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A subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f0401201.png" /> of a homogeneous space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f0401202.png" /> with fundamental group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f0401203.png" /> that can be included in a system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f0401204.png" /> of subsets of this space isomorphic to some space of a geometric object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f0401205.png" /> (see [[Geometric objects, theory of|Geometric objects, theory of]]). <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f0401206.png" /> is called the figure space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f0401207.png" />. The components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f0401208.png" /> are called the coordinates of the associated figure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f0401209.png" />. To each figure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012010.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012011.png" /> corresponds a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012012.png" /> of similar geometric objects. The rank, genre, characteristic, and type of a geometric object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012013.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012014.png" /> are called the rank, genre, characteristic, and type of the figure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012015.png" /> (the so-called arithmetic invariants of the figure, cf. [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012016.png" /> is called the stationarity system of equations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012017.png" />.
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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|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. [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012019.png" /> be two figures in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012020.png" />. If there is a mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012021.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012022.png" /> under which every geometric object corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012023.png" /> is covered by every geometric object corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012024.png" />, then one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012025.png" /> covers or induces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012026.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012027.png" /> is said to be covered or induced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012028.png" />). A figure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012029.png" /> of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012030.png" /> is called simple if it does not cover any other figure of lower rank. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012031.png" /> is called an inducing figure of index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012033.png" /> if there is a figure of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012034.png" /> that is covered by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012035.png" />, while the rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012036.png" /> of any other figure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012037.png" /> covered by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012038.png" /> does not exceed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012039.png" />. For example, a point, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012040.png" />-dimensional plane and a hyperquadric in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012041.png" />-dimensional projective space are simple figures, and a hyperquadric in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012042.png" />-dimensional affine space and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012043.png" />-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012044.png" /> quadric in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012045.png" />-dimensional projective space are inducing figures of indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012047.png" />, respectively.
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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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012048.png" />, is called a figure pair. The incidence coefficient of a figure pair is the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012049.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012050.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012051.png" />) is the rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012052.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012053.png" /> is the rank of the system of forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012056.png" />, that are the left-hand sides of the stationarity equations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012058.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012059.png" />, then the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040120/f04012060.png" /> is called non-incident.
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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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  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</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  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</TD></TR></table>

Revision as of 00:41, 27 November 2013

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

[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)
[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
How to Cite This Entry:
Figure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Figure&oldid=14577
This article was adapted from an original article by V.S. Malakhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article