Spinq structures
Masayoshi Nagase · Journal of the Mathematical Society of Japan · 1995
In this paper the notion of Spin-structure is introduced and some of the basic materials related to it will be discussed.TO explain the motivation briefly, let us take an $n$ -dimensional compact oriented Riemannian manifold $X$ .The reduced structure group $SO(n)(n\geqq 3)$ has the universal covering group Spin $(n)$ called the Spin group, together with the short exact sequence ([3, \S 5]).As is well-known, it plays a role of great importance parti- cularly in the study of the interrelations between topology, geometry and analysis.However, to our regret, it turns out apparently not always to be effective for researching into a complex manifold $X,$ $w_{2}(X)\equiv c_{1}(X)(mod 2)$ , be- cause there exists a Spin-structure on $X$ if and only if the second Stiefel-Whitney class vanishes, $w_{2}(X)=0$ .To avoid this disadvantage, the notion of $Spin^{c}-$ structure was introduced ([3, \S 5 Remark 4]).That is, using the unitary groupSince the existence can be characterized by the condition that $w_{2}(X)$ is the $mod 2$ reduction of an integral class, a complex structure certainly induces a Spin-structure.The study of complex manifolds using this structure is also too vast to survey here.Let us consider next the case where $X$ has an almost quaternionic structure.The so-called quaternionic K\"ahler manifolds are examples.The research in