(3+3+2) Warped-like Product Manifolds With Spin(7) Holonomy

Tez (Doktora) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 2008Thesis (PhD) -- İstanbul Technical University, Institute of Science and Technology, 2008Rieman holonomi grup teorisinde iki tane ayrıcalıklı durum vardır, bunlar 7-boyutlu manifoldlar üzerinde olan holonomi grubu ve 8-boyutlu manifoldlar üzerinde olan Spin(7) holonomi grubudur. Aynı zamanda bu holonomi gruplarına sahip Rieman manifoldları Ricci-düz uzaylardır. Bu tez çalışmasında, Spin(7) holonomisine sahip Rieman manifoldlarının yapısı araştırılıp, bu tip manifoldlar üzerindeki warped çarpım metriklerinin bir genelleştirilmesi çalışıldı. Spin(7) holonomi grubuna sahip manifoldları karakterize eden özel bir 4-form yapısının varlığıdır, Bonan form olarak adlandırılır ve Hodge anlamında kendine eş, Spin(7) invaryant ve kapalı bir formdur. Bonan formun yapısını açık şekilde elde etmek için Steiner üçlü sistem yardımıyla yeni bir Bonan form kurulum metodunu sunacağız. Literatürde 8-boyutlu Spin(7) holonomisine sahip manifoldlar üzerinde bir açık metrik yapısı örneği araştırılmış ve Yasui-Ootsuka tarafından manifoldu üzerinde verilen metrik yapı incelenmiştir. Hacim koruyan vektör alanları ve 2-vektör şartı adı verilen özel bir tensör formülünü sağlayan vektör alanları tahmini (ansatzını) kullanalarak Spin(7) invaryant metrik elde etmişlerdir. Bu tez çalışmasında warped çarpım metriğinin bir genelleştirmesi olarak warped-benzeri çarpım metriği ile adlandırdığımız diferansiyel form tahmini (ansatzını) kullanacağız. Çoklu-warped çarpım manifoldlarının bir genelleştirilmesini, lif metrik yapılarının diagonal olmamasına izin vererek warped-benzeri çarpım manifoldları olarak tanımlıyacağız. B baz manifoldu 2-boyutlu, lif uzayları i=1,2 tam, bağlantılı ve basit bağlantılı 3-boyutlu manifoldlar olan biçimindeki manifoldun Spin(7) holonomisine sahip olduğunda, lif uzaylarımız lerin e isometrik olduğunu ispatlayacağız. Sonra (3+3+2) warped-benzeri çarpım metrikleri sınıfı içerisinde ayar dönüşümleri kullanarak Yasui-Ootsuka çözümünü yeniden elde edeceğiz.In the theory of Riemannian holonomy groups there are two exceptional cases, the holonomy group in 7-dimensional and the holonomy group Spin(7) in 8-dimensional manifolds. Also Riemanian manifolds with these holonomy groups are Ricci-flat. In the present thesis, we investigate the structure of Riemannian manifolds with Spin(7) holonomy group and study a metric structure as a generalization warped product metrics on these type of manifolds. Spin(7) holonomy manifolds are characterized by the existence of a special 4-form called Bonan form which is a self-dual in the Hodge sense, Spin(7) invariant and closed form. Thus we review the methods of the explicit construction for the Bonan form and present a new method related to Steiner triple systems. We have surveyed an explicit metric structure of 8-dimensional manifold with Spin(7) holonomy in the literature and worked out the metric given by Yasui and Ootsuka on the manifold . They use vector fields ansatz to satisfy the concepts of volume-preserving vector fields and a specific tensor formula called the 2-vector condition. In our work, we use a differential form ansatz called warped-like product which is a generalization of warped product metric. We define warped-like product manifolds as a generalization of multiply-warped product manifolds, by allowing the fiber metric to be non block diagonal. We prove that if is a warped-like product manifold where the base B is two dimensional, the fibers i=1,2, are complete, connected and simply connected 3-manifolds and M has Spin(7) holonomy, then the s are isometric to . Then we recover the Yasui-Ootsuka solution in this class of (3+3+2) warped-like product metrics up to gauge transformations.DoktoraPh

Generalizations of warped product manifolds with Spin(7) holonomy

Riemann holonomi grupları teorisinde ayrıcalıklı iki grup yer almaktadır. Bunlar 7-boyutlu manifoldlar üzerinde G2 ve 8-boyutlu manifoldlar üzerinde Spin(7) holonomi gruplarıdır. Bu çalışmada, holonomi grubu Spin(7)'nin bir alt grubu olan Riemann manifoldlarının yapısı, özel bir durum için incelenmiştir.  Spin(7) holonomisine sahip manifoldlar, Bonan formu olarak adlandırılan bir 4-formun varlığı ile karakterize edilir. Bonan formu Hodge anlamında kendine eş,  Spin(7) invaryant ve kapalı bir    4-formdur. Çalışmada öncelikli olarak Bonan formunun oktonion çarpımı kullanılarak elde edilme yolu verilmiştir. Daha sonra, çoklu warped çarpım metriklerinin genellemeleri tartışılmış ve özel bir hal olan (3+3+2) warped-benzeri çarpım metriği tanım-lanmıştır. Bu metrik, literatürde Yasui-Ootsuka tarafından verilen Spin(7)manifoldu üzerindeki metriğin bir soyutlaması olarak düşünülmüş olup, warped çarpımların lif-taban dekompozisyonunu korumakta, ancak lif uzayındaki metriğin blok köşegen olmadığı durumu da içermektedir. Çalışmada elde edilen ana sonuç, 2 boyutlu bir taban üzerinde, 3 boyutlu, tam, bağlantılı ve basit bağlantılı liflerden oluşan (3+3+2) warped-benzeri bir çarpım manifoldunda, eğer Yasui-Ootsuka çalışmasında kullanılan Bonan formu kapalı ise, liflerin  S3’e isometrik olması gerektiğidir. Buradan, Yasui-Ootsuka çöz ümünün (3+3+2) warped-benzeri metrikler sınıfında, yukarıda belirlenmiş olan Bonan formuna karşılık gelen  Spin(7)yapıları içerisinde tek olduğunu göstermektedir. Anahtar Kelimeler: Holonomi, Spin(7) holonomi manifoldu, warped ve çoklu warped çarpım manifoldları, warped-benzeri çarpım manifoldu.The holonomy group of a Riemannian manifold was defined by Elie Cartan in 1923 and proved to be an efficient tool in the study of Riemannian manifolds (Kobayashi and Nomizu, 1969). Later, Berger (Berger, 1955) gave a list of the possible holonomy groups of irreducible, simply-connected and non-symmetric Riemannian manifolds. Berger's list (refined later by Alekseevski (1968) and Gray-Brown (1972)) includes the groups SO(n) in n-dimensions, U(n),SU(n) in 2n-dimensions, Sp(n),Sp(n)Sp(1) in 4n-dimensions and two exceptional cases, the holonomy group G2 in 7-dimensions and the holonomy group Spin(7) in 8-dimensions. After Berger introduced his classi-fication list, the existence of manifolds with the specified holonomy groups was an open problem. The existence of manifolds with exceptional holonomy was first demonstrated by Bryant (1987), complete examples were given by Bryant and Salamon (1989) and the first compact examples were found by Joyce (1996). The study of manifolds with exceptional holonomy and the construction of explicit examples is still an active research area in mathematics and physics. In the present work, we investigate the structure of Riemannian manifolds whose holonomy group is a subgroup of Spin(7), for a special case. Manifolds with Spin(7)  holonomy are characterized by the existence of a globally defined 4-form, called the Bonan form (Bonan, 1966) with the following properties i- self-duality in the Hodge sense, ii- Spin(7)  invariance, iii- closedness. We review the structure of the Bonan form and its explicit construction using the structure constants of the octonionic algebra. The starting  point of the present   research was an explicit example of  Spin(7) metric on S3 x S3 x R2 given by Yasui and Ootsuka (2001). We looked whether one could obtain other solutions by relaxing some of their assumptions, in particular without requiring the three dimensional submanifolds to be S3  The method used in (Yasui and Ootsuka, 2001) is based on the notion of "volume-preserving vector fields" and a specific tensor formula called the "2-vector condition". The construction of a metric with Spin(7) holonomy starts with an ansatz for an orthonormal frame which is shown to satisfy the conditions given in (Yasui and Ootsuka, 2001), provided that certain first order differential equations are satisfied. Then the solution of these equations gives a metric with Spin(7) holonomy on S3 x S3 x R2 that we call the "Yasui-Ootsuka solution". Inspired by the metric ansatz of Yasui-Ootsuka, we discuss a generalization of warped product metrics (O'Neil, 1983), by allowing the fiber metric to be non block diagonal in a multiply-warped product (Flores and Sanchez, 2002). We work with a spesific case that we call (3+3+2) warped-like product manifold M = F1 x F2 x B and a specific Spin(7) structure. We prove that, when  the base B is two  dimensional, the fibre F is a 6-manifold of the form   F = F1 x F2  such that Fi 's (i=1,2) are complete, connected and simply connected 3-manifolds and the metric is given by the (3+3+2) warped-like product, then the connection of the fibers is completely determined by the requirement that the Bonan 4-form given in the work by Yasui and Ootsuka (2001) be closed. With the global assumptions given above, it is concluded that the fibers (Fi,i = 1,2) are isometric to S3. It follows that the Yasui-Ootsuka solution is unique in the class of (3+3+2) warped-like product metrics admitting the Spin(7) structure determined by the Bonan form given in the work by Yasui-Ootsuka (2001). As the Bonan form  is a 4-form, then closedness of the Bonan form  gives 56 equations involving exterior derivatives of the basis 1-forms. In the case of the (3+3+2) warped-like product metric, there are 9 parameters on each 3-manifolds (Fi,i = 1,2). Hence there are totally 18 parameters and 56 equations mentioned above. Under some special conditions, it is not surprising to obtain a unique solution. Keywords: Holonomy, Spin(7) holonomy manifold, warped and multiply warped product manifolds, warped-like product manifolds

Fiber structures of special (4+3+1) warped-like manifolds with Spin(7) holonomy

L generalization of 8-dimensional multiply-warped product manifolds is considered as a special warped product, by allowing the fiber metric to be non-block diagonal. Motivating from the previous paper [S. Uguz and A. H. Bilge, (3 + 3 + 2) warped-like product manifolds with Spin(7) holonomy, J. Geom. Phys. 61 (2011) 1093-1103], we present a special warped product as a (4 + 3 + 1) warped-like manifold of the form M = F x B, where the base B is a 1-dimensional Riemannian manifold, and the fiber F is of the form F = F-1 x F-2 where F-i's (i = 1, 2) are Riemannian 4- and 3-manifolds, respectively. It is showed that the connection on M is entirely determined provided that the Bonan 4-form is closed. Assuming that the F-i's are complete, connected and simply connected, it is proved that the 3-dimensional fiber is isometric to S-3 with constant curvature k > 0. Finally, the geometric properties of the 4-dimensional fiber of M are studied