Deformation over a principal base

A deformation $F_t$ of a surface $F=F_0$ under which the directions of extremal deformation remain unaltered. The net formed by the curves which have the direction of extremal deformation is conjugate on each surface $F_t$ and is called the principal base of the deformation. For example, a helicoid has an infinite number of principal bases; surfaces of rotation and general canal surfaces allow a deformation over a principal base with geodesics as one family (see also Voss surface). The problem of investigating a deformation over a principal base was posed by K.M. Peterson [1]; in 1866 he established that if a surface $F$ is isometrically transformed into two surfaces $F'$ and $F''$ such that the directions of extremal deformation (and consequently, the base of the deformation, cf. Base of a deformation) from $F$ to $F'$ coincide with the directions of extremal deformation of $F$ to $F''$, then a deformation $F_t$ of the surface $F$ exists which includes $F'$ and $F''$, with the same directions of extremal deformation. In other words, if a conjugate net on $F$ serves as the base of two different deformations $F'$ and $F''$, then it is a principal base of deformation.

If the surfaces $F$, $F'$ and $F''$ are known, then all remaining surfaces $F_t$ obtainable by deforming $F$ over a principal base are determined by the following theorem: Let $\kappa$ be the normal curvature of $F$ in the direction of one of the two families of the principal base $\sigma$ at an arbitrary point $M\in F$, while $\kappa'$, $\kappa''$, $\kappa_t$ are the normal curvatures of the surfaces $F'$, $F''$ and $F_t$ at the corresponding points and in the corresponding directions, then the cross ratio $t=(\kappa^2,\kappa'^2,\kappa''^2,\kappa_t^2)$ is a constant quantity for all positions of $M$ on $F$.

A surface which allows a deformation over a principal base can be characterized by only the spherical image of the principal base: The equations which describe a deformation over a principal base are transformed so as to contain only the coefficients of the line element of the spherical image of the surface and take the form: $\partial\Gamma_{12}^1/\partial u=\partial\Gamma_{12}^2/\partial v=2\Gamma_{12}^1\Gamma_{12}^2$ (Kosser's equation), where $\Gamma_{12}^1$, $\Gamma_{12}^2$ are the Christoffel symbols of the third fundamental form of the surface, while the differentiation takes place along the coordinate lines $u,v$ which form the principal base of the deformation. The spherical image of the principal base of the deformation coincides with the spherical image of the asymptotic lines of the Bianchi surface which is the rotation indicatrix (or the adjoined surface) of the infinitesimal deformation of $F$ corresponding to the deformation over a principal base, as well as to the Clifford image of the asymptotic lines of a surface in an elliptic space (which is the rotations diagram of the deformation over a principal base of $F$).

Not all surfaces have a principal base; surfaces which allow of a deformation over a principal base form a special class of surfaces . A deformation over a kinematic base is a generalization of a deformation over a principal base; it is defined by the fact that the coefficients $b_{ij}$ of the second fundamental form satisfy the equation $b_{ij}A^{ij}=c$, where $A^{ij}$ is a non-degenerate tensor and $c$ is a function depending on the metric $g_{ij}$ of the surface $F$ and on its derivatives.

References

[1]	K.M. Peterson, Mat. Sb. , 1 (1866) pp. 391–438
[2]	V.F. Kagan, "Foundations of the theory of surfaces in a tensor setting" , 2 , Moscow-Leningrad (1948) (In Russian)
[3]	S.P. Finikov, "Deformation over a principal base and related problems in geometry" , Moscow-Leningrad (1937) (In Russian)
[4a]	N.N. Luzin, "Proof of a theorem in deformation theory" Izv. Akad. Nauk. SSSR Otd. Tekhn. Nauk , 2 (1939) pp. 81–105 (In Russian)
[4b]	N.N. Luzin, "Proof of a theorem in deformation theory" Izv. Akad. Nauk. SSSR Otd. Tekhn. Nauk , 7 (1939) pp. 115–132 (In Russian)
[4c]	N.N. Luzin, "Proof of a theorem in deformation theory" Izv. Akad. Nauk. SSSR Otd. Tekhn. Nauk , 10 (1939) pp. 65–84 (In Russian)