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I am interested in a complex derived by Eugenio Calabi in his article "On compact Riemannian manifolds with constant curvature".

The complex is referenced as "Calabi complex" in various citing publications, but the article itself remains nowhere to be found, not even on the AMS sites who published it some decades ago.

Do you know a modern exposition of the Calabi complex? I am particularly interested in the (covariant) form of the operators in the complex, which also appear in linearized elasticity.

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    $\begingroup$ You might ask Mike Eastwood or Andreas Cap. I think that this comes up as a BGG sequence for the appropriate groups, and I think that they have written about this in recent times. $\endgroup$
    – Robert Bryant
    Commented Apr 9, 2014 at 23:47
  • $\begingroup$ Thanks for the comment. Curiously, the publication year is either given as 1961 or 1971 in different citations, and the respective volume is missing in the AMS book store. $\endgroup$
    – shuhalo
    Commented Apr 10, 2014 at 9:58
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    $\begingroup$ The main result of Calabi's 1961 paper is a construction of a resolution of the sheaf of germs of Killing field on a Riemannian manifold of constant sectional curvature. The first operators come from the Lie derivative, and the linearized curvature. There is a 1975 paper by Bernard Bergery, Bourguignon, and Lafontaine called "Déformations localemente triviales des variétés riemanniennes" that contains a detailed exposition and some further results. There is a 1984 book of Gasqui and Goldschmidt that treats related material. $\endgroup$
    – Dan Fox
    Commented Apr 10, 2014 at 10:40
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    $\begingroup$ Calabi's original paper is available electronically from the AMS website ams.org/books/pspum/003 behind a paywall (you may perhaps have institutional access) $\endgroup$
    – YangMills
    Commented May 18, 2021 at 18:16

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A modern exposition with lots of explicit formulas can be found in my article

Khavkine, Igor, The Calabi complex and Killing sheaf cohomology, J. Geom. Phys. 113, 131-169 (2017). arXiv:1409.7212 ZBL1364.53066.

Ironically, while my paper first appeared on the arXiv a few months after this question, I've only noticed it just now.

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