Let V be a pseudo-Riemannian n-dimensional manifold or, more generally, let \((\xi ,Q)\) be a real fibre bundle whose base space is a paracompact space endowed with a non-degenerate quadratic form Q, (that is, with a structure group \({\mathrm {O}}(p,q),\)\(n=p+q\).) Let \(K_{p,q}\) denote the obstruction class for the existence of a \(\mathrm {Pin}(p,q)\)-spin structure on V or over \(\xi .\) Let \(K_{\mathrm {Conf}}(p,q)\) denote the obstruction class for the existence of a conformal spin structure in a strict sense on V or over \(\xi ,\) (simply: a \(C_{n}^{s}(p,q)\)-spin structure), if \(n=2r,\) or of a conformal special spin structure, if \(n=2r+1.\) This short self-contained paper will recall the determination of the obstruction class \(K_{p,q}\) on V, or over \(\xi ,\) for n even or odd. Then, the obstruction class \(K_{p+1,q+1}\) for the existence of a \(\mathrm {Pin}(p+1,q+1)\)-spin structure over \(\xi _{j}\), (Greub’s j-extension of \(\xi ,\) where j denotes the identity mapping from \({\mathrm {O}}(p,q)\) into \({\mathrm {O}}(p+1,q+1)\)), will be determined in order to express \(K_{\mathrm {Conf}}(p,q),\) for \(n=2r\) or \(n=2r+1,\) in terms of the Stiefel–Whitney classes \(w_{i}(p,q),\)\(i=1,2,\) of \(\xi \), decomposed as the Whitney sum \(\xi =\xi ^{+}\oplus \xi ^{-},\) where the restriction of Q to \(\xi ^{+}\) is positive definite and the restriction of Q to \(\xi ^{-}\) is negative. If \(n=2r,\) we find again results obtained in previous publications [4, 5, 7], by different methods.

中文翻译：

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严格意义上共形自旋结构存在的障碍类别

令V为伪黎曼n维流形，或更一般而言，令\ {{\ xi，Q} \}为实纤维束，其基空间是具有非简并二次形式Q的拟紧空间，（也就是说，具有结构组\（{\ mathrm {O}}（p，q），\）\（n = p + q \）。）让\（K_ {p，q} \）表示障碍物类别在V上或在\（\ xi。\）上存在\（\ mathrm {Pin}（p，q）\）- spin结构的情况下，让\（K _ {\ mathrm {Conf}}（p，q）\ ）表示在V或以上严格意义上存在共形自旋结构的障碍类别\（\ xi，\）（简单地：\（C_ {n} ^ {s}（p，q）\）- spin结构），如果\（n = 2r，\）或共形特殊自旋结构，如果\（n = 2r + 1。\）这个简短的独立论文将回顾在V上或在\（\ xi，\）上对n个偶数的障碍物类\（K_ {p，q} \）的确定。或奇怪。然后，在\（\ xi上存在\（\ mathrm {Pin}（p + 1，q + 1）\）- spin结构的障碍类\（K_ {p + 1，q + 1} \）_ {Ĵ} \） ，（Greub的Ĵ的-extension \（\ XI，\）其中Ĵ表示从身份映射\（{\ mathrm {O}}（p，q）\）变成\（{\ mathrm {O}}（p + 1，q + 1）\）），将被确定以表达\（K_ { \ mathrm {CONF}}（p，q），\）为\（N = 2R \）或\（N = 2R + 1，\）在施蒂费尔-惠特尼类的术语\（W_ {I}（p， q），\）\（I = 1,2，\）的\（\ XI \） ，分解为惠特尼总和\（\ XI = \ XI ^ {+} \ oplus \ XI ^ { - }，\）其中Q对\（\ xi ^ {+} \）的限制为正定，而Q对\（\ xi ^ {-} \）的限制为负。如果\（n = 2r，\） 我们再次发现以前的出版物[4、5、7]通过不同方法获得的结果。