Difference between revisions of "Semi-Euclidean space"
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− | Under these conditions, the semi-Euclidean space is said to have index | + | 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|semi-elliptic space]] or a [[Semi-hyperbolic space|semi-hyperbolic space]] with an improper absolute plane; these are spaces with projective metrics of the most general form. | In the projective classification, a semi-Euclidean space can be defined as a [[Semi-elliptic space|semi-elliptic space]] or a [[Semi-hyperbolic space|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 | + | 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 | + | The geometrical interpretation of Galileo–Newton mechanics leads to the semi-Euclidean space $ {} ^ {( 1 ) } R _ {n} $( |
+ | see [[#References|[2]]]). | ||
A semi-Euclidean space is a [[Semi-Riemannian space|semi-Riemannian space]] of curvature zero. | A semi-Euclidean space is a [[Semi-Riemannian space|semi-Riemannian space]] of curvature zero. | ||
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====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> D.M.Y. Sommerville, "Classification of geometries with projective metric" ''Proc. Edinburgh Math. Soc.'' , '''28''' (1910) pp. 25–41</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.P. Kotel'nikov, "The principle of relativity and Lobachevskii geometry" , ''In memoriam N.I. Lobachevskii'' , '''2''' , Kazan' (1926) (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> B.A. Rozenfel'd, "Non-Euclidean spaces" , Moscow (1969) (In Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> D.M.Y. Sommerville, "Classification of geometries with projective metric" ''Proc. Edinburgh Math. Soc.'' , '''28''' (1910) pp. 25–41</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.P. Kotel'nikov, "The principle of relativity and Lobachevskii geometry" , ''In memoriam N.I. Lobachevskii'' , '''2''' , Kazan' (1926) (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> B.A. Rozenfel'd, "Non-Euclidean spaces" , Moscow (1969) (In Russian)</TD></TR></table> | ||
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====Comments==== | ====Comments==== | ||
− | |||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> B.A. [B.A. Rozenfel'd] Rosenfel'd, "A history of non-euclidean geometry" , Springer (1988) (Translated from Russian)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> B.A. [B.A. Rozenfel'd] Rosenfel'd, "A history of non-euclidean geometry" , Springer (1988) (Translated from Russian)</TD></TR></table> |
Revision as of 08:13, 6 June 2020
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) |
Comments
References
[a1] | B.A. [B.A. Rozenfel'd] Rosenfel'd, "A history of non-euclidean geometry" , Springer (1988) (Translated from Russian) |
Semi-Euclidean space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-Euclidean_space&oldid=48651