Harmonik kaehler manifoldları için bir eğrilik özdeşliği
A Curvature identity for harmonic kaehlerian manifolds
- Tez No: 14129
- Danışmanlar: PROF.DR. ABDÜLKADİR ÖZDEĞER
- Tez Türü: Yüksek Lisans
- Konular: Matematik, Mathematics
- Anahtar Kelimeler: Kaehler manifoldu, Tensör alanı, Kaehler manifold, Tensor field
- Yıl: 1990
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
ÜZEI O- J H, F = (F- ) kompleks yapısına sahip, 2n-boyutlu bir harmo- nik Kaehler manifoldu olsun. H manifoldu üzerinde °hijk = RhiabRj k_RhabkRj i + Rhab.jRk i' Whijk“ Dhijk * WiV - DhkabFiaFjb + 2DhiabF/Fkb şeklinde tanımlanan D ve W tensör alanlarını gözönüne alalım. Burada iuvj.?GI\,1 II1UC OUI! I III IUI1UII u ve R.ab gaugbvR.. dır ı.ı y y ı uvj Bu çalışmada, D ve U tensör alanlarıyla ilgili olarak aşağıda ki iki teorimin ispatı verilmiştir. F= (F.J) kompleks yapısına sahip H n manifoldu üzerinde ta nımlı W = (W...*) tensör alanı Whijk = k(si j9hk ”9ik9hj + FijFhk“ FikFhj ”2Fhi V özdeşliğini gerçekler. Burada k = - ¦* - f(0) ve f de H n nin karak teristik fonksiyonudur. H harmonik Kaehler manifoldunun sabit holomorfik kesitsel eğriliğe sahip olması için gerek ve yeter şart, D= (Dı..k) tensör alanının J Dhijk = a Rhijk+ b^ıj9hk " 9ik9hj) özdeşliğini sağlamasıdır. Burada a ve b reel sabitlerdir. -iv-
Özet (Çeviri)
A CURVATURE IDENTITY FOR HARMONIC KAEHLERIAN MANIFOLDS SUMMARY The purpose of this work is to obtain a curvature identity for harmonic Kaehierian manifolds. Let Mn be a Riemannian manifold of dimension n- Let p Mn be the origin of a normal neighbourhood N. Denote. the, geodesic distance from p to any point p in N by s and let Q = y~ s^. ^ for every point p of Mn, Aft = f(ft), then Mn is called a harmonic manifold which we denote by Hn. The function f(fi) is called the characteristic function of Hn. It is well known that, a 2n-dimensional complex manifold Nrn with an almost complex structure F= (F-J) and a Riemannian metric (g-.) satisfying the conditions FfV“8iJ ¦ Fij-Via' FiJ = ”FJi ' \Fi-°- is called a Kaehierian manifold. Let Nrn be a 2n- dimensional Riemannian manifold and -L-J be the Christofrfel symbols. The Riemannian curvature tensor Rnl-,- » the Ricci tensor R.. and the scalar curvature R are defined, respec tively, by 1J «hi/“ 8h {io} ”3i <hj> + <îj> O - <hj> {ia>.“ij ~ Raij '.v-If a tensor field S = (S. ”.. ) satisfies the conditions v hij Shij ~ ' Sihj ' Shijk“ Shikj and S...k + S... k + S-t,,k = 0 hij ijh jhi it is called a generalized curvature tensor field, where a Shijk 9akShij In this work, two tensor fields denoted by D.... and W.... are defined and two identities satisfied by these tensors are obtained. The tensor field D.... is defined to be hijk Dhijk = Rhiab Rj k ”RhabkRj i + ^abj Rk i Using the tensor fields a.., &. ^.. 9 Yu,-,-^ which are defined“hijk = ^iab^k ' 3hijk = RhabiRjk Yhijk ~ %abiR5 k > the tensor field Dh-.. can be expressed in the form °hijk = ehijk ”Yhkji * yhjki ' The tensor fields a...., 3h.._.,,, yhUV satisfy the following conditions: T,1JK niJI< niJK“hijk = 'aihjk = _ahikj Thijk ~Yjkhi =Yihkj -vi-Using the first Bianchi identity Rhijk + Rijhk + Rjhik ~ ° one obtains 3hijk ~ ”“7”ahijk Thijk“ Tihjk ~ Yhijk ”Yhikj ~“BMjk By virtue of these conditions it can be shown that the tensor field Ek... is a generalized curvature tensor field. The Ricci tensor D... and the scalar curvature D with respect to D = (Dl.,-,- ) are given by n _ 3 p n abc _ nab D üij ~ T”Kiabc Kj K Raijb' and d = - Rah,HRabcd- RabRah o abed ab respectively. Let H be a 2n- dimensional harmonic Kaehlerian manifold with complex structure F= (fy*). The tensor field w“hi -k is defined as W - Dhijk * <^ab Fi V ”WF1 V ¦ 2DhiabFjaFkb. It is well known that any n-dimensional harmonic manifold satisfies the following equalities Rij = --T ^')9ij. *---T-'f (0) -vn-Yhijk 4 Yhjki +Yhkij + Yhikj + Yhkji + Yhjik = - -JL f(0)(ghigjk + ghjgki + g^.), and Rabc/bC<J ---f^l?(0)* -ÜSÜİ-MO)}. where (.) denotes the operation of taking the derivative with respect toa([4], [5])- The curvature tensor R = (K** ) of a 2n- dimensional Kaehlerian manifold with complex structure F= (F^) satisfies the conditions n D r Sp d D kc a n 3p K“Viljk ~ Khiab j fk. ”hia Fj ~“tiij ha ^abij- ”2Fi\j. ^“iabj ”Fi\j and Uhiabhj ^k ~ ujkabhh hi. Using the last equation, it is found that W = (VL.. ) is a generalized curvature tensor field. The Ricci tensor W-h and the scalar curvature W with respect to the tensor field W = (w...k) are given by J W-. = D.. + 3D. - FcaF.b ij ij Juabic j > and“ ”D - 3“abc/^ respectively, where F = g&Cfr_ ¦vin-Let H be a 2n- dimensional harmonic Kaehlerian manifold with complex structure F = (F-J). It is proved that the tensor field Wh-.. satifies the following condition. Whijk = k^ij9hk ~ SikShj + FijFhk ' FikFhj ”2FhiFok). where k = - -J - f (0) and f is the characteristic function of H n. Let u be a vector of the tangent space at the point p of the Kaehlerian manifold M. The sectional curvature K(p) at p e Mn is given by K(p) = -__mllk h J... gniuhu1g.kuJ*uk If the holomorphic sectional curvature is constant with respect to any vector at every point of the manifold, then the Kaehlerian manifold Mn is said to be of constant holomorphic sectional curvature. It is proved that if a Kaehlerian manifold has a constant holomorphic sectional curvature K at each point, then the curvature tensor of the manifold is of the form ^ijk =_4“ f9nk9ij ”9ikgnj + F^F.j - F.kFhj -2FhiFjk], and the scalar K is an absolute constant. Finally, it is proved that if H is a 2n- dimensional harmonic Kaehlerian manifold, then a necessary and sufficient condition for H^n to be of constant holomorphic sectional curvature is that the tensor field Dh^-k is expressible in the form °hijk - aRhijk + b(gij9hk“ 9ikghj). where a, b are real constants. To prove this theorem, we note that aij - F F b 9 - Mbhj -ix-”aur r a 9 Fuk =“Fk ' Furthermore, the harmonic Kaehlerian manifold H of constant holomorphic sectional curvature is an Einstein space: Rij = k(2n+2)g.d, R where k =. 2n(2n+2) In this case, D.... tensor field is obtained as ^» ”lS“ W* J(ntl) tgioghk ”9ik3hj>- -x-
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