Subprojective space

One of the generalizations of spaces of constant curvature (of projective space). One defines a $ k $- fold projective space with an affine connection and expresses its geodesic lines in some coordinate system by a system of $ ( n- 1) $ equations of which exactly $ k $ are linear. When $ k= n- 2 $, the geodesic lines are planar, and are situated in two-dimensional Euclidean planes, and the space is said to be subprojective if all these two-dimensional Euclidean planes pass through a common point or are parallel in one direction (the common point is infinitely distant).

Let $ A _ {n} $ be an $ n $- dimensional subprojective space with a torsion-free affine connection. With respect to a projective coordinate system $ x ^ {i} $ of $ A _ {n} $, the coefficients of the connection take the form

$$ \Gamma _ {jk} ^ {i} = x ^ {i} f _ {jk} + \delta _ {k} ^ {i} p _ {j} + \delta _ {j} ^ {i} p _ {k} ,\ \ f _ {jk} = f _ {kj} , $$

where $ \delta _ {j} ^ {i} $ are the Kronecker symbols and

$$ p _ {k} = \frac{1}{2} ( \Gamma _ {kk} ^ {k} - f _ {kk} x ^ {k} ). $$

In this coordinate system, all two-dimensional Euclidean planes on which the geodesic lines of $ A _ {n} $ are situated pass through the coordinate origin.

In general, in a subprojective space $ A _ {n} $ there exists a canonical coordinate system $ x ^ {i} $ in which the coefficients of the connection take the simplest form

$$ \Gamma _ {jk} ^ {i} = x ^ {i} f _ {jk} . $$

A Riemannian subprojective space $ V _ {n} $ is defined in the same way; its metric reduces to one of three possible forms:

$$ ds ^ {2} = g _ {\alpha \gamma } dx ^ \alpha dx ^ \gamma , $$

where

1) $ g _ {ik} = \partial _ {i} \tau \partial _ {k} \tau + ( \lambda / \theta ) \partial _ {ik} ( v/ \lambda ) $,

2) $ g _ {ik} = \partial _ {i} \tau \partial _ {k} \tau + c( \lambda / \theta ) \partial _ {ik} \lambda $, $ c = \textrm{ const } \neq 0 $; here $ \theta $ is an arbitrary function in the coordinates $ x ^ {i} $, $ \theta $ is a function of the variable $ \lambda / \theta $, $ v $ is a quadratic form in the $ x ^ {i} $, $ \lambda $ in 1) is a linear form and in 2) it is the square root of a quadratic form that is not a complete square.

3) The exceptional case

$$ \theta ds ^ {2} = a _ {\alpha \beta } dx ^ \alpha dx ^ \beta + 2 \ d \lambda dx ^ {n-1} + 2 dv dx ^ {n} , $$

where $ a _ {\alpha \beta } = \textrm{ const } $, $ \mathop{\rm det} | a _ {\alpha \beta } | \neq 0 $, $ \theta $ is a homogeneous function of degree one in $ x ^ {n-1} $ and $ x ^ {n} $, and $ \lambda $ and $ v $ are functions related by

$$ \frac{1}{2} a _ {\alpha \beta } x ^ \alpha x ^ \beta + \lambda x ^ {n-1} + vx ^ {n} = 0,\ \ \alpha , \beta = 1 \dots n- 2 . $$

The functions $ \lambda $ and $ v $ are not homogeneous of the first degree.

All three cases can be reduced to a uniform expression by the choice of coordinates $ z ^ {i} $:

1) $ ds ^ {2} = e ^ {- 2 \mu ( z ^ {1} ) } ( e _ {1} dz ^ {1 ^ {2} } + \dots + e _ {n} dz ^ {n ^ {2} } ) $,

2) $ ds ^ {2} = e ^ {- 2 \mu ( z) } \sum _ {i=1} ^ {n} ( e _ {i} dz ^ {i ^ {2} } ) $, $ z = ( \sum _ {i=1} ^ {n} e _ {i} dz ^ {i ^ {2} } ) ^ {1/2} $,

3) $ ds ^ {2} = e ^ {- 2 \mu ( z ^ {1} ) } ( 2 dz ^ {1} dz ^ {2} + \sum _ {i=3} ^ {n} e _ {i} dz ^ {i ^ {2} } ) $, $ i = 3 \dots n $ $ ( e _ {i} = \textrm{ const } ) $. All Riemannian subprojective spaces $ V _ {n} $ are conformal Euclidean spaces (cf. Conformal Euclidean space). Riemannian subprojective spaces belong to the class of semi-reducible Riemannian spaces and their metrics have a special structure.

Tensor criteria for conformal Euclidean subprojective spaces exist, distinguishing them from the class of all conformal Euclidean spaces. Every subprojective space $ V _ {n} $( apart from the case 3)) can be realized as a hypersurface in a Euclidean space $ E _ {n+} 1 $ in the case 1), or as a hypersurface of rotation in $ E _ {n+} 1 $ in the case 2). The converse is also true: Every hypersurface of rotation around a non-isotropic axis in a Euclidean space $ E _ {n+} 1 $, $ n > 2 $, is a Riemannian subprojective space with a metric of the form 2).

Motions in Riemannian subprojective spaces are defined in the usual way. The subprojective spaces $ V _ {n} $ are characterized by the fact that if $ V _ {n} $ is not a space of constant curvature, then it permits a maximal intransitive group of motions of order $ n( n- 1)/2 $, and, conversely, every Riemannian space $ V _ {n} $ that permits a maximal intransitive group of order $ n( n- 1)/2 $ is a subprojective space. Riemannian subprojective spaces $ V _ {n} $ are maximally-mobile non-Einsteinian spaces (spaces of constant curvature occupy the same position among the Einstein spaces).

The concept of a subprojective space permits the following generalizations: A space $ A _ {n} $ with an affine connection is called a generalized subprojective space if its geodesic lines lie in Euclidean planes $ E _ {r+} 1 $, $ 1 \leq r \leq n- 2 $, that pass through a fixed plane $ E _ {r-1} $( at a finite or infinite distance).

References

[1]	V.F. Kagan, "Subprojective spaces" , Moscow (1961) (In Russian)
[a1]	J.A. Schouten, "Ricci-calculus. An introduction to tensor analysis and its geometrical applications" , Springer (1954) (Translated from German)