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Let $(M,g)$ be a $d$-dimensional, oriented pseudo-Riemannian manifold, and $V_+$ 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.

Let $(M,g)$ be a $d$-dimensional, oriented pseudo-Riemannian manifold, and $V$ 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.

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 $g^\flat:TM\rightarrow T^*M$ 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)?

Let $(M,g)$ be a $d$-dimensional, oriented pseudo-Riemannian manifold, and $V_+$ 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.

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 linear isomorphism $g^\sharp:TM\rightarrow T^*M$ 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)?

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 $g^\flat:TM\rightarrow T^*M$ 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)?