A function of direction associated with n congruences of an orthogonal ennuple in a Riemannian hypersurface

Abstract

In a hypersurface Vn belonging to a Riemannian space Vn+1 we choose n congruences of an orthogonal ennuple of unit vectors of contravariant components λ h¦ i (h=1,2,...,n) and denote, by ωkk the normal curvature of the hypersurface in the direction of the unit vector of components λ k¦ i . In the present paper, we have shown that the expression $$\frac{\partial }{{\partial s_k }}\omega _{kk} - 2\mathop \Sigma \limits_{h = 1}^n \omega _{kh} \gamma _{hkk}$$ is a function of direction, where the symbol ∂/∂sk indicates the differentiation in the direction of the vector of components λ k¦ i and that ωkh (h≠k) and γlhk (l,h,k=1,2,...,n) are, respectively, the invariants of the geodesic torsion of the curve of the congruence with unit tangent vector of components λ k¦ i and Ricci's coefficients of rotation of the orthogonal ennuple.