Semi-Riemannian space

A space with a semi-Riemannian metric (with a degenerate metric tensor). A semi-Riemannian space is a generalization of the concept of a Riemannian space. The definition of a semi-Riemannian space can be expressed in terms of the concepts used in the definition of a Riemannian space. In the definition of a Riemannian space $ V _ {n} $ one uses as tangent space the space $ \mathbf R ^ {n} $ with a Euclidean metric, which is supposed to be invariant under parallel displacements of $ V _ {n} $( the metric tensor $ a _ {ij} $ of the space $ V _ {n} $ is absolutely constant). If the tangent space at every point of $ V _ {n} $ is equipped with the structure of a semi-Euclidean space $ R _ {n} ^ {m _ {1} \dots m _ {r - 1 } } $, then the metric of the space $ V _ {n} $ is degenerate, the metric tensor is also absolutely constant but is now degenerate, its matrix has rank $ m _ {1} $ and has a non-singular submatrix. One defines a second degenerate metric tensor in the $ ( n - m _ {1} ) $- plane $ ( a _ {ij} x ^ {j} = 0 ) $, which is called the zero $ ( n - m _ {1} ) $- plane of the tensor $ a _ {ij} $; its matrix also possesses a non-singular submatrix, etc. The last, $ r $- th metric tensor, defined in the zero $ ( n - m _ {r - 1 } ) $- plane of the $ ( r - 1) $- st tensor, is a non-degenerate tensor with a non-singular matrix. Such a space is called a semi-Riemannian space, and in this case it is denoted by the symbol $ V _ {n} ^ {m _ {1} \dots m _ {r - 1 } } $. Analogously one defines semi-Riemannian spaces of the form $ {} ^ {l _ {1} \dots l _ {r} } V _ {n} ^ {m _ {1} \dots m _ {r - 1 } } $, that is, when the tangent space has the structure of a semi-pseudo-Euclidean space $ {} ^ {l _ {1} \dots l _ {r} } R _ {n} ^ {m _ {1} \dots m _ {r - 1 } } $. The spaces $ V _ {n} ^ {m} $ and $ {} ^ {kl} V _ {n} ^ {m} $ are called quasi-Riemannian spaces.

As in a Riemannian space, one introduces the concept of curvature in a $ 2 $- dimensional direction. Semi-hyperbolic and semi-elliptic spaces are semi-Riemannian spaces of constant non-zero curvature, and a semi-Euclidean space is a semi-Riemannian space of constant curvature zero.

Thus, a semi-Riemannian space can be defined as a space of affine connection (without torsion) whose tangent spaces at every point are semi-Euclidean (or semi-pseudo-Euclidean), and where the metric tensor of the semi-Riemannian space is absolutely constant.

In a semi-Riemannian space, the differential geometry of lines and surfaces is constructed by analogy with the differential geometry of lines and surfaces in $ V _ {n} $, taking into account the special features of semi-Riemannian spaces indicated above. Surfaces of semi-hyperbolic and semi-elliptic spaces are themselves semi-Riemannian spaces. In particular, the $ m $- horosphere $ {} ^ {m+ 1 } {S _ {n} } $ in a semi-hyperbolic space is isometric to the semi-Riemannian space $ V _ {n - 1 } ^ {m, n - m - 1 } $, the metric of which can be reduced to the metric of the semi-elliptic space $ S _ {n - m - 1 } ^ {m} $; this fact is a generalization of the isometry of a horosphere in Lobachevskii space to a Euclidean space.

References

[1]	B.A. Rozenfel'd, "Non-Euclidean spaces" , Moscow (1969) (In Russian)

References

[a1]	B.A. [B.A. Rozenfel'd] Rosenfel'd, "A history of non-euclidean geometry" , Springer (1988) (Translated from Russian)
[a2]	B. O'Neill, "Semi-Riemannian geometry (with applications to relativity)" , Acad. Press (1983)