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Variation of Congruences of Curves of an Orthogonal Ennuple in a Riemannian Space

Published online by Cambridge University Press:  20 November 2018

T. K. Pan
T. K. Pan*
Affiliation:
University of California and University of Oklahoma
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Consider any three congruences of an orthogonal ennuple at a point of a Riemannian space. When one congruence is moved by local and a second congruence is moved by parallel displacement in the direction of the third congruence, the rate of change of cosine of the angle between the first two congruences is well known as Ricci's coefficient of rotation and has been extensively studied. It is the purpose of this note to investigate the corresponding rate of change, when the third congruence is replaced by an arbitrary one, in connection with parallelism and equidistance of congruences as studied by Miss Peters [2; 3].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 5 , 1953 , pp. 519 - 523
Copyright
Copyright © Canadian Mathematical Society 1953

References

1. Eisenhart, L. P., Riemannian geometry (Princeton, 1949).Google Scholar
2. Peters, R. M., Parallelism and equidistance in Riemannian geometry, Amer. J. Math., 57 (1935), 103–111.Google Scholar
3. Peters, R. M., Parallelism and equidistance of congruences, Amer. J. Math., 59 (1937), 564–574.Google Scholar