Generalizations of warped product manifolds with Spin(7) holonomy

Abstract

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 nonsymmetric 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 $G_2$ 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 $S^3xS^3 xR^2$ 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 $S^3$ . 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 $S^3xS^3 xR^2$ 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 =$F_1 x F_2 x B$ 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×F2 such that $F_i s$ (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 $S^3$ . 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 Ω is a 4-form, then closedness of the Bonan form (dΩ =0) 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 ( $F_i ,i$ = 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.

Özet

Riemann holonomi grupları teorisinde ayrıcalıklı iki grup yer almaktadır. Bunlar 7-boyutlu manifoldlar üzerinde $G_2$ 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 tanımlanmıştır. Bu metrik, literatürde Yasui-Ootsuka tarafından verilen $S^3xS^3 xR^2$ 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 $S^3$ ’e isometrik olması gerektiğidir. Buradan, Yasui-Ootsuka çözümünün (3+3+2) warped-benzeri metrikler sınıfında, yukarıda belirlenmiş olan Bonan formuna karşılık gelen Spin(7) yapıları içerisinde tek olduğunu göstermektedir.