Riemannian Vn Research Articles

Proportional nets in Weyl spaces

α-proportional and proportional nets in n-dimensional Weyl space Wn and Riemannian space Vn are introduced. The fundamental forms of the spaces Wn and Vn in the parameters of the α-proportional and proportional nets are found. The theorem - Weyl space \( W_n, n>2 \) which contains a proportional net is Riemannian Vn - is proved.

On the hyperasymptotic and hypergeodesic curvatures of a curve of a Finsier hypersurface

Mishra [2] defined the hyperasymptotic curves in a hypersurface Vn-1 of a Riemannian Vn. Amur [1] has also discussed a few properties of the hyperasymptotic curves and defined the hypergeodesics in a Vn-1 of a Riemannian Vn. Mishra and Sinha [3] have derived the equations of the hyperasymptotic curves in a hypersurface of a Finsler space Fn. In the present paper, we have derived the hyperasymptotic and hypergeodesic curves in the hypersurface of a Finsler space and studied some of their properties.

Union Curves in a Subspace Vn of a Riemannian Vm

A union curve on a surface in a euclidean 3-space, relative to a given congruence is characterized by the property that its osculating plane at each point contains the ray of the congruence through that point. Springer (2) and Pan (1) have studied union curves in a hypersurface Vn of a Riemannian Vn+1. In the present paper we proceed to obtain the equations of union curves in a subspace Vn of a Riemannian Vm.

Characteristic lines of a hypersurface Vn imbedded in a Riemannian Vn+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 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.

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.

VII.—Parallel Planes in a Riemannian Vn

SynopsisIn an earlier paper a description was given, in terms of classical projective geometry, of some of the properties of parallel fields of vector spaces (parallel planes) in a Riemannian Vn, and a detailed analysis was made of the case n = 4. The present paper contains the corresponding formulae for any n, though omits their projective interpretation. A parallel þ-plane is said to be of nullity q when the þ vectors of any normal basis contain q null and þ − q non-null vectors. The conditions of parallelism, namely that the co-variant derivatives of the basis-vectors should depend linearly upon these vectors, are examined for any þ and any q(<þ), and attention is thereafter mainly confined to the cases (i) n even, q = ½n − 1, p = ½n − 1 or ½n; (ii) n odd, q = ½(n − 3)) ,p = ½(n − 1), which possess exceptional features. In the former of these cases light is thrown upon the curious circumstance, noted in the previous paper, that the existence in a V4 of a null parallel i-plane necessitates the existence of parallel planes other than its conjugate. For a general n similar situations arise in the cases indicated.

XVIII.—The Riemann Tensor in a Completely Harmonic V4

If is a fixed point of a Riemannian Vn of fundamental tensor gij, and if s is the geodesic distance between it and a variable point (xi), then the Vn has been called centrally harmonic with respect to the base-point ifis a function of s only, and completely harmonic if this holds for every choice of base-point . A flat Vn (gij=δij) is obviously completely harmonic, since for such a space and

IX.—On the Line-Geometry of the Riemann Tensor

In this paper the curvature tensor Rijkl in a Riemannian Vn is used to define a quadratic complex of lines in an (n – I)-dimensional projective space Sn–1. Work in this direction has been done for a V4 by Struik (1927–28), Lamson (1930), and Churchill (1932). Of these, Struik and Lamson both use 3-dimensional projective geometry, the former for the purpose of defining sets of “principal directions” in V4 by means of the Riemann tensor, and the latter for the purpose of discussing some of the differential and algebraic consequences of the field equations of general relativity. Churchill considers the geometry of the Riemann tensor from the point of view of 4-dimensional Euclidean geometry. In this paper an indication is given of the nature of the general n-dimensional theory, which, by way of elementary illustration, is then applied in moderate detail to a V3. A few general formulae are also obtained for a V4.

On certain Quadric Hypersurfaces in Riemannian Space

The use of geodesic polar coordinates in the intrinsic geometry of a surface leads to the concept of a geodesic circle, i.e. the locus of points at a constant distance from the pole 0 along the geodesics through 0. A geodesic hypersphere is the obvious generalisation of this for a Riemannian Vn. We propose to consider more general central quadric hypersurfaces of Vn, which we define as follows. Let xi (i = 1, 2, …, n) be a system of coordinates in Vn, whose metric is gijdxidxj, and let aij be the components in the x's of a symmetric covariant tensor of the second order, evaluated at the point 0, which is taken as pole.