View source for Lacunary space

A space with an [[Affine connection|affine connection]] or a [[Riemannian space|Riemannian space]] of some definite  "degree of mobility of a space with an affine connectiondegree of mobility" . A lacunary space is defined by the order of the complete [[Group of motions|group of motions]], that is, by the largest number of parameters of that group for the given space. Thus, ordinary $n$-dimensional affine space admits a group of motions of maximal order $n^2+n$.

<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l0571003.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l0571004.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l0571005.png" /></td> <td colname="4" style="background-color:white;" colspan="1">Types of spaces and lacunae</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l0571006.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l0571007.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l0571008.png" /></td> <td colname="4" style="background-color:white;" colspan="1">Spaces of first lacunarity</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"></td> <td colname="4" style="background-color:white;" colspan="1">First lacuna</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l0571009.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710010.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710011.png" /></td> <td colname="4" style="background-color:white;" colspan="1">Spaces of second lacunarity</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710012.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710013.png" /></td> <td colname="3" style="background-color:white;" colspan="1"></td> <td colname="4" style="background-color:white;" colspan="1"></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710014.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710015.png" /></td> <td colname="3" style="background-color:white;" colspan="1"></td> <td colname="4" style="background-color:white;" colspan="1"></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">1</td> <td colname="2" style="background-color:white;" colspan="1">1</td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710016.png" /></td> <td colname="4" style="background-color:white;" colspan="1"></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"></td> <td colname="4" style="background-color:white;" colspan="1">Second lacuna</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710017.png" /></td> <td colname="4" style="background-color:white;" colspan="1">Spaces of third lacunarity</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710018.png" /></td> <td colname="4" style="background-color:white;" colspan="1"></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710019.png" /></td> <td colname="4" style="background-color:white;" colspan="1"></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"></td> <td colname="4" style="background-color:white;" colspan="1">Third lacuna</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710020.png" /></td> <td colname="4" style="background-color:white;" colspan="1">Spaces of fourth lacunarity</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710021.png" /></td> <td colname="4" style="background-color:white;" colspan="1"></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710022.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710023.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1">3</td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710024.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1">2</td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710025.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1">1</td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710026.png" /></td> </tr> </tbody> </table>

</td></tr> </table>

The orders of the complete groups of motions of other spaces with an affine connection belong to the interval of natural numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710029.png" />, but not every number of this interval can be the order of a complete group of motions. Intervals of greatest length composed of numbers that are not the order of a complete group of motions are called lacunae, and the complements of them with respect to the given interval are called intervals of condensation. A space is called a space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710031.png" />-th lacunarity if the order of its complete group of motions belongs to the interval of condensation having number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710032.png" />. The calculation is carried out with an interval of condensation containing the maximal order. For the well-known distribution of the number of motions of  "rigid"  bodies in spaces with an affine connection see Table 1. It deals with the orders of complete groups of motions — synonyms for the degree of mobility or the degree of freedom of rigid bodies. The question of determining the possible degrees of mobility of rigid and similarly varying bodies in Riemannian spaces has been partly solved for spaces with a metric of fixed sign only. In the general case the degrees of mobility and the Riemannian spaces realizing them that are known are exhibited in Table 2.

<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710033.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710034.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710035.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710036.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710037.png" /></td> <td colname="6" style="background-color:white;" colspan="1">Types of spaces and lacunae</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710038.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710039.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710040.png" /></td> <td colname="4" style="background-color:white;" colspan="1">15</td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710041.png" /></td> <td colname="6" style="background-color:white;" colspan="1">Spaces of first lacunarity (constant curvature)</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"></td> <td colname="4" style="background-color:white;" colspan="1"></td> <td colname="5" style="background-color:white;" colspan="1"></td> <td colname="6" style="background-color:white;" colspan="1">First lacuna</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710042.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710043.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710044.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710045.png" /></td> <td colname="6" style="background-color:white;" colspan="1">Spaces of second lacunarity</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710046.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710047.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710048.png" /></td> <td colname="5" style="background-color:white;" colspan="1"></td> <td colname="6" style="background-color:white;" colspan="1"></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">1</td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710049.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710050.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710051.png" /></td> <td colname="5" style="background-color:white;" colspan="1"></td> <td colname="6" style="background-color:white;" colspan="1"></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710052.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710053.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710054.png" /></td> <td colname="5" style="background-color:white;" colspan="1"></td> <td colname="6" style="background-color:white;" colspan="1"></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710055.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710056.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710057.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710058.png" /></td> <td colname="6" style="background-color:white;" colspan="1"></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"></td> <td colname="4" style="background-color:white;" colspan="1"></td> <td colname="5" style="background-color:white;" colspan="1"></td> <td colname="6" style="background-color:white;" colspan="1">Second lacuna</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"></td> <td colname="4" style="background-color:white;" colspan="1"></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710059.png" /></td> <td colname="6" style="background-color:white;" colspan="1">Spaces of third lacunarity</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"></td> <td colname="4" style="background-color:white;" colspan="1"></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710060.png" /></td> <td colname="6" style="background-color:white;" colspan="1"></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"></td> <td colname="4" style="background-color:white;" colspan="1"></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710061.png" /></td> <td colname="6" style="background-color:white;" colspan="1"></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"></td> <td colname="4" style="background-color:white;" colspan="1"></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710062.png" /></td> <td colname="6" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710063.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"></td> <td colname="4" style="background-color:white;" colspan="1"></td> <td colname="5" style="background-color:white;" colspan="1">3</td> <td colname="6" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710064.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"></td> <td colname="4" style="background-color:white;" colspan="1"></td> <td colname="5" style="background-color:white;" colspan="1">2</td> <td colname="6" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710065.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"></td> <td colname="4" style="background-color:white;" colspan="1"></td> <td colname="5" style="background-color:white;" colspan="1">1</td> <td colname="6" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710066.png" /></td> </tr> </tbody> </table>

</td></tr> </table>

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.P. Egorov,   "Motions in generalized differential-geometric spaces"  ''Progress in Math.'' , '''6'''  (1970)  pp. 171–228  ''Itogi Nauk. Algebra Topol. Geom. 1965''  (1967)  pp. 375–428</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.P. Egorov,   "Automorphisms in generalized spaces"  ''J. Soviet Math.'' , '''14'''  (1980)  pp. 1260–1287  ''Itogi Nauk. i Tekhn. Probl. Geom.'' , '''10'''  (1978)  pp. 147–191</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.P. Egorov,   "Geometry" , Moscow  (1979)  (In Russian)</TD></TR></table>



====Comments====
The Riemannian manifolds of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710069.png" /> with group of isometries of (the maximal) dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710070.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710071.png" />-dimensional Euclidean space, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710072.png" />-dimensional sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710073.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710074.png" />-dimensional real projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710075.png" />, and an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710076.png" />-dimensional simply-connected hyperbolic space.

For a list of Riemannian manifolds of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710077.png" /> with a  "degree of mobility"  at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710078.png" />, and the corresponding groups cf. [[#References|[a1]]], p. 54.

At the other end of the scale there is, for example, the theorem of Bochner, [[#References|[a4]]], that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710079.png" /> is a compact Riemannian manifold with negative Ricci tensor, then its group of isometries is finite.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710080.png" /> be a manifold of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710081.png" />. For each Riemannian metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710082.png" /> let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710083.png" /> be the group of isometries of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710084.png" />. The degree of symmetry of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710086.png" /> is the maximum of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710087.png" />.

For a description of the possible isometry groups of dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710088.png" /> cf. [[#References|[a2]]], [[#References|[a3]]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Kobayashi,   "Transformation groups in differential geometry" , Springer  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Kobayashi,   T. Nagano,   "Riemannian manifolds with abundant symmetries" , ''Differential geometry (in honor of K. Yano)'' , Kinokuniya  (1972)  pp. 195–220</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Wakakuwa,   "On <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710089.png" />-dimensional Riemannian spaces admitting some groups of motions of order less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057100/l05710090.png" />"  ''Tôhoku Math. J.'' , '''6'''  (1954)  pp. 121–134</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  S. Bochner,   "Vector fields and Ricci curvature"  ''Bull. Amer. Math. Soc.'' , '''52'''  (1946)  pp. 776–797</TD></TR>
</table>

{{TEX|want}}