Infinitesimal rigidity of Euclidean submanifolds
Annales de l'Institut Fourier, Volume 40 (1990) no. 4, p. 939-949

A submanifold M n of the Euclidean space R n is said to be infinitesimally rigid if any smooth variation which is isometric to first order is trivial. The main purpose of this paper is to show that local or global conditions which are well known to imply isometric rigidity also imply infinitesimal rigidity.

Une sous-variété M n de l’espace euclidien R n est dite infinitésimalement rigide si toute déformation différentiable isométrique au premier ordre est triviale. Nous montrons ici que certaines conditions locales ou globales bien connues pour entraîner la rigidité isométrique entraînent aussi la rigidité infinitésimale.

@article{AIF_1990__40_4_939_0,
     author = {Dajczer, M. and Rodriguez, L. L.},
     title = {Infinitesimal rigidity of Euclidean submanifolds},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {40},
     number = {4},
     year = {1990},
     pages = {939-949},
     doi = {10.5802/aif.1242},
     zbl = {0727.53011},
     mrnumber = {92d:53048},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1990__40_4_939_0}
}
Dajczer, M.; Rodriguez, L. L. Infinitesimal rigidity of Euclidean submanifolds. Annales de l'Institut Fourier, Volume 40 (1990) no. 4, pp. 939-949. doi : 10.5802/aif.1242. http://www.numdam.org/item/AIF_1990__40_4_939_0/

[A] C. B. Allendoerfer, Rigidity for spaces of class greater than one, Amer. J. Math., 61 (1939), 633-644. | JFM 65.0802.01 | MR 1,28g | Zbl 0021.15803

[CD] M. Do Carmo and M. Dajczer, Conformal rigidity, Amer. J. of Math., 109 (1987), 963-985. | MR 89e:53016 | Zbl 0631.53043

[DG] M. Dajczer and D. Gromoll, Real Kaehler submanifolds and uniqueness of the Gauss map, J. Diff. Geometry, 22 (1985), 13-28. | MR 87g:53088b | Zbl 0587.53051

[DR1] M. Dajczer and L. Rodriguez, Rigidity of real Kaehler submanifolds, Duke Math. J., 53 (1986), 211-220. | MR 87g:53089 | Zbl 0599.53005

[DR2] M. Dajczer and L. Rodriguez, Hypersurfaces which make a constant angle, in "Differential Geometry", Longman Sc. & Tech., Harlow, 1990. | Zbl 0723.53004

[GR] R. A. Goldstein and P.J. Ryan, Infinitesimal rigidity of submanifolds, J. Diff. Geometry, 10 (1975), 49-60. | MR 51 #1675 | Zbl 0302.53029

[S] R. Sacksteder, On hypersurfaces with no negative sectional curvature, Amer. J. Math., 82 (1960), 609-630. | MR 22 #7087 | Zbl 0194.22701

[Y] K. Yano, Infinitesimal variations of submanifolds, Kodai Math. J., 1 (1978), 30-44. | MR 58 #7504 | Zbl 0388.53017