Bir yüzey üzerinde teğetsel olmayan bir vektör alanına bağlı eşit-eşlenik diyagonal eğriler ve hiperyüzeylerde eşit-eşlenik eğri çiftleri

Abstract

ÖZET 3 Bu çalışmada, önce, 3-boyutlıı Öklid uzayındaki C sınıfından bir S yüzeyi Üzerinde tanımlı, teğetsel olmayan C sınıfından bir A birim vektör alanına bağlı olarak genelleştirilmiş eşit-eşlenik diyagonal eğriler veya kısaca A-diyagonel eğriler tanımlanmış, bun ların diferansiyel denklemi elde edilmiş ve bazı temel özellikleri üzerinde durulmuştur. iki A-diyagonal eğri ailesi ile iki A-eğrilik çizgisi ailesi nin oluşturduğu 4-1 U dokunun al ti geni iği incelenmiş ve bununla geodezik paralellik arasında bir ilişkinin var olduğu görül müş tür. Ayrıca, S yüzeyinin A~eğrilik çizgilerinin oskülatör düzlemle rinin arakesid doğrularının oluşturduğu A kongüransına bağlı iki diya gonal eğri ailesinin geodezik paralellerden ibaret olduğu sabit orta lama eğrilikli Liouville yüzeyleri tayin edilmiştir, (n+1)- boyutlu E, öklid uzayındaki bir hiperyüzeye ait eşit- eşlenik u£., u. (Htm) vektörlerinin belirdiği orientasyona bağlı Riemann eğriliği K^m, bu doğrultular arasındaki açı 6. ve bu <Joğ- rultulardaki normal eğrilik P olsun. Hiperyüzeye ait eşit-eşlenik eğri çiftlerinin p£ =<£m sin^e^ bağıntısı ile karakterize edildiği gösterildikten sonra, karşılıklı eşit-eşlenik n-li bir sistem için teşkil edilen 2 cotg2% A.m-1 K£nj toplamının hiperyüzey için bir invaryant olduğu ispatlanmıştır. Son olarak, hiperyüzeye ait üç doğrultuya bağlı Codazzi fonksi yonu elde edilmiştir. 4V

SUMMARY EQUI-CONJUGATE DIAGONAL CURVES ON A SURFACE ASSOCIATED WITH A NON-TANGENTIAL VECTOR FIELD AND PAIRS OF EQUI-CONJUGATE CURVES IN A HYPERSURFACE 3 Let S be a surface of class C in 3-diraensional Euclidean space and let it be given by the vector equation xsx(u, v) in an orthogonal cartesian coordinate system. Let A be a non-tangential, unit vector field of class C2 defined on S. Suppose that the vector field A(or, equivalently the congruence A) is given by x` 3? A=£ x, *. mx0 * nn (x,= -, x0r - -) 12 1 ^ 2 /Ğ with reference to the trihedron (x,, 5L, `n). We know that the generalized normal curvature of a curve C on S, denoted by ç-., is defined [l] as the negative of the tendency of the congruence A in the direction of C. Then, for p, we find that 2 2 adu «-(b+b')dudv+cdv p v 1 A EduN-2Fdudv*Gd/ (a=xu-Au, b=tu-t, b'=x;<Tu, c= x^) In this paper the following results are obtained: If we take, along the tangents at a point P on a surface S, the length PQ=/ iR^i (RA= 1/p^) then the locus of the point Q becomes a conic. This conic is named as the generalized Dupin indicatrix relative to the congruence A. Then the equation of this conic is obtained as E /EG G (x,y) being the coordinates of Q in a curvilinear coordinate system with the origin P.The directions that I coincide with the corijugate-ftadii of generalized Dupin indicatrix are defined as the generalized conjugate directions. In _JT], the conjugacy of the directions of the unit tangent vector a and 6*, at a point P on S, relative to the congruence A is defined as follows: These directions are called generalized conjugate directions, If the resolved part in the direction of b* of the derived vector of A in the direction of £ is the negative of the resolved part in the direction of a of the derived vector of A in the direction of b. It is not difficult to see that these two definitions are equivalent. Using these definitions, curves which are tangent to equi- conjugate radii of generalized Dupin indicatrix are named as generalized diagonal curves relative to the congruence A or, in short, the A-diagonal curves. Also the differential equations of these A-diagonal curves are found to be {2a[ Ec-aG*F(b«-b')] -E(b*b'}2 }du2« 2[4Fac-(b+b') (Ec*aG)] du dv * {2c[ F(b*b') - Ec*aG] - G(b+b'}2} dv2= 0 The conditions for A-diagonal curves to be parametric lines are found as Ec-aG =0, b+b' = 0 It is proved that A »diagonal curves form a harmonic division with both A-asmyptotic lines and A-curvature lines. Conversely, it is shown that a net of curves forming harmonic division with both A-asymptotic lines and A-curvature lines are composed of A-diagonal curves. It is shown that the A-diagonal curves and A-curvature lines will form a hexagonal 4-web, if and only if the A-diagonal curves form a rhombic net. It is proved that a necessary condition for the two families of A-diagonal curves to be geodesic parallels is that the A-curvature lines and the A-diagonal curves form a hexagonal 4-web. viUsing the inyariant derivatives the congrunce A, composed of the lines of intersection of the Osculating planes of the A-curvature lines is determined as -? qr -* qr ¦+ rr.+ Ar - iü- x, - ^- x0 + - - n a 1 a c a as /(qr)2 ¦ (qr)2 ¦ {rrf where, r, r and q, q indicate respectively the normal and geodesic curvatures of the curves v= constant and u= constant. It is known that a Liouville surface of constant mean curvature is isothermic to a surface of revolution. Using this fact the invariants of the Liouville surfaces of constant mean curvature on which the two families of A-diagonal lines, associated with the above congruence, are geodesic parallels are determined. with coordinates x (i=l,2,...,n), with coordinates y^otr 1,2,...,n+l ). Let V be a hypersurface, immersed in Riemannian space V, Let the fundamental metrics of V and Vp4.-be respectively g--dx1dxJ and a^g dyadya and let Q^j ¦LL_ -`----¦¦- -j second fundamental form of V`. be the coefficients of the Let us consider the geodesic coordinate system on Vn with pole 0. Let Sn be the Euc]idşan space, tangent to Vn at 0 and having the same metric g1-,-dx1dxJ..If the Reimannian coordinates of a point P on Sn is denoted by y1, then the equation defines a hyperquadric in the Euclidean space S-. This hyperquadric can be considered as the generalization of Dupin indicatrix for a surface. The conjugate radii of generalized Dupin indicatrix having the same length are called equi-conjugate radii and the curves tangent to these radii at 0 are called an equi-conjugate pair of curves of the hypersurface. Let ujjj and u^. (^^k) be the unit tangent vectors of an equi-conjugate* pair of curves on the hypersurface. Let us denote, respectively, the normal curvatures in the direction of these vectors, the angle between these directions and the Riemannian curvature for the orientation determined by these vectors by vnpnlk' pnl£ ' 6k£ ' Kk£ ' An equi-conjugate pair of curves in a Euclidean hypersurface V is characterized by ^ = ha sin2 ek£ where pnlk= pnir pn ' It is shown that the expression 2 n n cotg e^ evaluated for the equi-conjugate ennuple of.congruences at a point P of a Euclidean hypersurface, is equal to ft^J fi^1` g-j£ g.-m. Hence it is seen that the above metioned sura is independent of the chosen equi-conjugate ennuple and, consequently is an invariant for the hypersurface. Let z^j, z^, z^be the contravariant components of the vectors 1*(, tl, Trin Vn, relative to the Rieraannian space vn*l. Let us denote the absolute derivate of VN in the direction of 5,1 ^ 5(VN) 6Sql where VN= y`. g1J n`.. The expression »I »J M = `^h.k 5p' Cql 5rl - * -* - ? is called the Codazzi function for the directions Ç., £,,, I,. pi q ı r i vniBy taking pr q= r in the Codazzi function, the Laguerre function of direc£ion,_^found by Nirmala is obtained. If, inparticular, the vectors TDi » T i = ~C ı are perpendicular the Darboux function of the direction qL rrelative to the direction T f is found. Let T., e^ı (p,Jt= l,2,...,n) be the unit vectors at a point P of the hypersurface in the direction of the curves of a system of congruences and in the direction of an orthogonal ennuple, respectively. Let the Laguerre function in the direction of `e*£, the Darboux function of the direction ~t^ with respect to It ^ and the Codazzi function for the directions `e*^, `e^i,`^! be denoted by L£, dUm and C^, respectively. Then it is proved that t n n n £l Sl^ql^ruV £ ^=1cplilcqİJZCr!fîiD££nıf £ ^cp/lcq/mCrlmDlmm (Mm.) (Mm) n n + £ m=l CP^CqimCrlAm^+ £ n,St=1 cplJlcq!mcrltC£tm (M m) (Mmfit) = -c(^p» £rl» £qJ) U