Figure

A subset $F$ of a homogeneous space $E^n$ with fundamental group $G$ that can be included in a system $R(F)$ of subsets of this space isomorphic to some space of a geometric object $\Phi$ (see Geometric objects, theory of). $R(F)$ is called the figure space of $F$. The components of $\Phi$ are called the coordinates of the associated figure $F$. To each figure $F$ in $E^n$ corresponds a class $\\{ \Phi \\}$ of similar geometric objects. The rank, genre, characteristic, and type of a geometric object $\Phi$ in $\\{ \Phi \\}$ are called the rank, genre, characteristic, and type of the figure $F$ (the so-called arithmetic invariants of the figure, cf. [2]). For example, a circle in three-dimensional Euclidean space is a figure of rank 6, genre 1, characteristic 1, and type 1; a point in three-dimensional projective space is a figure of rank 3, genre 0, characteristic 2, and type 1. The completely-integrable system of Pfaffian equations defining the geometric object $\Phi$ is called the stationarity system of equations of $F$.

Let $F$ and $\bar F$ be two figures in $E^n$. If there is a mapping of $R(F)$ onto $R(\bar F)$ under which every geometric object corresponding to $\bar F$ is covered by every geometric object corresponding to $F$, then one says that $F$ covers or induces $\bar F$ ($\bar F$ is said to be covered or induced by $F$). A figure $F$ of rank $N$ is called simple if it does not cover any other figure of lower rank. $F$ is called an inducing figure of index $\bar N < N$ if there is a figure of rank $\bar N$ that is covered by $F$, while the rank $N'$ of any other figure $F'$ covered by $F$ does not exceed $\bar N$. For example, a point, a $p$-dimensional plane and a hyperquadric in an $n$-dimensional projective space are simple figures, and a hyperquadric in an $n$-dimensional affine space and a $d$-dimensional $(d \le n-2)$ quadric in an $n$-dimensional projective space are inducing figures of indices $n$ and $(d+2)(n-d-1)$, respectively.

An ordered set of two figures, $F = (F_1, F_2)$, is called a figure pair. The incidence coefficient of a figure pair is the number $k = N_1 + N_2 - N$, where $N_i$ $(i = 1, 2)$ is the rank of $F_i$, and $N$ is the rank of the system of forms $\Omega^{J_1}, \Omega^{J_2}$, $J_i = 1, \dots, N_i$, that are the left-hand sides of the stationarity equations of $F_1$ and $F_2$. If $k=0$, then the pair $(F_1, F_2)$ is called non-incident.

#### References[edit]

[1] |  G.F. Laptev, "Differential geometry of imbedded manifolds. Group-theoretical method of differential-geometric investigation" Trudy Moskov. Mat. Obshch. , 2 (1953) pp. 275–383 (In Russian)

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[2] |  V.S. Malakhovskii, "Differential geometry of manifolds of figures and of figure pairs" Trudy Geom. Sem. Inst. Nauchn. Inform. Akad. Nauk SSSR , 2 (1969) pp. 179–206 (In Russian)
[3] |  V.S. Malakhovskii, "Differential geometry of lines and surfaces" J. Soviet Math. , 2 (1974) pp. 304–330 Itogi Nauk. i Tekhn. Algebra. Topol. Geom. , 10 (1972) pp. 113–158