• Characteristic lines of a hypersurface Vn imbedded in a Riemannian Vn+1

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      https://www.ias.ac.in/article/fulltext/seca/044/02/0053-0062

    • Keywords

       

      Principal Curvature; Principal Direction; Normal Curvature; Characteristic Line; Characteristic Direction

    • Abstract

       

      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 Vn imbedded in a Riemannian Vn+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.

    • Author Affiliations

       

      M K Singal1 Ram Behari1

      1. University of Delhi, India
    • Dates

       
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