f-Kenmotsu Manifoldlar ve Ricci Solitonlar Üzerine
Tolga Demirli
YÜKSEK LİSANS TEZİ
Matematik ve Bilgisayar Bilimleri Anabilim Dalı Haziran 2014
On the Ricci Solitons and f-Kenmotsu Manifolds
Tolga Demirli
MASTER OF SCIENCE THESIS
Department of Mathematics and Computer Sciences June 2014
f-Kenmotsu Manifoldlar ve Ricci Solitonlar Üzerine
Tolga Demirli
Eskişehir Osmangazi Üniversitesi Fen Bilimleri Enstitüsü Lisansüstü Yönetmeliği Uyarınca
Matematik ve Bilgisayar Bilimleri Anabilim Dalı Geometri Bilim Dalında
YÜKSEK LİSANS TEZİ Olarak Hazırlanmıştır
Danışman: Doç. Dr. Cumali Ekici
Haziran 2014
ONAY
Matematik ve Bilgisayar Bilimleri Anabilim Dalı Yüksek Lisans öğrencisi Tolga Demirli’nin YÜKSEK LİSANS tezi olarak hazırladığı “f-Kenmotsu Manifoldlar ve Ricci Solitonlar Üzerine” başlıklı bu çalışma, jürimizce lisansüstü yönetmeliğin ilgili maddeleri uyarınca değerlendirilerek kabul edilmiştir.
Danışman : Doç. Dr. Cumali EKİCİ
İkinci Danışman : -
Yüksek Lisans Tez Savunma Jürisi:
Üye : Doç. Dr. Cumali EKİCİ
Üye : Prof. Dr. Ali GÖRGÜLÜ
Üye : Doç. Dr. İbrahim GÜNALTILI
Üye : Yrd. Doç. Dr. Melih TURĞUT
Üye : Doç. Dr. Özcan GELİŞGEN
Fen Bilimleri Enstitüsü Yönetim Kurulu’nun ... tarih ve ...
sayılı kararıyla onaylanmıştır.
Prof. Dr. Nimetullah BURNAK Enstitü Müdürü
v ÖZET
Bu tez çalışmasının amacı semi-simetrik metrik olmayan koneksiyonlu f- Kenmotsu manifoldlarında eğrilik tensörlerini, Ricci solitonlarını ve bazı eğrilik şartlarını incelemektir.
Öncelikle, Ricci solitonlar ve f-Kenmotsu manifoldlar üzerine daha önceden yapılan çalışmalar hakkında tarihsel bilgiler verilmiştir. Ardından, geriye kalan çalışmalar iki kısma ayrılırsa ilk kısımda bu tez çalışmasında kullanılacak olan temel kavramlar ve bazı teoremler ele alınmıştır. Devamında Kenmotsu manifoldlarında bazı eğrilik şartlarına göre Ricci solitonların λ sabitine bağlı olarak daralan, genişleyen ve değişmeyen olma durumları gösterilmiştir. Bunun yanı sıra f- Kenmotsu manifoldlar üzerine bazı tanımlar ve teoremler, eğrilik şartları ve bu şartlara göre Ricci solitonların hesaplamalarından oluşmaktadır. İkinci kısımda ise;
semi-simetrik metrik olmayan koneksiyonlu f-Kenmotsu manifoldlar üzerine bazı tanımlar ve teoremler verilip, eğrilik şartları ve bu şartlara göre Ricci solitonların λ sabiti ile f fonksiyonuna bağlı olarak sınıflandırılması yapılmıştır. Ayrıca Ricci semi- simetrik f-Kenmotsu manifoldunun bir Einstein manifoldu olduğu gösterilmiştir.
Anahtar Kelimeler: Hemen hemen değme metrik manifoldu, Eğrilikler, Metrik koneksiyon, Kenmotsu manifoldu, f-Kenmotsu manifoldu, Einstein manifoldu, Ricci solitonlar.
vi
SUMMARY
The aim of this thesis study is to examine Ricci solitons on f-Kenmotsu manifold with the semi-symmetric non-metric connection and some curvature conditions on these manifolds.
Firstly, a short historical brief is given about previous studies related to f- Kenmotsu manifolds and Ricci solitons. Then, if rest part of this study is divided into two sections, firstly fundemantal notions and some theorems, that be used on this study, were indicated. After that, it is investigated that according to some curvature conditions Ricci solitons on Kenmotsu manifolds which are classified as shrinking, expanding and steady with respect to λ constant. Furthermore, some definitions and theorems, curvature conditions and Ricci solitons' calculations according to these conditions on f-Kenmotsu manifolds are claimed. In second section, some definitions and theorems are given related to f- Kenmotsu manifolds with semi-symmetric non-metric connection, curvature conditions and according to these conditions Ricci solitons are classified regards to f function and λ constant.
Moreover, it is indicated that if f- Kenmotsu manifold is Ricci semi-symmetrik then it is also named as Einstein manifold.
Keywords: Almost almost contact metric manifold, Curvatures, Metric connection, Kenmotsu manifold, f-Kenmotsu manifold, Einstein manifold, Ricci solitons.
vii
TEŞEKKÜR
f-Kenmotsu Manifoldlar ve Ricci Solitonlar Üzerine adlı bu tez çalışmamda, bana yardım ederek beni yönlendiren ve gerekli her türlü olanağı sağlayan danışmanım Sayın
Doç. Dr. Cumali EKİCİ
hocama, çalışmam süresince her zaman yanımda olan değerli arkadaşlarıma ve her şeyim aileme destekleri için çok teşekkür ederim.
ESKİŞEHİR, 2014 Tolga DEMİRLİ
viii
İÇİNDEKİLER Sayfa ÖZET ... v
SUMMARY ... vi
TEŞEKKÜR ... vii
SİMGELER DİZİNİ ... x
1. GİRİŞ ... 1
2. TEMEL KAVRAMLAR ... 4
2.1. Riemann Manifoldlar ve Koneksiyonlar ... 4
2.2. Eğrilik Tensörleri ... 8
2.3. Hemen Hemen Değme Manifoldlar ... 11
2.4. Kenmotsu Manifoldları ve Ricci Soliton ... 13
2.5. Semi-simetrik Metrik Olmayan Koneksiyonlu Kenmotsu Manifoldlar ... 16
3. KENMOTSU MANİFOLDLARINDA RICCI SOLİTONLAR VE BAZI EĞRİLİK ŞARTLARI ... 19
3.1. Daralan Ricci Solitonlar ... 19
3.2. Genişleyen Ricci Solitonlar ... 24
3.3. Değişmeyen Ricci Solitonlar ... 32
4. f-KENMOTSU MANİFOLD VE RICCI SOLİTONLAR ... 36
4.1. f-Kenmotsu Manifoldlar ... 36
4.2. Ricci-semisimetrik 3-boyutlu f-Kenmotsu Manifoldlar ... 59
4.3. f-Kenmotsu Manifoldunda Ricci Solitonlar ... 60
ix
İÇİNDEKİLER (devam)
5. SEMİ-SİMETRİK METRİK OLMAYAN KONEKSİYONLU f-KENMOTSU
MANİFOLD VE RICCI SOLİTONLAR ... 65 5.1. Semi-simetrik Metrik Olmayan Koneksiyonlu f-Kenmotsu Manifoldlar ... 65 5.2. Ricci-semisimetrik 3-boyutlu Metrik Olmayan Koneksiyonlu f-Kenmotsu Manifoldlar ... 80
5.3. Semi-simetrik Metrik Olmayan Koneksiyonlu f-Kenmotsu Manifoldunda Ricci Solitonlar ... 81
SONUÇ VE ÖNERİLER ... 86
KAYNAKLAR DİZİNİ ... 87
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