View source for Deformation over a principal base

A deformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d0307201.png" /> of a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d0307202.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d0307203.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d0307204.png" /> is isometrically transformed into two surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d0307205.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d0307206.png" /> such that the directions of extremal deformation (and consequently, the base of the deformation, cf. [[Base of a deformation|Base of a deformation]]) from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d0307207.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d0307208.png" /> coincide with the directions of extremal deformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d0307209.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d03072010.png" />, then a deformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d03072011.png" /> of the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d03072012.png" /> exists which includes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d03072013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d03072014.png" />, with the same directions of extremal deformation. In other words, if a [[Conjugate net|conjugate net]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d03072015.png" /> serves as the base of two different deformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d03072016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d03072017.png" />, then it is a principal base of deformation.

If the surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d03072018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d03072019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d03072020.png" /> are known, then all remaining surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d03072021.png" /> obtainable by deforming <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d03072022.png" /> over a principal base are determined by the following theorem: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d03072023.png" /> be the normal curvature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d03072024.png" /> in the direction of one of the two families of the principal base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d03072025.png" /> at an arbitrary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d03072026.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d03072027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d03072028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d03072029.png" /> are the normal curvatures of the surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d03072030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d03072031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d03072032.png" /> at the corresponding points and in the corresponding directions, then the cross ratio <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d03072033.png" /> is a constant quantity for all positions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d03072034.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d03072035.png" />.

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: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d03072036.png" /> (Kosser's equation), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d03072037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d03072038.png" /> are the Christoffel symbols of the third fundamental form of the surface, while the differentiation takes place along the coordinate lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d03072039.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d03072040.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d03072041.png" />).

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d03072042.png" /> of the second fundamental form satisfy the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d03072043.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d03072044.png" /> is a non-degenerate tensor and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d03072045.png" /> is a function depending on the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d03072046.png" /> of the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030720/d03072047.png" /> 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>