Laguerre's function in a weyl hypersurface

SUMMARY LAGUERRE'S 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 g%j and a symmetric connection satisfying the compatibility given by the equation Vk9ij - 2Tkgij = 0, where Tk denotes a covariant vector and Vfcfifjj denotes usual covariant derivative. Under renormalization of the fundamental tensor of the form ğij = X2gij the complementary vector T,- is transformed by the law Ti = Ti + 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 J. Let A be a satellite of gij with weight {k}. d{A given by the equation diA = diA-kTiAr is said to be the prolonged derivative of A and VaA, given by the equation VaA = VaA - kTsA is called the prolonged covariant derivative of A. The prolonged covariant derivatives of the vector fields V* and their recip rocals Vi are, respectively, given by * Ot Ot (T IT From these formulas, it follows that Tic COS are, respectively, the curvature tensor of Wn and the angle between the directions determined by V and V. Let Wn (gij,Tk) be a hypersurface, with coordinates ul(i = 1, 2,..., n), of a Weyl space Wn+i (ffaft, Tc) with coordinates xa(a = 1, 2,..., rc+1) Suppose that the metrics of Wn and W"+i are elliptic and that they are given, respectively, by gijduxdui and gabdxadxb which are connected by the relations 9ij = 9ahX*xhi (a,b= l,2,---,n + l;i,j = l,2,---,n) where zf denotes the covariant derivative of xa with respect to u\ The pro longed covariant derivative of A, relative to Wn, and Wn+i, are related by VkA = x%VcA (jfc = l,2,---,n;c= l,2,---,n + 1). Let na be the contravariant components of the vector field in W"+i normal to W", and let it be renormalized by the condition gabnanb = 1. The moving frame {aj^ra,,} on Wn, reciprocal to the moving. frame. {xf,na} is defined by the relations nana = 1, naXi = 0, naxxa = 0, x1xJa = 8\. On the other hand, we have the covariant derivatives of x" with respect to u* Vkx* = wikna. We have the prolonged covariant derivatives of na with respect to uk Vfcna = -gilWjkx*. Let C : u* = u% (s) be any curve in Wn passing through a point P and m', m°, the contravariant components of the tangent vector to the curve C in Wn and Wn+i which are renormalized by the conditions gijin'm1 - 1 and flfatmam6 = 1, respectively. Hence m Vn in = -Kn viis said to be the tendency of the n in the direction of the vector m in Wn, where Kn is the normal curvature of the curve C. Consider a curve C in W"with tangent vector of components m\ and prin cipal vector of components m\ relative to Wn> According to [6], at any point P on the geodesic tangent to the curve C, we have the following Prenet formulae: - m!j. = Kxmx+1 - Kx-\mxx_x (x = 1, 2, ?.., n + 1) where Kq = Kn+i = 0, and the vectors mi,m2,...,mn+i are mutually orthog onal in Wn+i. In particular -mi = Kim\ where K\ is the first curvature of the curve Cand j^ is the symbol of the prolonged derivative in the direction of the curve C. The normal curvature vector of C in the direction of ra° is denoted by Kn. We find Kn = wijm\m{. On the other hand, the function Tg = Wijm\m32 may be regarded as the invariants of the geodesic torsion of the curve C relative to the normal component n°. Let Aa be the contravariant components in Wn+i, of the tangent vector A to a curve of the congruence (A) such that one curve of the congruence passes through each point of Wn. Resolving A tangentially and normally to Wn, we have Aa = xft1 + rna viiwhere t = gah\a\h = gımtltm + r2. Choose a curve C : ul = ul (s') in Wn such that 6j and ba are the contravariant components of the tangent vectors to the curve C in Wnand Wn+i which are normalized by the conditions gijb%V == 1 abd gacbabc - 1, respectively. Hence the prolonged covariant derivative of A° with respect to the curve C is given by and the prolonged covariant derivative of Aa with respect to the curve C" is given by k\a- =(KX\nn* + KMgz*)t. where (i) K\in, K\ig are the normal curvature and geodesic curvature of the congru ence (A) with respect to C and K\in, K\ig are the normal curvature and geodesic curvature of the same congruence with respect to C". (ii) n is the vector along the normal curvature vector of the congruence (A) which is normalized by gabnanh = 1. z, "z are the vectors along the geodesic curvature vector of congruence (A) with respect to C and C which are normalized by gabzazb = 1 and ğabZazb - 1> respectively. (iii) Vj is the symbol for the prolonged covariant differentiation. We have ex pressed the prolonged covariant derivative of A with respect to the curve C in the following form: - - = rnJ V,-A OS In addition, the prolonged covariant derivative A with respect to the curve C is given by Ö4~ = VVjV ?. OS viiiLet us define the operator D D = xtfiVj where the operator Vj is the symbol of the prolonged covariant differentiation. We find that Kn =wikm\m\ Kgml2 =mJVfcmj where Kn, Kg are the normal curvature and geodesic curvature of the congru ence (A) with respect to C, respectively. Hence, we obtained mDm = Knna + Kgba Similarly, we have mD\ = {KX\nna+KMgza)t where K\ın, K\ig are the normal curvature and geodesic curvature of the con gruence (A) with respect to C. We obtained KX\gtzl = m> (v/ - rghlwhj) and Kx\nt = bi(İ7jr + tlwjl) Kx\gtzl = V (Vjt1 - rg^Wi/j where K\ın, K\inaxe the normal curvatures of the congruence (A) with respect to C and C, respectively and K\tg, K\ig are the geodesic curvatures of the congruence (A) with respect to C and C", respectively. ixLet gj = Vjt1 - rgdWij, Sj = Wjit1 + Vjr. Then we have İ7jXc = q'jxl + Sjnc Hence rri- 8s m-z - m = -mhm^mk (Vkqhj - 2Tkqhj - SjWkh) ? We obtained -m'mkmk(Vkqhi - sjWkh) = - rn- - m - 2Tkqhjm^mhmk. We shall call --m*mhmk (Vkqhj - SjWkh) as the the generalised Laguerre function for the direction ra and we may denote this by Lxi- A curve in Wn for which the tangent vector m satisfies the equation L\ı = 0 as generalised La guerre line. Let s is the arc length along the curve C. Hence L\i = ??- ? " + KxigKg cos + KxigKgt cos 0 + KxinKnt - 9hlTjtlKg (bh + V) - gMmhmjmkVk (Tjt1) - 2Tkqhjmhmjmk where 8m,_,- _, Is DXrh = KxigKgt cos 6, mPA-r- = KxinKnt + KxigKgt cos as When A is normal to Wn and Pi = Tfcmf, VfcliTn = Vfe/<"n hence mkm{mİVkWhj = mf V*^ - 2/^r, + 2P1KnWe may denote this by L. Consider the Riemannien space, T* = 0, Pi =0, Tg = - r2 and K\ = Kg. Hence C can be put in the following form [8]:" SKn 6s £=-^- + 2KgT2 XI