Nonexistence of $4$-dimensional almost Kaehler manifolds of constant curvature
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- by David E. Blair PDF
- Proc. Amer. Math. Soc. 110 (1990), 1033-1039 Request permission
Abstract:
It is shown that in dimension 4 there are no almost Kaehler manifolds of constant curvature unless the constant is 0, in which case the manifold is Kaehlerian. This was previously shown in dimensions $\geq 8$ by Z. Olszak and remains open in dimension 6.References
- D. E. Blair and S. Ianuş, Critical associated metrics on symplectic manifolds, Nonlinear problems in geometry (Mobile, Ala., 1985) Contemp. Math., vol. 51, Amer. Math. Soc., Providence, RI, 1986, pp. 23–29. MR 848929, DOI 10.1090/conm/051/848929
- F. Brackx, Richard Delanghe, and F. Sommen, Clifford analysis, Research Notes in Mathematics, vol. 76, Pitman (Advanced Publishing Program), Boston, MA, 1982. MR 697564 R. Feuter, Die funktionentheorie der differentialgleichungen $\Delta u = 0$ und $\Delta \Delta u = 0$ mit vier reellen variablen, Comment. Math. Helv. 7 (1935), 307-330.
- S. I. Goldberg, Integrability of almost Kaehler manifolds, Proc. Amer. Math. Soc. 21 (1969), 96–100. MR 238238, DOI 10.1090/S0002-9939-1969-0238238-1
- Ognian T. Kassabov, Almost Kähler manifolds of constant antiholomorphic sectional curvature, Serdica 9 (1983), no. 4, 372–376 (1984). MR 745268
- Ognian T. Kassabov, Conformal flat $AK_2$-manifolds, Pliska Stud. Math. Bulgar. 9 (1987), 12–16. MR 892675
- Zbigniew Olszak, A note on almost Kaehler manifolds, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26 (1978), no. 2, 139–141 (English, with Russian summary). MR 493830
- A. Sudbery, Quaternionic analysis, Math. Proc. Cambridge Philos. Soc. 85 (1979), no. 2, 199–224. MR 516081, DOI 10.1017/S0305004100055638
- Kentaro Yano, Differential geometry on complex and almost complex spaces, International Series of Monographs in Pure and Applied Mathematics, Vol. 49, The Macmillan Company, New York, 1965. A Pergamon Press Book. MR 0187181
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 110 (1990), 1033-1039
- MSC: Primary 53C55; Secondary 53C15
- DOI: https://doi.org/10.1090/S0002-9939-1990-1043404-2
- MathSciNet review: 1043404