View source for Conformal-differential geometry

A branch of [[Conformal geometry|conformal geometry]] in which the geometric quantities that are invariant under conformal transformations are studied by the methods of analysis, in the first instance, differential calculus.

In the conformal plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c0247401.png" /> each point or circle is defined by a vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c0247402.png" />, where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c0247403.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c0247404.png" />, are the so-called [[Tetracyclic coordinates|tetracyclic coordinates]]. For a point one has

<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/c/c024/c024740/c0247405.png" /></td> </tr></table>

and for a circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c0247406.png" />. The conformal-differential geometry in the plane studies sequences and congruences of circles. To a sequence of circles corresponds a curve in three-dimensional hyperbolic space and to a congruence of circles — a surface. A sequence is given by a parametrization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c0247407.png" />. The parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c0247408.png" /> can be specialized to

<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/c/c024/c024740/c0247409.png" /></td> </tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c02474010.png" /> being the angle between two infinitesimally-close circles in the sequence. Of special significance in the theory of sequences are the two branches of the envelope of this sequence, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c02474011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c02474012.png" />, their osculating circles. As in ordinary differential geometry of curves, one can write derivation formulas for a sequence of circles by decomposing the derivatives of the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c02474013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c02474014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c02474015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c02474016.png" /> in terms of themselves:

<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/c/c024/c024740/c02474017.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/c/c024/c024740/c02474018.png" /></td> </tr></table>

Two invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c02474019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c02474020.png" /> can be obtained. The invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c02474021.png" /> is expressed in terms of the angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c02474022.png" /> between the osculating circles of the envelope: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c02474023.png" />. The theory of curves in the conformal plane is constructed from the theory of sequences of circles: each curve is regarded as an envelope, namely the sequence with invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c02474024.png" />. If, furthermore, the invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c02474025.png" /> is constant, then the curve turns out to be the isogonal trajectory of a pencil of circles, that is, a loxodrome.

In the three-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c02474026.png" /> the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c02474027.png" /> defines a sequence of spheres. In its study its enveloping surface, the so-called [[Canal surface|canal surface]], plays an important role. Each sequence of spheres is characterized by three invariants, which are expressed in terms of certain angles determined by the spheres of the sequence.

A congruence of circles in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c02474028.png" /> is given by a parametrization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c02474029.png" />. On the surface corresponding to it in the hyperbolic space, the polar normalization is conveniently introduced by taking as the normal of the first kind the line orthogonal to the tangent plane of the surface at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c02474030.png" />, and as the normal of the second kind, the polar of the normal of the first kind with respect to the absolute <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c02474031.png" /> (see [[#References|[3]]]). In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c02474032.png" /> the normalization of a congruence corresponds to the normalization of the surface: Associated with each circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c02474033.png" /> is the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c02474034.png" /> orthogonal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c02474035.png" /> and to each infinitesimally-close circle, and two circles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c02474036.png" />, defining the pencil of the circles, conjugate to the pencil <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c02474037.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c02474038.png" />. On <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c02474039.png" /> invariants of the congruence of circles are defined corresponding to the invariants of the associated surface in hyperbolic space, and special types of congruences are singled out.

In the theory of surfaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c02474040.png" /> a framing of the surface is introduced by means of normal circles orthogonal at each point to all tangent spheres of the surface; associated with each point is a conformal frame consisting of a point of the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c02474041.png" />, two coordinate spheres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c02474042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c02474043.png" />, defining normal circles, a tangent sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c02474044.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c02474045.png" />, and the point of intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c02474046.png" /> of this sphere and the normal circle. In the general theory of normalization of surfaces one uses the isomorphism of the theory of normalized surfaces of a conformal space and the theory of interior polar normalizations of the absolute of a hyperbolic space. The interior geometry of a normalized surface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024740/c02474047.png" /> is a Weyl geometry whose first fundamental tensor is the same as the tensor of the angular metric of the surface, while the second fundamental tensor is the normalizer defining the supporting coordinates of the sphere.

The results obtained for a normalized surface are valid also for a normalized conformal space.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Darboux,   "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , '''1–4''' , Gauthier-Villars  (1887–1896)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Blaschke,   "Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einstein's Relativitätstheorie" , '''3. Differentialgeometrie der Kreisen und Kugeln''' , Springer  (1929)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.P. Norden,   "Spaces with an affine connection" , Nauka , Moscow-Leningrad  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  G.V. Bushmanova,   A.P. Norden,   "Elements of conformal geometry" , Kazan'  (1972)  (In Russian)</TD></TR></table>