Weyl hiperyüzeylerinde genelleştirilmiş darboux fonksiyonu
Darboux function in a weyl hypersurface
- Tez No: 39834
- Danışmanlar: DOÇ.DR. AYNUR UYSAL
- Tez Türü: Yüksek Lisans
- Konular: Matematik, Mathematics
- Anahtar Kelimeler: Darboux fonksiyonu, Hiperyüzey, Türevler, Weyl uzayları, Darboux function, Hypersurface, Derivatives, Weyl spaces
- Yıl: 1994
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
ÖZET Bu çalışmanın birinci bölümünde, Weyl uzayının metrik tensörüne ait uyduların genelleştirilmiş türevleri ve genelleştirilmiş kovaryant türevleri tanımlanarak, rali bir şebekeye ait vektör alanlarına ve bunların karşıtlarına ait türev formülleri verilmiştir. ikinci bölümde, Weyl uzayının birinci ve ikinci cins Chebyshev şebekesi tanımları verilerek bu tür şebekelerin, kendilerine karşı gelen eğrilikler yardımıyla karakterizasyonları üzerinde durulmuştur. Bu bölümde ayrıca s -8rK. + 2Ç K, + 2 P K. ifadesinin bir doğrultu fonksiyonu olduğu ispatlanmıştır. rr T re T rr Üçüncü bölümde ise, W“hiperyüzeyinin her noktasından bir eğrisi geçmek üzere Wn+ı Weyl uzayının (A) kongrüans eğrisine ait A teğet vektörünün kontravaryant bileşeninin C ve C”eğrileri doğrultusundaki genelleştirilmiş ko varyant türevleri bulunmuş, ayrıca Vj genelleştirilmiş kovaryant türev sembolü olmak üzere D operatörü tanımlanmıştır. Bu operatör yardımıyla Wn+ı'e ait bir (A) kongrüansınm A teğet vektörü için t^ doğrultusuna göre V doğrultusun daki genelleştirilmiş Darboux fonksiyonunun ifadesi çıkarılmıştır. Bu ifadeden yararlanarak A'nıu Wn'e ait normaller kongrüansı olması halinde genelleştiril miş Darboux fonksiyonunun özel hali elde edilmiştir. iv
Özet (Çeviri)
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<lmb = 1, respectively. VIHence m W m = -Kn ? is 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. Let V% and Va (r = 1, 2,...,n) be the contravariant components of the n independent vector fields V of an orthogonal ennuple in Wn and Wn+i which satisfy the conditions gijVzVi = 1 and gabVaVb - 1, respectively. E any one of the vector fields of the net ( V, V, ?. ?, V ) is parellelly translated \1 2 n J along the lines determined by the remaining fields of the net, such a net is called a chebyshev net of the first kind. A net is said to be a Chebyshev net of the second kind, if any of its (n- l)-dimensional area elements determined by the fields of the net is parellelly translated along the lines determined by the remaining field of the net. Coressponding to these two kinds of Chebyshev nets, the Chebyshev cur vatures of the first and second kind are, respectively, defined by T = fkYk (o,6,c,fc = l,2,---,n;o^6) and p = fkVk (a,6,fc = l,2,---,n;o^6). 6 o a. In addition we have C=TkVk r r r s where Ç axe the geodesic curvatures of the net considered. Then we have VcVcVa = JCna + TV's? p r rp rp 3 and VtlVhVö= K,nb+ÇV° T T rr T3 VIIwhere K, = WikVlVk may be regarded as the invariants of the geodesic tor- rp r p sion of the curve of the congruence with tangent vector V and the functions T K, = WijVlVJ is the normal curvature of Wn in the direction of the vector of rr * r r components Vk. T It is shown that the expression -6TK, + 2CK + 2PK, rr Tra T rr is a function of direction. Let Ac 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 Ac = tmxcm+rnc. Let qf = Vjtm - rgkmwjk and Sj = tmWjrn + V7t. Hence, V,AC = qfxcm + Sjnc. Let us define the operator D, D = xtflVj where the operator V7- is the symbol of the prolonged covariant differentiation. We choose n congruences of an orthogonal ennuple of the vector field V(l,2,...,n) which are normalized by gijV*V3 - 1, in the hypersurface T T T Wn (gij,Tk) of the Weyl space Wn+X {gab,Tc). Consider F^j^F where 5^ is the arc length along a curve c& of the orthog onal ennuple in Wn. In this work, the two following theorems are obtained: vmWe have obtained rrr (v*«-> - 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
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