# 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)