Semi-pseudo-Riemannian space

A manifold with a degenerate indefinite metric. The semi-pseudo-Riemannian manifold $ {} ^ {l _ {1} \cdots l _ {r} } V _ {n} ^ {m _ {1} \cdots m _ {r - 1 } } $ is defined as an $ n $-dimensional manifold with coordinates $ x ^ {i} $ in which there are given $ r $ line elements

$$ ds _ {a} ^ {2} = \ \sum _ { i,j= 1} ^ { {m _ a} - m _ {a- 1} } g _ {( a) ij } \ dx ^ {i + m _ {a- 1} } dx ^ {j + m _ {a- 1} } , $$

where $ 0 = m _ {0} < m _ {1} < \dots < m _ {r} = n $; $ a = 1, \dots, r $; and where the index of the quadratic form $ g _ {( a) ij } $ is $ l _ {a} $. The line element $ ds _ {a} ^ {2} $ is defined for those vectors for which all components with indices smaller than $ m _ {a - 1 } + 1 $ or larger than $ m _ {a} $ vanish. If $ l _ {1} = l _ {2} = \dots = 0 $, a semi-pseudo-Riemannian space is a semi-Riemannian space. The spaces $ V _ {n} ^ {m} $ and $ {} ^ {kl} {V _ {n} ^ {m} } $ are quasi-Riemannian spaces. The basic concepts of differential geometry (for example, curvature) are defined in semi-pseudo-Riemannian spaces similarly to Riemannian spaces (see [1]).

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)