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Calculation of the Ricci-tensor of a special \(K3\) imbedded in \(\mathbb{CP}^3\) for the metric induced by the canonical metric. (Calcul du tenseur de Ricci d’une \(K3\) particulière plongée dans \(\mathbb{CP}^3\) pour la métrique induite par la métrique canonique.) (French) Zbl 0747.53039

Sémin. Théor. Spectrale Géom., Chambéry-Grenoble 1983-1984, Exp. No. IX, 23 p. (1984).
More than 25 years ago B. Smyth has shown in his doctoral dissertation, written under the direction of K. Nomizu the following theorem [see Ann. Math. (2) 85, 246–266 (1967; Zbl 0168.19601)]: If \(n\) is at least three, a smooth hypersurface of degree \(d\) in projective \(n\)-space, equipped with the Kählerian metric induced from the Fubini- Study metric, is Einsteinian if and only if \(d\) is one or two. Therefore the Fubini-Study metric induces a non-Einsteinian metric on a smooth quartic surface in projective three-space. But since such a surface is \(K3\), we know from Yau’s proof of Calabi’s conjecture [Commun. Pure Appl. Math. 31, 339-441 (1978; Zbl 0362.53049)] the existence of an Einsteinian metric on quartic surfaces.
In the present paper the Ricci tensor of the induced metric of a quartic surface is explicitly described in local coordinates. The calculation uses the second fundamental form and standard computations from S. Kobayashi and K. Nomizu [Foundations of differential geometry. Vol. I, New York: Interscience Publishers (1963; Zbl 0119.37502), Vol. II (1969; Zbl 0175.48504)].

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C55 Global differential geometry of Hermitian and Kählerian manifolds
14D20 Algebraic moduli problems, moduli of vector bundles
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