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Curvature invariants and applications. (English) Zbl 0789.53038
Szenthe, J. (ed.) et al., Differential geometry and its applications. Proceedings of a colloquium, held in Eger, Hungary, August 20-25, 1989, organized by the János Bolyai Mathematical Society. Amsterdam: North- Holland Publishing Company. Colloq. Math. Soc. János Bolyai. 56, 137-154 (1992).
This paper is a survey about curvature invariants in the sense of representation theory and its applications. In §§2-3 the author provides definitions and a brief history on curvature invariants, in particular, on quadratic invariants. In §4 she explains some results of B. Y. Chen, K. Ogiue, herself and N. Blazić concerning the necessary and sufficient conditions for a Kähler manifold to be biholomorphically covered by a complete simply-connected complex space-form in terms of the first Chern class, the Bochner-Kähler metric, and/or the complex conharmonic curvature tensor. In §5 the author introduces the results of A. Gray-L. Vanhecke and of N. Blazić on the volume conjecture together with a result of I. Dimitric in the same direction. In §6 she introduces Blazić’s extension of Chen-Ogiue’s inequality between the first and the second Chern numbers of the tangent bundle of a Kähler manifold and a similar inequality of M. Lübke for an Einstein-Hermitian vector bundle over a Kähler manifold. In the last section she explains her own results on symmetric connections and their applications to projective geometry.
For the entire collection see [Zbl 0764.00002].
53C55 Global differential geometry of Hermitian and Kählerian manifolds