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 -dimensional projective space , one should distinguish in this space an -dimensional hyperplane , called the ideal hyperplane, and establish in this hyperplane an elliptic polar correspondence of points and -dimensional hyperplanes (that is, a polar correspondence under which no point belongs to the -dimensional plane corresponding to it).

Suppose that is a subset of the projective space obtained by removing from it an ideal hyperplane; and let be points in . Two segments and are said to be congruent if there exists a projective transformation taking the points and to the points and , respectively, and preserving the polarity .

The concept of congruence of segments thus defined allows one to introduce a metric of a Euclidean space in . For this, in the projective space a system of projective coordinates is introduced with the basis simplex , where the point does not not belong to the ideal hyperplane while the points do. Suppose that the point in this coordinate system has the coordinates , and that the points , , have the coordinates

Then the elliptic polar correspondence defined in the hyperplane can be written in the form

The matrix of this correspondence is symmetric, and the quadratic form

corresponding to it is positive definite. Let

be two points in (that is, , ). One may set

Then the distance between the points and is defined by

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

where and are the points of intersection of the straight line with the hypersurface and is a positive number related to the curvature of the Lobachevskii space. To introduce an elliptic metric in the projective space , one considers an elliptic polar correspondence in this space. Two segments and are said to be congruent if there exists a projective transformation taking the points and to the points and , respectively, and preserving the polar mapping (that is, for any point and its polar , the polar of the point is ). If the elliptic polar correspondence is given by the relations

then the matrix is symmetric and the quadratic form corresponding to it is positive definite. Now, if

then

where is the bilinear form given by the matrix .

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 -dimensional oval surface of order two. In the case of a hyperbolic determination of a metric, the absolute is an oval -dimensional real hypersurface of order two. In the case of an elliptic determination of a metric, the absolute is an imaginary -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)