# Geometry Seminar

Spin(7) Geometry with Torsion & Vector Fields

Eyup Yalcinkaya

12:00 PM - 1:00 PM

Hylan 202 and Zoom (Id: 722 375 5052)

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, some of almost Hermitian 6-manifold can be classified by the structure of M.

Event contact: skleene at ur dot rochester dot edu

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