View source for Manifold of figures (lines, surfaces, spheres)

A manifold whose constituent elements are various figures in a certain [[Homogeneous space|homogeneous space]]. From an analytic point of view the simplest figures are algebraic curves and surfaces. So the manifolds that have been investigated (as a rule, in Euclidean, affine and projective spaces) mainly have as constituent elements points, straight lines, planes, circles, spheres, conics, quadrics, and multi-dimensional analogues of these.

An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m0622202.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m0622203.png" /> of figures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m0622204.png" /> of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m0622205.png" /> is defined by a closed system of differential equations

<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/m/m062/m062220/m0622206.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</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/m/m062/m062220/m0622207.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m0622208.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m0622209.png" />) are the left-hand sides of the isotropy equations of a figure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222011.png" /> are the Pfaffian forms arising upon closing the Pfaffian equations of the system (1), with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222012.png" /> (cf. [[Pfaffian form|Pfaffian form]]; [[Pfaffian equation|Pfaffian equation]]). After carrying out a sequence of prolongations of the system (1), one obtains a sequence of fundamental objects of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222013.png" />, selects from it a basic object of the manifold, and carries out an invariant construction of the differential geometry of the manifold.

The differential geometry of ruled manifolds has been developed in depth. The simplest manifolds with non-linear constituent elements are manifolds of conics. With every one-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222014.png" /> of conics in a three-dimensional space (Euclidean, affine or projective) there is associated a torse <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222015.png" />, which is the envelope of the planes of the conics. A manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222016.png" /> is called focal or non-focal, depending on whether a generator of the torse is tangent to the conics or not. A two-dimensional manifold (a congruence) of conics in a three-dimensional space has, in general, six focal surfaces and six focal families. All the conics in the congruence are tangent to these surfaces. Congruences of conics with indefinite focal families (every two adjacent conics of which intersect up to second order) are characterized by all the conics in the congruence belonging to one quadric. Congruences of conics whose planes form a one-parameter family have one quadruplicate focal family to which correspond four focal points of intersection of two adjacent conics belonging to one plane. The two other focal points are points of intersection of a conic with a characteristic of its plane.

A congruence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222017.png" /> of quadrics in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222018.png" /> has, in general, eight focal surfaces, to which all the quadrics in the congruence are tangent. A point of a quadric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222019.png" /> of the congruence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222020.png" /> defined along any direction by the system of equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222021.png" />, is called a focal point of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222023.png" /> of this quadric. A focal point of the second order is a fourfold point of the first order; a focal point of third order is a focal point of arbitrary order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222024.png" />. On each conic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222025.png" /> of a three-dimensional manifold (a complex) of conics there are, in general, six invariant points (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222027.png" />-focal points of the conic). For each conic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222028.png" /> of a complex whose planes form a two-parameter family, there is a unique conic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222029.png" /> passing through the characteristic point of the plane of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222030.png" /> and the four points of intersection of the conic with an adjacent conic in the same plane. The geometric properties of multi-parameter families of conics depend essentially on the number of parameters that characterize the planes of the conics of these families.

An immediate generalization of a conic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222031.png" /> is a quadratic element — an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222032.png" />-dimensional non-degenerate quadric in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222033.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222034.png" />). An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222035.png" />-parameter family of quadratic elements whose hyperplanes form an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222036.png" />-parameter family is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222038.png" />-manifold in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222039.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222040.png" />-manifolds, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222041.png" />, are the most general one-parameter families, congruences and complexes of conics in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222042.png" />. With each local quadratic elements of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222043.png" />-manifold for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222044.png" /> is associated an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222045.png" />-dimensional characteristic subspace and an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222046.png" />-dimensional polar subspace. The rank of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222048.png" />-manifold is the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222049.png" /> equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222050.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222051.png" /> is the dimension of the subspace in which the characteristic subspace intersects its polar subspace.

The differential geometry of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222052.png" />-manifold can be regarded as the geometry of some regular hypersurface of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222053.png" />-dimensional centro-projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222054.png" /> in which the original <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222055.png" />-dimensional space plays the role of the fixed point.

A quadric of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222056.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222057.png" /> always lies in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222058.png" />-dimensional subspace. Algebraic surfaces of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222059.png" /> do not have this property, in general. In investigating manifolds of algebraic surfaces it turns out to be advisable to select families of surfaces that lie in planes of dimension higher by one. A non-degenerate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222060.png" />-dimensional algebraic surface of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222061.png" /> belonging to a hyperplane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222062.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222063.png" /> is called a plane algebraic element of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222065.png" />. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222066.png" />-dimensional manifold of plane algebraic elements of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222067.png" /> whose hyperplanes form an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222068.png" />-parameter family is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222070.png" />-manifold. A fundamental object of the first order is the basic object of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222071.png" />-manifold.

In applications of geometry to hydromechanics, field theory and differential equations one uses manifolds of lines and surfaces that are not algebraic, in general. The study of such manifolds is of interest for geometry in its own right. A curvilinear congruence in a three-dimensional space is a two-dimensional family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222072.png" /> of curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222073.png" /> such that through each point of the space passes, in general, a unique curve of the family. For fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222075.png" /> a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222076.png" /> of the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222077.png" /> is distinguished; for fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222078.png" /> and variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222079.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222080.png" /> a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222081.png" /> is distinguished which is called a transversal surface of the congruence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222082.png" />. Points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222083.png" /> at which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222084.png" /> are called focal points. The set of focal points of the curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222085.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222086.png" /> is called the focal surface of the congruence. A surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222087.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222088.png" /> on which the curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222089.png" /> have an envelope is called a principal surface of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062220/m06222090.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> V.S. Malakhovaskii, "Differential geometry of manifolds of figures" ''J. Soviet Math.'' , '''21''' : 2 (1983) pp. 127–150 ''Itogi Nauk. i Tekhn. Probl. Geom.'' , '''12''' (1981) pp. 31–60</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) {{MR|0507725}} {{ZBL|0408.14001}} </TD></TR></table>