Spin structures have wide applications to mathematical physics,
in particular to quantum field theory. For the special class Spin(7)
geometry, there are different approaches. One of them is constructed
by holonomy groups. According to the Berger classification (1955), the
Spin(7) group is one of these holonomy classes. Firstly, it is presented
its properties. After that, torsion which is another important term
in superstring theory will be geometrically introduced and related to
Spin(7) geometry.
Let M be an 8-dimensional manifold with the Riemannian met-
ric g and structure group G ⊂ SO(8). The structure group G ⊂
Spin(7), then it is called M admits Spin(7)-structure. M. Fernan-
dez [1] classifies the all types of 8-dimensional manifolds admitting
Spin(7)-structure. In general, torsion-free Spin(7) manifold are stud-
ied considerably.
On the other hand, manifolds admitting Spin(7)-structure with tor-
sion have rich geometry as well. Locally conformal parallel structures
has been studied for a long time with K ̈ahler condition is the oldest
one. By means of further groups whose holonomy is the exceptional,
the choices of the G2 and Spin(7) deserves to attention. Ivanov [3], [4],
[5] introduces a condition when 8-dimensional manifold admits locally
conformal parallel Spin(7) structure.
Salur and Yalcinkaya [6] studied almost symplectic structure on
Spin(7)-manifold with 2-plane field. Then, Fowdar [2] studied Spin(7)
metrics from K ̈ahler geometry. In this research, we introduce 8-manifold
equipped with locally conformal Spin(7)-structure with 2-plane field.
Then, almost Hermitian 6-manifold can be classified by the structure
of M.
Keywords: Spin(7) structure, Torsion , Almost Hermitian structure
Primary 53D15; Secondary 53C29.
References
[1] M. Fernandez, A Classification of Riemannian Manifolds with
Structure Group Spin(7), Annali di Mat. Pura ed App., vol (143),
(1986), 101—122.
[2] U. Fowdar Spin(7) metrics from K ̈ahler Geometry, arXiv:2002.03449, (2020)
[3] S. Ivanov, M. Cabrera, SU(3)-structures on submanifolds of a
Spin(7)-manifold, Differential Geometry and its Applications,V 26 (2),
(2008) 113–132
[4] S. Ivanov, M. Parton and P. Piccinni, Locally conformal parallel
G2 and Spin(7)manifolds Mathematical Research Letters, V 13, (2006), 167–177
[5] S. Ivanov Connections with torsion, parallel spinors and geometry
of Spin(7) manifolds,math/0111216v3.
[6] S. Salur and E. Yalcinkaya Almost Symplectic Structures on
Spin(7)-Manifolds,
Proceedings of the 2019 ISAAC Congress (Aveiro, Portugal), 2020)
