View source for Rotations diagram

A surface in the elliptic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082690/r0826901.png" /> defined by isometric smooth surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082690/r0826902.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082690/r0826903.png" /> in the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082690/r0826904.png" /> in the same way as the [[Rotation indicatrix|rotation indicatrix]] is defined for infinitesimal deformations in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082690/r0826905.png" />. L. Bianchi was the first to mention surfaces in elliptic space coincident with the rotations diagram, in examining the spherical representation of a deformation base for surfaces, where he showed that it coincides with the representation in the sense of Clifford for the asymptotic lines in the rotations diagram.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082690/r0826906.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082690/r0826907.png" /> be isometric smooth identically-oriented surfaces. At the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082690/r0826908.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082690/r0826909.png" /> corresponding in isometry, the trihedra formed by the tangent vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082690/r08269010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082690/r08269011.png" /> to the corresponding isometric pairs of curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082690/r08269012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082690/r08269013.png" /> and the normals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082690/r08269014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082690/r08269015.png" /> are equal, i.e.

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082690/r08269016.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082690/r08269017.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082690/r08269018.png" /></td> </tr></table>

and therefore one of them is obtained from another by rotation around an axis with direction unit vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082690/r08269019.png" /> through an angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082690/r08269020.png" /> (defined up to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082690/r08269021.png" />). Let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082690/r08269022.png" /></td> </tr></table>

be the [[Quaternion|quaternion]], equal in modulus to 1 and defined up to its sign, representing this rotation. The set of such quaternions parametrized by the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082690/r08269023.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082690/r08269024.png" />) defines a set of points in the elliptic space, which is called the rotations diagram for the isometric surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082690/r08269025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082690/r08269026.png" />. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082690/r08269027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082690/r08269028.png" /> are isometric pieces of cylinders, then the rotations diagram is a part of a Clifford surface, and a minimal Clifford surface corresponds to circular cylinders. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082690/r08269029.png" />, there is an elliptic plane outside the rotations diagram, and under a geodesic mapping of the elliptic space into the Euclidean space,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082690/r08269030.png" /></td> </tr></table>

the image of the rotations diagram is the rotation indicatrix for some infinitesimal deformation of the median surface corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082690/r08269031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082690/r08269032.png" /> (see [[Cohn-Vossen transformation|Cohn-Vossen transformation]]) (it is regular if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082690/r08269033.png" />).

The properties of the rotations diagrams for isometric surfaces of positive Gaussian curvature are analogous to those of rotation indicatrices; for example, the specific internal curvature of the rotations diagram is always negative, and therefore it plays the same part as the rotation indicatrix when examining the isometry of convex surfaces.



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Spivak,   "A comprehensive introduction to differential geometry" , '''1979''' , Publish or Perish  pp. 1–5</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.V. Pogorelov,   "Intrinsic geometry of surfaces" , Amer. Math. Soc.  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.V. Pogorelov,   "Extrinsic geometry of convex surfaces" , Amer. Math. Soc.  (1973)  (Translated from Russian)</TD></TR></table>