CALABI-WEILRIGIDITY-InstitutFourier.PDFVIP

CALABI-WEIL RIGIDITY G´erard BESSON We based these lectures on the approach developed by Raghunatan ([Rag]). The reader can also see the original papers by A. Weil (see [We1], [We2]). The text which follows is neither intended to be original nor exhaustiv. It aims at presenting in a very elementary way the theory of infinitesimal rigidity as described in [Rag]. Introduction A trivial example The group Z can be viewed as a subgroup of the group of translations of the real line R, and of infinitely many ways. More precisely let t R and let us call Tt the translation defined by T (x) = x + t for x R. t Translations are isometries of the euclidean structure on R, so that we can define a family of morphisms t : Z Isom(E) n T n (x x + nt). t Such morphisms are called representations of Z as isometries of R. We thus get a deformation of the canonical representation . 1 This deformation is not trivial in the sense that there does not exist, for t close to 1, an isometry gt of R such that n Z , (n) = g (n)g1 . t t 1 t 1 11 00 Another trivial example Again Z2 can be viewed as a subgroup of the 2 2 2 group of translations of R , the quotient space being a torus R /Z . The translations of R2 are isometries with respect to the usual euclidean struc- ture. The orbit of the origin