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VII.—Parallel Planes in a Riemannian Vn

Published online by Cambridge University Press:  14 February 2012

H. S. Ruse
H. S. Ruse
Affiliation:
The University, Leeds
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Synopsis

In an earlier paper a description was given, in terms of classical projective geometry, of some of the properties of parallel fields of vector spaces (parallel planes) in a Riemannian Vn, and a detailed analysis was made of the case n = 4. The present paper contains the corresponding formulae for any n, though omits their projective interpretation. A parallel þ-plane is said to be of nullity q when the þ vectors of any normal basis contain q null and þ − q non-null vectors. The conditions of parallelism, namely that the co-variant derivatives of the basis-vectors should depend linearly upon these vectors, are examined for any þ and any q(<þ), and attention is thereafter mainly confined to the cases (i) n even, q = ½n − 1, p = ½n − 1 or ½n; (ii) n odd, q = ½(n − 3)) ,p = ½(n − 1), which possess exceptional features. In the former of these cases light is thrown upon the curious circumstance, noted in the previous paper, that the existence in a V4 of a null parallel i-plane necessitates the existence of parallel planes other than its conjugate. For a general n similar situations arise in the cases indicated.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 63 , Issue 1 , 1950 , pp. 78 - 92
Copyright
Copyright © Royal Society of Edinburgh 1950

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References

page 78 note * Walker, A. G., “On Parallel Fields of Partially Null Vector Spaces”, Quart. Journ. Math. (Oxford), XX, 1949, 135–145.CrossRefGoogle Scholar

page 78 note † Ruse, H. S., “On Parallel Fields of Planes in a Riemannian Space”, Quart. Journ. Math. (Oxford), XX, 1949, 218–234CrossRefGoogle Scholar.

page 79 note * Wong, Y. C., “Quasi-orthogonal Ennuple of Congruences in a Riemannian Space”, Ann. of Math., XLVI, 1945, 158–173CrossRefGoogle Scholar.

page 79 note † The idea of representing suffix-ranges by single capital letters is due to A. G. Walker.

page 81 note * The notation (α, β) ∈R is an abbreviation for “α ∈ R and β ∈ R”.

page 81 note † Wong, loc. cit. This use of nullity is different from that of § I, where it was used in connection with parallel planes. The latter meaning is the one to be understood in this paper, except in the present instance, which is the only occasion upon which the word is used in Wong's sense.

page 82 note * Cf. Wong, loc. cit., equation (3.2).

page 84 note * The vectors are auxiliary, and pairs of them could be replaced by pairs of auxiliary null vectors defined in terms of them by formulae similar to (2.6), in which case (3.5) would be modified in form.

page 92 note * Walker, A. G., “Canonical Form for a Riemannian Space with a Parallel Field of Null Planes”, Quart. Journ. Math. (Oxford, Second Series), I, 1950, 69–79CrossRefGoogle Scholar; and a further paper, at present in the press (May 1950), to appear in the same Journal.