Weyl hiperyüzeylerinde Chebyshev şebekeleri
Chebyshev nets in weyl hypersurfaces
- Tez No: 39830
- Danışmanlar: PROF. DR. ABDULKADİR ÖZDEĞER
- Tez Türü: Yüksek Lisans
- Konular: Matematik, Mathematics
- Anahtar Kelimeler: Chebyshev şebekeleri, Hiperyüzey, Weyl uzayları, Chebyshev networks, Hypersurface, 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
Bilindiği gibi, konform bir gv metrik tensörüne ve bu tensörle ^kgİJ=2Tlcgil şeklinde bir uygunluk koşulunu gerçekleyen simetrik bir konneksiyona sahip n-boyutlu bir Wn manifolduna Weyl uzayı denir. Burada Tk kovaryant bir vektörü, Vfc&,- ise alışılmış kovaryant türevi göstermektedir. Wn Weyl uzaymda bağımsız v1 (r = 1,2,..., m) vektör alanları n-boyutlu bir (v,v,...,v) 1 2 şebekesi belirler. Bu çalışmanın birinci bölümünde, bir Weyl uzayının metrik tensörüne ait uyduların genelleştirilmiş türevleri ve genelleştirilmiş kovaryant türevleri tanımlanarak, n-li bir şebekeye ait vektör alanlarına ve bunların karşıtlarına ait türev formülleri verilmiştir. Çalışmanın ikinci bölümünde, Weyl uzaylarında birinci ve ikinci cins Chebyshev şebekesi tanımları verilerek bu tür şebekelerin, kendilerine karşı gelen eğrilikler yardımıyla karakterizasyonlan üzerinde durulmuştur. Bu bölümde, ayrıca, bu cins şebekelere sahip Weyl hiperyüzeyleri ile ilgili olarak, bilinen üç teoremin ispatlarına yer verilmiştir. WH+l Weyl uzayının Wn hiperyüzeyine ait genelleştirilmiş metriksel Chebyshev, kuvvetli-metriksel Chebyshev ve genelleştirilmiş eşit-uzaklıklı şebekelerin ele alındığı üçüncü bölümde, bu çeşit şebekelerle ilgili üç yeni teorem ispatlanmıştır.
Özet (Çeviri)
An n-dimensional manifold Wn is said to be a Weyl space, if it has a conformal metric tensor gtj and a symmetric connection satisfying the compatibility condition given by the equation V^. =2Tkgv, where Tk denotes a covariant vector and Vt&, denotes usual covariant derivative. In n-dimensional Weyl space Wn, the independent vector fields v'(r=l,2,...,n) determine an n-dimensional net (v,v,...,v). r 12» o Let A be a satellite of gtj with weight {k}. dt A, given by the equation o o is said to be the prolonged derivative of A and V4 A, given by the equation o VsA = VsA-hTsA, is called the prolonged covariant derivative of A. The prolonged covariant derivatives of the vector fields v' and their a a reciprocals vt are, respectively, given by a a a vi v' = Tk v' > Vfc vi i =~ Tk vi (i,k,a,<r= 1,2,-,»). a a a <r From these formulas, it follows that a Tk cosp = 0, ^*+»V*]=0' -V-V{Jk-[ + Tik Tsl =0 a a a where R jM and a are, respectively, the curvature tensor of Wn and the angle between the directions determined by v and v. a a If any one of the vector fields of the net (v,v,...,v) is parallelly 12 n translated 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 parallelly translated along the lines determined by the remaining field of the net. Corresponding to these two kinds of Chebyshev nets, the Chebyshev curvatures of the first and second kind are, respectively, defined by r= Tkvk (a,*,c,* = l,2,...,/i; a*b) ab a * and b b © The Chebyshev nets of the first and second kind, are also characterized by the corresponding Chebyshev vectors as follows: The vectors defined by a' ' = tv,bi = pvi.* sk r r are, respectively, called the first and second Chebyshev vectors of the net (v,v,...,v). 1 2 n On the other hand, the net (v,v,...,v) whose Chebyshev vectors of the second kind satisfy the condition 1 2 n s ZA,=0 -VI-is said to be a b-net The net (v,v,...,v) will be called a c-net if its geodesic vectors satisfy 1 2 n the condition £ c' = 0 *=1 * In [6], Tsareva and Zlatanov studied the Chebyshev nets of the first and second kind. They also gave some characterizations of b-nets and c-nets. In the same paper, they studied the subnet (v,v) of the net (v,v,...,v) by using the r s '“ tensors and the affinor defined, respectively, by I 2 and rs r s r s aik ~ Vi Vk+Vk Vi » a* = V' V* + V* V' rs r s r s k b k a. = v. v + v v.. rs s r The following results concernings such subnets are obtained in [6]: (a) İy+Â/=-V*(® «*) OS (b) If the area element (v,v,..., v, v,...,v) is parallelly translated along the lines 12 r-\ r+l n (v), then s @ (c) If the subnet (v,v) is geodesic, then b,+b,=-Vk(kaJlakJ) rs rs (d) If the subnet (v,v) is geodesic and area element (v,v,..., v, v,...,v) is r s 1 2 r-l r+l n parallelly translated along the lines (v), then r b\=-Vkia\a% re re J -vu-By means of these results, the following two corollaries are easily obtained: If the net (v,v,...,v) contains a Chebyshev subnet (v,v) and area \ 1 n r s elements (v,v,..., v, v,...,v) and (v,v,..., v, v,...,v) are parallelly translated 1 2 r-1 r+1 n 12 s-\ s+l n along the lines (v) and (v), respectively, then the tensors of the subnet (v,v) r s r s satisfy the condition V*(a,j*) = 0. If the net (v,v,...,v) contains â Chebyshev and geodesic subnet (v,v) I 2 n r s and area elements (v,v,..., v, v,...,v) and (v,v,..., v, v,...,v) are parallelly 1 2 r-1 r+1 n 1 2 s-1 «+1 n translated along the lines (v) and (v), respectively, then the affinor of the r s subnet (v,v) satisfies the condition r s Vfc(a{aj) = 0. In [9], n-dimensional nets in the hypersurface Wn of the Weyl space Wn+X are considered and the following results and corollaries are obtained: 1)-If a”and a' are, respectively, the components of the Chebyshev rp rp vector fields of the first kind of the net (v,v,...,v) with respect to Wn+, and 1 2 n Wn, then the relation x'o“ = a' a rp rp holds. From this it follows that, if the net (v,v,...,v) in Wn is a Chebyshev net 1 2 of the first kind with respect to Wn+X, it is also a Chebyshev net of the first -VUl-kind with respect to Wn. Furthermore, if the net is a Chebyshev net of the first kind relative to Wn, then the Chebyshev vector fields of the first kind relative to W”+l are normal to Wn 2)- If ba and bt are, respectively, the components of the Chebyshev vector fields of the second kind of the net (v,v,...,v) with respect to Wn+J and I 2 n Wn, then the relations r i r hold. As a consequence of this theorem we conclude that, if the net (v,v,...,v) in Wn is a Chebyshev net of the second kind with respect to Wn+1, 1 2 n it is also a Chebyshev net of the second kind with respect to Wn. Furthermore, if the net is a Chebyshev net of the second kind relative to Wn, then the Chebyshev vector fields of the second kind relative to Wn+X are normal to Wn. If the net (v,v,...,v) in Wn is a b-net with respect to Wn+i, it is also a b-net with respect to Wn with £0' =o. Furthermore, if the net (v,v,...,v) is a i 1 2 n b-net with respect to Wn, then the vector field £ ba is normal to Wn 3)- If c“ and c' are, respectively, the components of the geodezic vector r r fields of the net (v,v,...,v) with respect to Wn+l and W”, then they are related 1 2 n by the equations x İ ca = c1 and Ica = Zicif +1 d x *. -IX-According to this theorem, if the net (v,v,...,v) in Wn is a c-net with 1 2 respect to Wn+X, it is also a c-net with respect to Wn with £«.=(). Moreover, r“. if the net (v,v,...,v) is a c-net with respect to Wn, then the vector field £ca \ 1 n r r is normal to W.. Let W”(gv,Tk) be a hypersurface of (n+l)-dimensional Weyl space d a (a=l,2,...,n), the condition KASabX) and let (v,v,...,v) be a net belonging to W"(gij,Tk). If for a fixed 12 n V[,v,] = 0 holds, the net (v,V,...,v) is said to be a generalized metrically a-Chebyshev net. I 2 n If the above condition holds for each a (a=l,2,...,n), the net (v,v,...,v) is said 1 2 n to be strongly-metrically Chebyshev. If a net (v,v,...,v) belonging to the hypersurface Wn satisfies the condition 12 n o Vt(v<+v,+...+vl) = 0 [ik], it is said to be a generalized equidistant Chebyshev net. In this work, the following three theorems concerning these special nets are obtained: If the net (v,v,...,v) belonging to the hypersurface Wn of Wn+X is 1 2 n generalized metrically a-Chebyshev with respect to Wn+l, it is also generalized metrically a -Chebyshev with respect to the hypersurface Wn. A strongly-metrically net (v,v,...,v) belonging to the Weyl space W is a 1 2 n generalized equidistant net. -X-If the net (v,v,...,v) belonging to the hypersurface Wn of Wn+X is 12 n generalized equidistant with respect to Wn+X, it is also generalized equidistant with respect to the hypersurface Wn.
Benzer Tezler
- Weyl hiperyüzeylerinde genelleştirilmiş darboux fonksiyonu
Darboux function in a weyl hypersurface
FÜSUN ÖZEN
- Weyl hiperyüzeylerinde genelleştirilmiş laguerre fonksiyonu
Laguerre's function in a weyl hypersurface
SEZGİN ALTAY
- Weyl uzaylarında bazı özel eğri şebekeleri
Some special nets of curves in weyl spaces
NİL KOFOĞLU
Doktora
Türkçe
1997
Matematikİstanbul Teknik ÜniversitesiMatematik Ana Bilim Dalı
PROF. DR. ABDÜLKADİR ÖZDEĞER
- Weyl-wigner-groenewold-moyal kuantizasyonu
Weyl-wigner-groenewold-moyal quantization
İLHAMİ BUĞDAYCI
Yüksek Lisans
Türkçe
1998
Fizik ve Fizik MühendisliğiAnkara ÜniversitesiFizik Ana Bilim Dalı
PROF. DR. ABDULLAH VERÇİN