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Let $(M,g)$ be a $d$-dimensional, oriented pseudo-Riemannian manifold, and $V_+$ V$the subbundle of$E=TM\oplus T^*M$given by the graph of the musical linear isomorphism$g^\flat:TM\rightarrow T^*M$associated to the metric$g$. The nondegeneracy of$g$entails that$E$decomposes as the Whitney sum$E=V\oplus V'$, where$V'$is the graph of$-g^\flat=(-g)^\flat$. Indeed, the projections $P_\pm(X\oplus\xi)=\frac{1}{2}(X\pm g^\sharp(\xi))\oplus(\xi\pm g^\flat(X))$ satisfy$P_-=\mathbb{1}-P_+$,$P_+(E)=V$and$P_-(E)=V'$, where$g^\sharp:T^*M\rightarrow TM$is the musical linear isomorphism associated to$g^{-1}$. Moreover,$E$carries a canonical, pseudo-Riemannian metric$h$of signature$0$$h(X_1\oplus\xi_1,X_2\oplus\xi_2)=\frac{1}{2}(\xi_1(X_2)+\xi_2(X_1))$ such that$(\mathbb{1}\oplus g^\flat)^*(h|_V)=g$. It can be shown that$E$admits a spin structure associated to$h$- for example, the (space of sections of the) exterior algebra bundle$\Lambda^*T^*M$is a Clifford module, and its tensor product$\Lambda^*T^*M\otimes(\Lambda^dT^*M)^{1/2}$with the real line bundle of half-densities over$M$is a spinor bundle. Moreover, the space of spin structures on$(E,h)$is an affine space modelled on the group$H^1(M,\mathbb{Z}_2)$of real line bundles over$M$, which maps the corresponding spinor bundles onto each other by tensoring (the above preliminary results can be found on Chapter 2 of Marco Gualtieri's PhD thesis on generalized complex geometry, arXiv:math.DG/0401221). Question(s): if$(M,g)$admits a spin structure, does a choice of spin structure on$(TM\oplus T^*M,h)$descend by restriction to$V$to a choice of spin structure on$(M,g)$? Does this establish a one-to-one correspondence between both sets of spin structures? If so, how does this generalize to, say, generalized Riemannian metrics (i.e. rank-$d$subbundles$W$of a twisting of$TM\oplus T^*M$by a Cech 1-cocycle$B$with values at closed 2-forms, such that the restriction of$h$to$W$is positive definite)? In other words, I want to know if, in the above convext, there is a specific converse to the well-known property that, given two pseudo-Riemannian vector bundles$V,V'$and their Whitney sum$E=V\oplus V'$, a choice of spin structure for any two of these bundles uniquely determines a spin structure on the remaining one. In the case$V$and$V'$are Riemannian, this is Proposition 2.1.15, pp. 84-85 of H. B. Lawson and M.-L.Michelsohn, "Spin Geometry" (Princeton, 1989). See also M. Karoubi, "Algèbres de Clifford et K-Théorie". Ann. Sci. Éc. Norm. Sup. 1 (1968) 161-270. More precisely, here we use the fact that there is a canonical isomorphism between$Spin(p,q)$and$Spin(q,p)$which covers the canonical isomorphism between$SO(p,q)$and$SO(q,p)$(see M. Karoubi, ibid.) to establish a canonical one-to-one correspondence between the set of spin structures on$(M,g)$and the set of spin structures on$(M,-g)$. This correspondence, on its turn, is used to induce spin structures on$V$and$V'$from that of$(M,g)$together with the orientation-preserving, isometric bundle isomorphisms$\mathbb{1}\oplus(\pm g^\flat)$. What I want to know is if, among the pairs of spin structures on$V$and$V'$which determine the given spin structure on$E$in the above fashion, there is (only?) one pair which, once pulled back to$M$, is related by the above one-to-one correspondence between$Spin(p,q)$- and$Spin(q,p)$-structures. 6 Expanded explanation of context and question made more precise using input from comments Let$(M,g)$be a$d$-dimensional, oriented pseudo-Riemannian manifold, and$V$V_+$ the subbundle of $TM\oplus E=TM\oplus T^*M$ given by the graph of the "musical" musical linear isomorphism $g^\flat:TM\rightarrow T^*M$ associated to the metric $g$. We know The nondegeneracy of $g$ entails that $E$ decomposes as the bundle Whitney sum $TM\oplus T^*M$ E=V\oplus V'$, where$V'$is the graph of$-g^\flat=(-g)^\flat$. Indeed, the projections $P_\pm(X\oplus\xi)=\frac{1}{2}(X\pm g^\sharp(\xi))\oplus(\xi\pm g^\flat(X))$ satisfy$P_-=\mathbb{1}-P_+$,$P_+(E)=V$and$P_-(E)=V'$, where$g^\sharp:T^*M\rightarrow TM$is the musical linear isomorphism associated to$g^{-1}$. Moreover,$E$carries a natural canonical, pseudo-Riemannian metric$h$of signature$(d,d)$and 0$
$h(X_1\oplus\xi_1,X_2\oplus\xi_2)=\frac{1}{2}(\xi_1(X_2)+\xi_2(X_1))$
such that $(\mathbb{1}\oplus g^\flat)^*(h|_V)=g$. It can be shown that $E$ admits a spin structure associated to $h$ - for example, the (space of sections of the) exterior algebra bundle $\Lambda^*T^*M$ is a Clifford module, and its tensor product $\Lambda^*T^*M\otimes(\Lambda^dT^*M)^{1/2}$ with the real line bundle of half-densities over $M$ is a spinor bundle. Moreover, the space of spin structures on $(TM\oplus T^*M,h)$ (E,h)$is an affine space modelled on the group$H^1(M,\mathbb{Z}_2)$of real line bundles over$M$, which maps the corresponding spinor bundles onto each other by tensoring (the above preliminary results can be found on Chapter 2 of Marco Gualtieri's PhD thesis on generalized complex geometry, arXiv:math.DG/0401221). In other words, I want to know if, in the above convext, there is a specific converse to the well-known property that, given two pseudo-Riemannian vector bundles$V,V'$and their Whitney sum$E=V\oplus V'$, a choice of spin structure for any two of these bundles uniquely determines a spin structure on the remaining one. In the case$V$and$V'$are Riemannian, this is Proposition 2.1.15, pp. 84-85 of H. B. Lawson and M.-L.Michelsohn, "Spin Geometry" (Princeton, 1989). See also M. Karoubi, "Algèbres de Clifford et K-Théorie". Ann. Sci. Éc. Norm. Sup. 1 (1968) 161-270. More precisely, here we use the fact that there is a canonical isomorphism between$Spin(p,q)$and$Spin(q,p)$which covers the canonical isomorphism between$SO(p,q)$and$SO(q,p)$(see M. Karoubi, ibid.) to establish a canonical one-to-one correspondence between the set of spin structures on$(M,g)$and the set of spin structures on$(M,-g)$. This correspondence, on its turn, is used to induce spin structures on$V$and$V'$from that of$(M,g)$together with the orientation-preserving, isometric bundle isomorphisms$\mathbb{1}\oplus(\pm g^\flat)$. What I want to know is if, among the pairs of spin structures on$V$and$V'$which determine the given spin structure on$E$in the above fashion, there is (only?) one pair which, once pulled back to$M$, is related by the above one-to-one correspondence between$Spin(p,q)$- and$Spin(q,p)$-structures. 5 small notation fix Let$(M,g)$be a$d$-dimensional, oriented pseudo-Riemannian manifold, and$V$the subbundle of$TM\oplus T^*M$given by the graph of the "musical" linear isomorphism$g^\sharp:TM\rightarrow g^\flat:TM\rightarrow T^*M$associated to the metric$g$. We know that the bundle$TM\oplus T^*M$carries a natural pseudo-Riemannian metric$h$of signature$(d,d)$and admits a spin structure associated to$h$- for example, the (space of sections of the) exterior algebra bundle$\Lambda^*T^*M$is a Clifford module, and its tensor product$\Lambda^*T^*M\otimes(\Lambda^dT^*M)^{1/2}$with the real line bundle of half-densities over$M$is a spinor bundle. Moreover, the space of spin structures on$(TM\oplus T^*M,h)$is an affine space modelled on the group$H^1(M,\mathbb{Z}_2)$of real line bundles over$M$, which maps the corresponding spinor bundles onto each other by tensoring (the above preliminary results can be found on Chapter 2 of Marco Gualtieri's PhD thesis on generalized complex geometry, arXiv:math.DG/0401221). Question(s): if$(M,g)$admits a spin structure, does a choice of spin structure on$(TM\oplus T^*M,h)$descend by restriction to$V$to a choice of spin structure on$(M,g)$? Does this establish a one-to-one correspondence between both sets of spin structures? If so, how does this generalize to, say, generalized Riemannian metrics (i.e. rank-$d$subbundles$W$of a twisting of$TM\oplus T^*M$by a Cech 1-cocycle$B$with values at closed 2-forms, such that the restriction of$h$to$W\$ is positive definite)?