Darboux function in a weyl hypersurface

An n-dimensionai manifold Wn is said to be a Weyl space, if it has a confor- mal metric tensor gij and a symmetric connection satisfying the compatibility given by the equation Vkgij - 2Tkgij = 0, where Tk denotes a covariant vector and VkÇij denotes usual covariant derivative. Under renormalization of the fundamental tensor of the form ğij = X2gij the complementary vector Ti is transformed by the law T{ = T{ + di In A, where A is a function of the point. In n-dimensional Weyl space Wn, the independent vector fields V* (r = 1, 2,..., n) determine an n-dimensional net ( V, V,..., V ) \X 2 n J Let ibea satellite of g^ with weight {k}. d{A given by the equation diA = diA - kTiA, is said to be the prolonged derivative of A and VaA, given by the equation V3A = VSA - kT, is called the prolonged covariant derivative of A. The prolonged covariant derivatives of the vector fields V1 and their recip- a locals Vi are, respectively, given by VkVi=fkV\VkVi=fkVi(i,k,a,a = l,2,-..,n) a a a a s From these formulas, it follows that Tfc cos v3 = 0,where R\ki and a? are, respectively, the curvature tensor of Wn and the angle ıra between the directions determined by V and V. J a a Let Wn (gij,Tk) be a hypersurface, with coordinates ul(i = 1, 2,..., n), of a Weyl space Wn+i (#06, Tc) with coordinates xa(a = 1, 2,..., n+1) Suppose that the metrics of Wn and Wn+i are elliptic and that they are given, respectively, by gijdtfdui and gabdxadxb which are connected by the relations gij = gabxtxj (a,b= 1,2, ??.,n + l;i,j = 1,2,...,n) where x* denotes the covariant derivative of xa with respect to «!. The pro longed covariant derivative of A, relative to Wn, and Wn+\, are related by Vfc A = xckVcA. (k = 1, 2,..., n; c = 1, 2,..., n + 1) Let na be the contravariant components of the vector field in Wn+i normal to Wra, and let it be renormalized by the condition gabnanb = 1. The moving frame {xza,na} on Wn, reciprocal to the moving frame {xf,na} is defined by the relations nana = 1, nax1 = 0, nax\ = 0, x\x3a = Sf. uk On the other hand, we have the covariant derivatives of x" with respect to Vfc;r? = wikna. We have the prolonged covariant derivatives of na with respect to uk Vfcna = -gjlWjkxf. Let C : uJ = uJ (5) be any curve in Wn passing through a point P and m% ma, the contravariant components of the tangent vector to the curve C in Wn and Wn+i which are renormalized by the conditions gijmlm? = 1 and gaf,m - v*> = f-^f + 2YTr^ where ]P = TkVk. We shall call VJ VkVt (VkÇej - SjWke) as the generalised Darboux function of the direction V with respect to the direction V, h ^ L We shall denote it by VX\hhi- When A is normal to Wn, r = 1, t = 0 while VfcÇej = - VfctUej. Hence, 2>&w = --Ai- +)CÇ+lcf- 2P/C. Ssh '' E i>, ih h hi Similary, we obtained vkih = -YjYJkVkWei vlhh = -v^YVkW^- When A is normal to Wra, then Vhhi = "Dihh V= ^hih- If space is a EucHdean space, hence Vhhi = Vhih. IX