M K Singal
Articles written in Proceedings – Section A
Volume 42 Issue 6 December 1955 pp 309-316
Generalization of normal curvature of a curve in a Riemannian V_{n}
In the present paper a congruence of curves through points of a hypersurface V_{n} imbedded in a Riemannian V_{n+1} has been considered. In analogy with the normal curvature of a curve C in V_{n}, the generalized normal curvature of C at any point of it, relative to the curve of the congruence through that point, has been defined as the negative of the resolved part along C, of the derived vector of the unit tangent to the curve of the congruence through the point along C. The concepts of normal curvature of a hypersurface, principal directions, principal curvatures, lines of curvature, conjugate directions, asymptotic directions and asymptotic lines have been generalized and generalizations of several known theorems on the curvature of a hypersurface V_{n} in V_{n+1} have been obtained.
Volume 44 Issue 2 August 1956 pp 53-62
Characteristic lines of a hypersurface V_{n} imbedded in a Riemannian V_{n+1}
Upon a surface of positive Gaussian curvature there exists a unique conjugate system for which the angle between the directions at any point is the minimum angle between the conjugate directions at that point. This system of lines is called characteristic lines. In the present paper characteristic lines of a hypersurface V_{n} imbedded in a Riemannian V_{n+1} have been studied. It has been proved that characteristic directions are linear combinations of the principal directions corresponding to any two distinct values of the principal curvatures. It has also been proved that the normal curvatures in the two characteristic directions lying in the pencil determined by the principal directions corresponding to two distinct values of the principal curvatures are equal, each being equal to the harmonic mean between the principal curvatures. The directions for which the ratio of the geodesic torsion and normal curvature is an extremum have also been studied and it has been shown that the directions for which the ratio of the geodesic torsion and the normal curvatures is an extremum are characteristic directions.
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