Semi-Euclidean space

A real affine $ n $- space equipped with a scalar product of vectors such that, relative to a suitably chosen basis, the scalar product $ ( \mathbf x , \mathbf x ) $ of any vector with itself has the form

$$ ( \mathbf x , \mathbf x ) = - \sum _ {i=1}^ { l } ( x ^ {i} ) ^ {2} + \sum _ { j=l+ 1} ^ { n-d} ( x ^ {j} ) ^ {2} . $$

Under these conditions, the semi-Euclidean space is said to have index $ l $ and deficiency $ d $ and is denoted by $ {} ^ {l + ( d ) } R _ {n} $. If $ l = 0 $, the expression for the scalar product of a vector with itself is a semi-definite quadratic form and the space is called an $ n $- space of deficiency $ d $, denoted by $ {} ^ {( d ) } R _ {n} $.

In the projective classification, a semi-Euclidean space can be defined as a semi-elliptic space or a semi-hyperbolic space with an improper absolute plane; these are spaces with projective metrics of the most general form.

One defines a semi-non-Euclidean space as a metric $ n $- space which is a hypersphere with identified antipodal points in the semi-Euclidean space of index $ l $ and deficiency $ d $. Thus, semi-elliptic and semi-hyperbolic spaces may be interpreted as hyperspheres of the above type (i.e. as semi-non-Euclidean spaces) in semi-Euclidean spaces of appropriate index and deficiency.

The geometrical interpretation of Galileo–Newton mechanics leads to the semi-Euclidean space $ {} ^ {( 1 ) } R _ {n} $( see [2]).

A semi-Euclidean space is a semi-Riemannian space of curvature zero.

References

[1]	D.M.Y. Sommerville, "Classification of geometries with projective metric" Proc. Edinburgh Math. Soc. , 28 (1910) pp. 25–41
[2]	A.P. Kotel'nikov, "The principle of relativity and Lobachevskii geometry" , In memoriam N.I. Lobachevskii , 2 , Kazan' (1926) (In Russian)
[3]	B.A. Rozenfel'd, "Non-Euclidean spaces" , Moscow (1969) (In Russian)
[a1]	B.A. [B.A. Rozenfel'd] Rosenfel'd, "A history of non-euclidean geometry" , Springer (1988) (Translated from Russian)