# Projective determination of a metric

An introduction in subsets of a projective space, by methods of projective geometry, of a metric such that these subsets become isomorphic to a Euclidean, hyperbolic or elliptic space. This is achieved by distinguishing in the class of all projective transformations (cf. Projective transformation) those transformations that generate in these subsets a group of transformations isomorphic to the corresponding group of motions. The presence of motions allows one "to lay off" segments of a straight line from a given point in a given direction, thereby introducing the concept of the length of a segment.

To obtain the Euclidean determination of a metric in the $n$- dimensional projective space $P$, one should distinguish in this space an $( n - 1 )$- dimensional hyperplane $\pi$, called the ideal hyperplane, and establish in this hyperplane an elliptic polar correspondence $\Pi$ of points and $( n - 2 )$- dimensional hyperplanes (that is, a polar correspondence under which no point belongs to the $( n - 2 )$- dimensional plane corresponding to it).

Suppose that $E _ {n}$ is a subset of the projective space $P$ obtained by removing from it an ideal hyperplane; and let $X, Y , X ^ \prime , Y ^ \prime$ be points in $E _ {n}$. Two segments $XY$ and $X ^ \prime Y ^ \prime$ are said to be congruent if there exists a projective transformation $\phi$ taking the points $X$ and $Y$ to the points $X ^ \prime$ and $Y ^ \prime$, respectively, and preserving the polarity $\Pi$.

The concept of congruence of segments thus defined allows one to introduce a metric of a Euclidean space in $E _ {n}$. For this, in the projective space $P$ a system of projective coordinates is introduced with the basis simplex $OA _ {1} {} \dots A _ {n}$, where the point $O$ does not not belong to the ideal hyperplane $\pi$ while the points $A _ {1} \dots A _ {n}$ do. Suppose that the point $O$ in this coordinate system has the coordinates $0 \dots 0 , 1$, and that the points $A _ {i}$, $i = 1 \dots n$, have the coordinates

$$x _ {1} = 0 \dots x _ {i-} 1 = 0 ,\ x _ {i} = 1 , x _ {i+} 1 = 0 \dots x _ {n+} 1 = 0 .$$

Then the elliptic polar correspondence $\Pi$ defined in the hyperplane $\pi$ can be written in the form

$$u _ {i} = \sum _ { j= } 1 ^ { n } a _ {ij} x _ {j} ,\ \ i = 1 \dots n .$$

The matrix $( a _ {ij} )$ of this correspondence is symmetric, and the quadratic form

$$Q ( x _ {1} \dots x _ {n} ) = \sum a _ {ij} x _ {i} x _ {j}$$

corresponding to it is positive definite. Let

$$X = ( a _ {1} : \dots : a _ {n+} 1 ) \ \ \textrm{ and } \ Y = ( b _ {1} : \dots : b _ {n+} 1 )$$

be two points in $E _ {n}$( that is, $a _ {n+} 1 \neq 0$, $b _ {n+} 1 \neq 0$). One may set

$$\frac{a _ 1}{a _ n+} 1 = x _ {1} \dots \frac{a _ n}{a _ n+} 1 = x _ {n} ;$$

$$\frac{b _ 1}{b _ n+} 1 = y _ {1} \dots \frac{b _ n}{b _ n+} 1 = y _ {n} .$$

Then the distance $\rho$ between the points $X$ and $Y$ is defined by

$$\rho ( X , Y ) = \sqrt {Q ( x _ {1} - y _ {1} \dots x _ {n} - y _ {n} ) } .$$

For a projective determination of the metric of the $n$- dimensional hyperbolic space, in the $n$- dimensional projective space $P$ a set $U$ of interior points of a real oval hypersurface $S$ of order two is considered. Let $X , Y , X ^ \prime , Y ^ \prime$ be points in $U$; then the segments $XY$ and $X ^ \prime Y ^ \prime$ are assumed to be congruent if there is a projective transformation of the space $P$ under which the hypersurface $S$ is mapped onto itself and the points $X$ and $Y$ are taken to the points $X ^ \prime$ and $Y ^ \prime$, respectively. The concept of congruence of segments thus introduced establishes in $U$ the metric of the hyperbolic space. The length of a segment in this metric is defined by

$$\rho ( X , Y ) = c | \mathop{\rm ln} ( XYPQ ) | ,$$

where $P$ and $Q$ are the points of intersection of the straight line $XY$ with the hypersurface $S$ and $c$ is a positive number related to the curvature of the Lobachevskii space. To introduce an elliptic metric in the projective space $P$, one considers an elliptic polar correspondence $\Pi$ in this space. Two segments $XY$ and $X ^ \prime Y ^ \prime$ are said to be congruent if there exists a projective transformation $\phi$ taking the points $X$ and $Y$ to the points $X ^ \prime$ and $Y ^ \prime$, respectively, and preserving the polar mapping $\Pi$( that is, for any point $M$ and its polar $m$, the polar of the point $\phi ( M)$ is $\phi ( m)$). If the elliptic polar correspondence $\Pi$ is given by the relations

$$u _ {i} = \sum _ { j= } 1 ^ { n+ } 1 a _ {ij} x _ {j} ,\ \ i = 1 \dots n + 1 ,$$

then the matrix $( a _ {ij} )$ is symmetric and the quadratic form corresponding to it is positive definite. Now, if

$$X = ( x _ {1} : \dots : x _ {n+} 1 ) ,\ \ Y = ( y _ {1} : \dots : y _ {n+} 1 ) ,$$

then

$$\rho ( X , Y ) = \mathop{\rm arccos} \frac{| B ( X , Y ) | }{\sqrt {B ( X , X ) } \sqrt {B ( Y , Y ) } } ,$$

where $B$ is the bilinear form given by the matrix $( a _ {ij} )$.

In all the cases considered (if a real projective space is completed to a complex projective space), under the projective transformations defining the congruence of segments, that is, under motions, some hypersurfaces of the second order remain invariant; these are called absolutes. In the case of a Euclidean determination of a metric, the absolute is an imaginary $( n - 2 )$- dimensional oval surface of order two. In the case of a hyperbolic determination of a metric, the absolute is an oval $( n - 1 )$- dimensional real hypersurface of order two. In the case of an elliptic determination of a metric, the absolute is an imaginary $( n - 1 )$- dimensional oval hypersurface of order two.

#### References

 [1] N.V. Efimov, "Higher geometry" , MIR (1980) (Translated from Russian) [2] N.A. Glagolev, "Projective geometry" , Moscow (1963) (In Russian) [3] H. Busemann, P.J. Kelly, "Projective geometry and projective metrics" , Acad. Press (1953)