Bir doğru kongrüansının incelenmesinde diferansiyel formların kullanılması

Abstract

Üç bölümden oluşan bu çalışmanın 1. bölümünde, 3-boyutlu Öklid Uzayında doğru kongrüanslarını diferansiyel formlar yardımıyla incelenmesi problemi ele alınmış ve bu çeşit kongrüanslara ait L, II. ve III. esas formları ile ilgili bazı temel tanım ve teoremler verilmiştir. Bilhassa, kongrüansın orta-zarf yüzeyine bağlı olarak ortaya çıkan ve klasik kongrüans teorisinde yer almayan III. esas form ve buna bağlı K* karışık eğriliği ve H*, karışık ortalama eğriliği ile ilgili olarak, verilen bir yüzeyi orta-zarf yüzeyi ve H*, fonksiyonunu karışık ortalama eğrilik olarak kabul eden kongrüansın belirlenmesi problemi ele alınmıştır. ikinci bölümde, referans yüzeyinin eğrilik çizgilerinin rektifiyan düzlemlerinin arakesit doğrularının belirlendiği T* kongrüansı gözönüne alınmış ve birinci bölümde ele alman problem T* için incelenmiştir. Üzerinde T* doğru kongrüansma bağlı 1-parametreli bir hiperasimptotik eğri ailesi ile iki eğrilik çizgileri ailesinin 3-lü altıgen doku teşkil ettiği ds2 = f(a(u) + fi(v))(du2 + dv2) metriğine sahip minimal yüzeyler belirlenmiştir. Ayrıca bu şekilde tanımlı hiperasimptotik eğri ailesinin denklemi bulunmuştur.

In this work, we consider a rectilinear congruence V in three-dimen sional Euclidean space satisfying the conditions, a) The spherical representation of T is one-to-one, b) The middle envelope OM = M(u,v) of F has no umbilical or parabolic points, c) There is a one-to-one correspondence between the points of the middle envelope OM and points of the middle surface OP = P(u,v) of r. Let the middle surface OP = P(u, v) of the congruence be the surface of reference and G be a simply connected domain in the uu-plane. We as sociate a trihedron { et(u,v), e2(u,v), e3(u,v)} of mutually orthogonal unit vectors e{, with the congruence T. The change of position of P and of the e, is given by equations of the form 3 d^ = Xla' ?" !=1 3 de; =^2w.t e(, j = 1,2,3 8=1 where crf. w.. denotes linear differential forms. In Chapter I, we first obtain the first and the second fundamental quadratic forms of the congruence as I: = (e*iT.)2 = «£+«£ 2 I 1",".. ",, I »n",,2 II : = ( e3, d e3, dP) = lw* + 2mw31w32 + nw v Let we denote, by j- and j-, the extremum values of j = T. The curvature and the mean curvature of T are defined as. 1/1 1 ^ l+n 2\d'dJ 2 Ruled surfaces, corresponding to the directions of the extremum valu es of j on the surface of reference are called (Sannia) principal surfaces. The equation of the principal surfaces is obtained as mw23l + (n - l)w31w33 ~ "»«& = 0 Since the normals to the middle envelope OM are parellel to the corre sponding rays of the congruence, { ex(u,v), e2(u,v), e3(it,u)} can be used as the moving frame of OM. Consequently, d M = p e, + <7 e! 2, where /?, a are certain differential 1-forms. Apart from the first and the second quadratic differential forms stu died in the classical theory of the congruences. We consider the so-called third quadratic differential form defined by[4]. Ill := {dP, dM, e3). Ruled surfaces corresponding to the directions of the extremum values of -^ = ^p are called third principal surfaces of the equation which is [2/3h - mfo + r,)]{w2n - w2J + 2{al - 6n)w3lw32 = 0 Any ray of T passing through a point (u0, v0) of the surface of reference is called as third umbilical ray, if it satisfies the following conditions: al - 6n = 0, 2/?/i-m(r1 +r2) = 0. VI Let we denote, by ^- and -ş-, the extremum values of -^ = ^. The mixed curvature and the mixed mean curvature of V are denned as K* : = -l-.-k- 11 1 In this work, the problem of determining a rectilinear congruence which admits a given surface as its middle envelope and the function H* as its mixed mean curvature is studied. The solution of this problem is reduced to solving the following system of two differential equations involving partial derivatives of the first order of s and g. V.s + V^g-Ys-Xg =r,+r2, r2V2s - rx Vtg - Xrts + Yr2g = -2H* This system of equations has infinitely many solutions, but we can find a unique solution of the system under appropriate initial conditions. Let {s0(u,v),g0(u,v)} be a particular solution of the system. Then, we have Vi5o + v20o ~Yso -Xga =ri+r2> r2V2s0 -r.V.g, -XrlSo+Yr2g0 = -2H*. In this case, the solution of the problem is reduced to solving the differ ential equation rt Vt Vxtf + r2V2V2^ - Yr7Vrf - Xr, V2 = 0 where xj) is the desired solution. In particular, suppose that the middle envelope of the given congru ence is minimal. Then the above problem reduces to solving the differ ential equation v1v^-v2v2^ + rv1v»-xv2^ = o On the other hand, the spherical image of the lines of curvature of any minimal surfaces form an isothermic net-. "Thus we find Consequently, the characteristic curves of this hyperbolic equation are obtained as the spherical images of the asymptotic lines of the middle envelope. vii In addition to the above results, we obtain an analogue of the Euler equation of the surface theory. Namely, we have Ö) D J =^«»V+^-«nV / n l 2 Consider the rectilinear congruence T*, formed by the intersections of the rectifying planes of the lines of curvature. on the reference surface. In Chapter II, assuming that the middle surface and the middle en velope of T* are minimal, we obtain the system of differential equations (j, fj, (j,2 r corresponding to the above system. In Chapter three, the minimal surfaces, with metric c^2 = /(a(u) + /3(.u))(du2 + dv2) the two families of lines of curva ture and a one-parameter family of hyperasymptotic lines of T* form a hexagonal 3-web are determined. In order to simplify the calculations, the lines of curvature of the surface reference are taken as the paremetric lines and the differential equations of the hyperasymptotic lines is reduced to [rq2 + rq2 ) sin (p cos (p -