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On a Generalization of the First Curvature of a Curve in a Hypersurface of 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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The unit tangent vector at a point of a curve in a hypersurface of a Riemannian space has two derived vectors along the curve, one with respect to the Riemannian space in which the hypersurface is imbedded and one with respect to the hypersurface itself. When the former vector is decomposed along the directions normal and tangent to the hypersurface, its tangential component, which is called the first curvature vector of the curve at the point in the hypersurface, is exactly the latter vector.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 6 , 1954 , pp. 210 - 216
Copyright
Copyright © Canadian Mathematical Society 1954

References

1. Dekker, D. B., Hypergeodesic curvature and torsion, Bull. Amer. Math. Soc, 55 (1949), 1151–1168.Google Scholar
2. Eisenhart, L. P., Riemannian geometry (Princeton, 1949).Google Scholar
3. Eisenhart, L. P., Non-Riemannian geometry (Amer. Math. Soc, 1949).Google Scholar
4. Springer, C. E., Union curves and union curvature, Bull. Amer. Math. Soc, 51 (1945), 686–691.Google Scholar
5. Wilczynski, E. J., Some generalizations of geodesies, Trans. Amer. Math. Soc, 23 (1922), 223–239.Google Scholar