolds with constant holomorphic sectional curvature, J. Differential Geometry 9 (1974), 123 – 134. · Zbl 0277.53036
[20] J. A. Schouten, Der Ricci-Kalkül, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 10, Springer-Verlag, Berlin-New York, 1978 (German). Eine Einführung in die neueren Methoden und Probleme der mehrdimensionalen Differentialgeometrie; Reprint of the 1924 original. · JFM 49.0736.02
[21] I. M. Singer and J. A. Thorpe, The curvature of 4-dimensional Einstein spaces, Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press, Tokyo, 1969, pp. 355 – 365. · Zbl 0199.25401
[22] Malladi Sitaramayya, Curvature tensors in Kaehler manifolds, Trans. Amer. Math. Soc. 183 (1973), 341 – 353. · Zbl 0266.53023
[23] Franco Tricerri and Lieven Vanhecke, An example of a 6-dimensional flat almost Hermitian manifold, Colloq. Math. 40 (1978/79), no. 1, 111 – 117. · Zbl 0409.53046
[24] Franco Tricerri and Lieven Vanhecke, Flat almost Hermitian manifolds which are not Kähler manifolds, Tensor (N.S.) 31 (1977), no. 3, 249 – 254. · Zbl 0375.53037
[25] F. Tricerri and L. Vanhecke, Decomposition of a space of curvature tensors on a quaternionic Kähler manifold and spectrum theory, Simon Stevin 53 (1979), no. 1-2, 163 – 173. · Zbl 0409.53034
[26] Izu Vaisman, On locally conformal almost Kähler manifolds, Israel J. Math. 24 (1976), no. 3-4, 338 – 351. · Zbl 0335.53055 · doi:10.1007/BF02834764
[27] Izu Vaisman, Some curvature properties of locally conformal Kähler manifolds, Trans. Amer. Math. Soc. 259 (1980), no. 2, 439 – 447. · Zbl 0435.53044
[28] Lieven Vanhecke, The Bochner curvature tensor on almost Hermitian manifolds, Rend. Sem. Mat. Univ. e Politec. Torino 34 (1975/76), 21 – 38. · Zbl 0408.53015
[29] Lieven Vanhecke, On the decomposition of curvature tensor fields on almost Hermitian manifolds, Differential geometry (Proc. Conf., Michigan State Univ., East Lansing, Mich., 1976) Dept. Math., Michigan State Univ., East Lansing, Mich., 1976, pp. 16 – 33. · Zbl 0343.53020
[30] L. Vanhecke and F. Bouten, Constant type for almost Hermitian manifolds, Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.) 20(68) (1976), no. 3-4, 415 – 422 (1977). · Zbl 0366.53034
[31] Lieven Vanhecke, On the decomposition of curvature tensor fields on almost Hermitian manifolds, Differential geometry (Proc. Conf., Michigan State Univ., East Lansing, Mich., 1976) Dept. Math., Michigan State Univ., East Lansing, Mich., 1976, pp. 16 – 33. · Zbl 0343.53020
[32] Lieven Vanhecke, The Bochner curvature tensor on almost Hermitian manifolds, Geometriae Dedicata 6 (1977), no. 4, 389 – 397. · Zbl 0383.53022 · doi:10.1007/BF00147777
[33] L. Vanhecke and D. Janssens, The Bochner curvature tensor on almost Hermitian manifolds, Hokkaido Math. J. 7 (1978), no. 2, 252 – 258. · Zbl 0399.53020 · doi:10.14492/hokmj/1381758450
[34] Hermann Weyl, The classical groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Their invariants and representations; Fifteenth printing; Princeton Paperbacks. · Zbl 1024.20501
[35] Y. L. Xin, Remarks on characteristic classes of four-dimensional Einstein manifolds, J. Math. Phys. 21 (1980), no. 2, 343 – 346. · Zbl 0443.53036 · doi:10.1063/1.524420
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