The Journal of the Indian Mathematical Society

Ram Behari ¹, Nirmala Prakash ¹

Affiliations
1 University of Delhi, Delhi 6, IN

Abstract

The derived vector of a vector v belonging to a vector field along a Curve C in Riemannian or Finslerian space is known as the absolute curvature vector of the field with respect to the curve C. This curvature vector has components along the tangent and the normal. The tangential component for Riemannian space was studied by W. C. Graustein (1932) and R. M. Peters (1935) (1937). The normal component for Riemannian spaces was studied by T. K. Pan (1952). Y. Nagata extended the results for Finsler spaces in Cartan's sense. In the present paper it has been studied for Finsler spaces of more general character.

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Ram Behari ¹

Affiliations
1 St. Stephen's College, Delhi, IN

Abstract

The object of this paper is to obtain both analytically and geometrically the following theorem :-

There exist ∞² ruled surfaces of an ordinary congruence the osculating quadrics of which are equilateral, but there are only ∞¹ such ruled surfaces if the congruence is normal.

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Ram Behari ¹

Affiliations
1 University of Delhi, IN

Abstract

The problem of infinitesimal deformation of a ruled surface or the equivalent problem of determining a surface which corresponds to a given ruled surface with orthogonality of corresponding elements has been investigated by Goursat, Darboux and Haag.

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Ram Behari ¹

Affiliations
1 University of Delhi, IN

Abstract

The relation between the theory of line congruences and Gauss's theory of surfaces has been noted by Kummer. He mentions the analogy between Hamilton's formula p = p₁ cos²θ + p₂ sin²θ and Euler's formula 1/R = cos²θ/R₁+sin²θ/R₂. The analogy between the total and mean curvatures of the surface, and the total and mean parameters of the congruence and some other analogies have been mentioned by Sannia.

The object of this paper is to point out some other similarities between the two theories.

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Ram Behari ¹

Affiliations
1 University of Delhi, IN

Abstract

Taking a curve on a surface as the base curve, x, y, z, the coordinates of a point P on the base curve and X, Y, Z, the direction cosines of the normal to the surface at P, it has been proved in a previous paper that Laguerre's function.

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Ram Behari ¹

Affiliations
1 University of Delhi, IN

Abstract

Consider a thin pencil formed by rays adjacent to a ray l of the rectilinear congruence given by

ξ=x+tX, η=y+tY, ζ=z+tZ

where x, y, z, and X, Y, Z are functions of two parameters u and v. Let C be the closed curve on the surface of reference which forms the boundary of the area dS on it cut off by the pencil. Let (x, y, z) be the point where the ray l meets C.

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Ram Behari ¹

Affiliations
1 University of Delhi, IN

Abstract

Let a rectilinear congruence be defined by the co-ordinates (x, y, z) of a point M on the surface of reference S and by the direction cosines (X, Y, Z) of the line l passing through M, where x, y, z; X, Y, Z are functions of the two parameters u and v. The co-ordinates of any point P of the ray are given by

ξ=x+tX, η=y+tY, ζ=z+tZ,

where t is the distance of P measured from the point M.

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Ram Behari ¹

Affiliations
1 University of Delhi, IN

Abstract

Consider a rectilinear congruence defined by the co-ordinates (x, y, z) of a point M on the surface of reference S and by the direction cosines (X, Y, Z) of the line l passing through M, where x,y,z; X,Y,Z are functions of the two parameters u and v.

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Ram Behari ¹

Affiliations
1 University of Delhi, IN

Abstract

If we take a closed curve C and through points of it draw lines, then the surface generated by these lines is a ruled surface.

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