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)

References

[a1]	S. Kobayashi, "Transformation groups in differential geometry" , Springer (1972)