Encounter with a geometer: Eugenio Calabi.

*(English)*Zbl 0926.53001
de Bartolomeis, Paolo (ed.) et al., Manifolds and geometry. Proceedings of a conference, held in Pisa, Italy, September 1993. Cambridge: Cambridge University Press. Symp. Math. 36, 20-60 (1996).

Trying to mimic the title of the paper under review, we can qualify it as the meeting of two big geometers: Marcel Berger, a geometer as Eugenio Calabi, presents here his brillant comments on the geometrical work of Calabi (in occasion of his 70th birthday celebration). The very large scope of Calabi’s work makes serious difficulties for the selection of material, but we can be sure that Berger’s choice is the best possible.

The diastasis is an invariant, introduced by Calabi, which helps us to understand the nature of the Riemannian metric in the case of Kähler analytic manifolds. A wonderful exposition of “the diastasis story” is given by Berger, who adds a new question, namely about the comparison of diastasis with the various types of Kobayashi metrics when they exist.

The greater part of Berger’s review is devoted to space forms (i.e., constant sectional curvature spaces), to which Calabi has made important contributions. After a historical introduction, the author recalls the glorious time of the Kodaira seminar in Princeton during 1958-59, and the whole story about the Selberg conjecture concerning the rigidity of higher dimension space forms of negative curvature. Calabi’s attempt to prove that Selberg’s hypothesis is wrong led him finally to a proof that there are no nontrivial deformations of space forms when the dimension is greater than 2. The proof is sketched here, namely the computation of the first cohomology group \(H^1(M,K)\) of a compact space \(M\) with values in the sheaf \(K\) of germs of Killing vector fields on \(M\). More generally, it is remarked that Calabi’s ideas in his space form paper [Proc. Sympos. Pure Math. 3, 155-180 (1961; Zbl 0129.14102)] influenced mathematicians like A. Weil, S. S. Chern and Y. Matsushima.

Another outcome of Calabi’s theory (his famous exact sequence) concerns the so-called (by J. Gasqui and H. Goldschmidt) “infinitesimal deformations” of Riemannian manifolds all of whose geodesics are periodic. On \(\mathbb{K} P\) \((\mathbb{K}=\mathbb{R},\mathbb{C},\mathbb{H},C a)\) the only known metrics with this property are the standard ones. It was proved that there are no nontrivial infinitesimal deformations. J.-P. Bourguignon gave a simple and conceptual proof based on Calabi’s ideas. This proof is exposed here.

Periodic geodesics on convex surfaces are exposed carefully. It is because of the joint paper of E. Calabi and J.-G. Cao [J. Differ. Geom. 36, 517-549 (1992; Zbl 0768.53019)], where an important contribution to the geometry of spheres \((S^2,g)\) was given (having in mind the classical theorems of Weyl, Alexandrov, Nirenberg, Pogorelov). The fundamental result in the Calabi-Cao paper says that a periodic geodesic of smallest length in a smooth surface of non-negative curvature is always simple. The ideas of the proof of this theorem are sketched with remarkable geometric lucidity and elegance.

Other topics covered in this paper include: generalized space forms (the famous E. Calabi-E. Vesentini joint paper in [Ann. Math., II. Ser. 71, 472-507 (1960; Zbl 0100.36002)], embedding minimal spheres into spheres, harmonic 1-forms, extremal systolic surfaces and their relations with a physically motivated geometric question studied by these two very respectable geometers.

For the entire collection see [Zbl 0840.00037].

The diastasis is an invariant, introduced by Calabi, which helps us to understand the nature of the Riemannian metric in the case of Kähler analytic manifolds. A wonderful exposition of “the diastasis story” is given by Berger, who adds a new question, namely about the comparison of diastasis with the various types of Kobayashi metrics when they exist.

The greater part of Berger’s review is devoted to space forms (i.e., constant sectional curvature spaces), to which Calabi has made important contributions. After a historical introduction, the author recalls the glorious time of the Kodaira seminar in Princeton during 1958-59, and the whole story about the Selberg conjecture concerning the rigidity of higher dimension space forms of negative curvature. Calabi’s attempt to prove that Selberg’s hypothesis is wrong led him finally to a proof that there are no nontrivial deformations of space forms when the dimension is greater than 2. The proof is sketched here, namely the computation of the first cohomology group \(H^1(M,K)\) of a compact space \(M\) with values in the sheaf \(K\) of germs of Killing vector fields on \(M\). More generally, it is remarked that Calabi’s ideas in his space form paper [Proc. Sympos. Pure Math. 3, 155-180 (1961; Zbl 0129.14102)] influenced mathematicians like A. Weil, S. S. Chern and Y. Matsushima.

Another outcome of Calabi’s theory (his famous exact sequence) concerns the so-called (by J. Gasqui and H. Goldschmidt) “infinitesimal deformations” of Riemannian manifolds all of whose geodesics are periodic. On \(\mathbb{K} P\) \((\mathbb{K}=\mathbb{R},\mathbb{C},\mathbb{H},C a)\) the only known metrics with this property are the standard ones. It was proved that there are no nontrivial infinitesimal deformations. J.-P. Bourguignon gave a simple and conceptual proof based on Calabi’s ideas. This proof is exposed here.

Periodic geodesics on convex surfaces are exposed carefully. It is because of the joint paper of E. Calabi and J.-G. Cao [J. Differ. Geom. 36, 517-549 (1992; Zbl 0768.53019)], where an important contribution to the geometry of spheres \((S^2,g)\) was given (having in mind the classical theorems of Weyl, Alexandrov, Nirenberg, Pogorelov). The fundamental result in the Calabi-Cao paper says that a periodic geodesic of smallest length in a smooth surface of non-negative curvature is always simple. The ideas of the proof of this theorem are sketched with remarkable geometric lucidity and elegance.

Other topics covered in this paper include: generalized space forms (the famous E. Calabi-E. Vesentini joint paper in [Ann. Math., II. Ser. 71, 472-507 (1960; Zbl 0100.36002)], embedding minimal spheres into spheres, harmonic 1-forms, extremal systolic surfaces and their relations with a physically motivated geometric question studied by these two very respectable geometers.

For the entire collection see [Zbl 0840.00037].

Reviewer: S.Dimiev (Sofia)

##### MSC:

53-03 | History of differential geometry |

01A60 | History of mathematics in the 20th century |

53C55 | Global differential geometry of Hermitian and Kählerian manifolds |

53C20 | Global Riemannian geometry, including pinching |

53C22 | Geodesics in global differential geometry |

53C45 | Global surface theory (convex surfaces à la A. D. Aleksandrov) |