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Union Curves of a Hypersurface

Published online by Cambridge University Press:  20 November 2018

C. E. Springer
C. E. Springer*
Affiliation:
University of Oklahoma
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1. Introduction. A curve on an ordinary surface is a union curve if its osculating plane at each point contains the line of a specified rectilinear congruence through the point. The author has obtained the differential equations of union curves on a metric surface in ordinary space and has exhibited certain generalizations for union curves of known results concerning geodesic curves on a surface. It is the purpose of the present paper to develop the differential equations of the union curves of a hypersurface Vn immersed in a Riemannian manifold Vn+1 of n + 1 dimensions. The osculating plane to a curve on a surface is generalized to a totally geodesic surface the straight lines of which are geodesies in the space Vn+1. A formula is given for the union curvature vector of a curve in Vn.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 2 , 1950 , pp. 457 - 460
Copyright
Copyright © Canadian Mathematical Society 1950

References

1 Sperry, P., Properties of a certain protectively defined two-parameter family of curves on a general surface, Amer. J. of Math., vol. 40 (1928), p. 213.Google Scholar

2 Springer, C. E., Union curves and union curvature. Bull. Amer. Math. Soc, vol. 51 (1945), pp. 686–691.CrossRefGoogle Scholar

3 Weatherburn, C. E., Riemannian Geometry and the Tensor Calculus (Cambridge University Press, 1938).Google Scholar