Geodesic
Bianchi congruences of two-dimensional surfaces in $E^4$
Sbornik. Mathematics, Tome 196 (2005) no. 10, pp. 1473-1493 
V. A. Gorkavyy. Bianchi congruences of two-dimensional surfaces in $E^4$. Sbornik. Mathematics, Tome 196 (2005) no. 10, pp. 1473-1493. http://geodesic.mathdoc.fr/item/SM_2005_196_10_a2/
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Voir la notice de l'article provenant de la source Math-Net.Ru

Pseudospherical Bianchi congruences in Euclidean 4-space $E^4$ are considered. The focal surfaces of such congruences are shown to have a constant negative Gaussian curvature. A geometric and an analytic description of special pseudo-spherical surfaces in $E^4$ admitting a Bianchi congruence are obtained.

[1] Aminov Yu. A., Geometriya podmnogoobrazii, Naukova dumka, Kiev, 2000

[2] Tenenblat K., Transformations of manifolds and applications to differential equations, Longman, London, 1998 | MR | Zbl

[3] Tenenblat K., Terng C.-L., “Backlund theorem for $n$-dimensional submanifolds of $R^{2n-1}$”, Ann. of Math. (2), 111 (1980), 477–490 | DOI | MR | Zbl

[4] Terng C.-L., “A higher dimension generalisation of the Sine-Gordon equation and its soliton theory”, Ann. of Math. (2), 111 (1980), 491–510 | DOI | MR | Zbl

[5] Aminov Yu., Sym A., “On Bianchi and Backlund transformations of two-dimensional surfaces in $E^4$”, Math. Phys. Anal. Geom., 3:1 (2000), 75–89 | DOI | MR | Zbl

[6] Eisenhart L. P., A treatise on the differential geometry of curves and surfaces, Ginn, New York, 1909 | Zbl

[7] Shefel S. Z., “Geometricheskie svoistva pogruzhennykh mnogoobrazii”, Sib. matem. zhurn., 26:1 (1985), 170–188 | MR | Zbl

[8] Borisenko A. A., Nikolaevskii Yu. A., “Mnogoobraziya Grassmana i grassmanov obraz podmnogoobrazii”, UMN, 46:2 (1991), 41–83 | MR | Zbl