View source for Deformation over a principal base

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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|Voss surface]]). The problem of investigating a deformation over a principal base was posed by K.M. Peterson [[#References|[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|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|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|Bianchi surface]] which is the [[Rotation indicatrix|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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K.M. Peterson,   ''Mat. Sb.'' , '''1'''  (1866)  pp. 391–438</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.F. Kagan,   "Foundations of the theory of surfaces in a tensor setting" , '''2''' , Moscow-Leningrad  (1948)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.P. Finikov,   "Deformation over a principal base and related problems in geometry" , Moscow-Leningrad  (1937)  (In Russian)</TD></TR><TR><TD valign="top">[4a]</TD> <TD valign="top">  N.N. Luzin,   "Proof of a theorem in deformation theory"  ''Izv. Akad. Nauk. SSSR Otd. Tekhn. Nauk'' , '''2'''  (1939)  pp. 81–105  (In Russian)</TD></TR><TR><TD valign="top">[4b]</TD> <TD valign="top">  N.N. Luzin,   "Proof of a theorem in deformation theory"  ''Izv. Akad. Nauk. SSSR Otd. Tekhn. Nauk'' , '''7'''  (1939)  pp. 115–132  (In Russian)</TD></TR><TR><TD valign="top">[4c]</TD> <TD valign="top">  N.N. Luzin,   "Proof of a theorem in deformation theory"  ''Izv. Akad. Nauk. SSSR Otd. Tekhn. Nauk'' , '''10'''  (1939)  pp. 65–84  (In Russian)</TD></TR></table>