Geodesic
Classification of multidimensional submanifolds in Euclidean space with a totally geodesic Gauss image
Sbornik. Mathematics, Tome 76 (1993) no. 1, pp. 225-246 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A complete classification is given of submanifolds that have a totally geodesic Gauss image in the ambient Grassmann manifold. The arguments are based on a study of the tangent space to the Gauss image of the submanifold and Cartan's theorem on totally geodesic submanifolds of symmetric spaces. 
@article{SM_1993_76_1_a12,
     author = {Yu. A. Nikolaevskii},
     title = {Classification of multidimensional submanifolds in {Euclidean} space with a~totally geodesic {Gauss} image},
     journal = {Sbornik. Mathematics},
     pages = {225--246},
     year = {1993},
     volume = {76},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1993_76_1_a12/}
}
TY  - JOUR
AU  - Yu. A. Nikolaevskii
TI  - Classification of multidimensional submanifolds in Euclidean space with a totally geodesic Gauss image
JO  - Sbornik. Mathematics
PY  - 1993
SP  - 225
EP  - 246
VL  - 76
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SM_1993_76_1_a12/
LA  - en
ID  - SM_1993_76_1_a12
ER  - 
%0 Journal Article
%A Yu. A. Nikolaevskii
%T Classification of multidimensional submanifolds in Euclidean space with a totally geodesic Gauss image
%J Sbornik. Mathematics
%D 1993
%P 225-246
%V 76
%N 1
%U http://geodesic.mathdoc.fr/item/SM_1993_76_1_a12/
%G en
%F SM_1993_76_1_a12
Yu. A. Nikolaevskii. Classification of multidimensional submanifolds in Euclidean space with a totally geodesic Gauss image. Sbornik. Mathematics, Tome 76 (1993) no. 1, pp. 225-246. http://geodesic.mathdoc.fr/item/SM_1993_76_1_a12/

[1] Khelgason S., Differentsialnaya geometriya i simmetricheskie prostranstva, Mir, M., 1964 | Zbl

[2] Chen B.-Y., Yamaguchi S., “Classification of surfaces with totally geodesic Gauss image”, Indiana Univ. Math. Journal, 32:1 (1983), 143–154 | DOI | MR | Zbl

[3] Chen B.-Y., Yamaguchi S., “Submanifolds with totally geodesic Gauss image”, Geom. Dedicata, 15:3 (1984), 313–322 | MR | Zbl

[4] Pak J. S., Kim J. J., “Isotropic immersions with totally geodesic Gauss image”, Tensor, 43:2 (1986), 167–174 | MR | Zbl

[5] Muto J., “Submanifolds of a Euclidean space with homothetic Gauss map”, J. Math. Soc. Japan, 32:3 (1980), 531–555 | DOI | MR | Zbl

[6] Vilms J., “Submanifolds of a Euclidean space with parallel second fundamental form”, Proc. Amer. Math. Soc., 32 (1972), 263–267 | DOI | MR | Zbl

[7] Ferus D., “Immersions with parallel second fundamental form”, Math. Zeit., 140:1 (1974), 87–93 | DOI | MR | Zbl

[8] Ferus D., “Symmetric submanifolds of Euclidean space”, Math. Ann., 247:1 (1980), 81–93 | DOI | MR | Zbl

[9] Takeuchi M., Kobayashi S., “Minimal imbeddings of R-spaces”, J. Diff. Geom., 2:2 (1968), 203–215 | MR | Zbl

[10] Kobayashi S., “Isometric imbeddings of compact symmetric spaces”, Tohoku Math. J., 20:1 (1968), 21–25 | DOI | MR | Zbl

[11] Nagano T., “Transformation groups on compact symmetric spaces”, Trans. Amer. Math. Soc., 118:6 (1965), 428–453 | DOI | MR | Zbl

[12] Kobayasi Sh., Nomidzu K., Osnovy differentsialnoi geometrii, T. 2, Nauka, M., 1981

[13] Gantmakher F. R., Teoriya matrits, Nauka, M., 1969 | MR

[14] Adams J. F., Lax P. D., Phillips R. S., “On matrices whose real linear combinations are nonsingular”, Proc. Amer. Math. Soc., 16:2 (1965), 318–322 | DOI | MR