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.



Generalized geometry and spin structures
OK, I am making an assumption: *I can re-interpret the problem (using the musical isomorphism) as $V$ being the diagonal embedding of $TM$ inside $TM\oplus TM$ and studying spin structures on them*.


**Actually, it turns out** *(see comments)* **that this "re-interpretation" is slightly different from the original construction. But the main bulk still goes through:**


I will restrict to $\dim M=3$, in which case our (closed oriented) manifold is always spinnable. A spin structure $\mathfrak{s}$ on $M$ induces a canonical spin structure $\mathfrak{S}\_0=\mathfrak{s}\oplus\mathfrak{s}$ on $TM\oplus TM$, and this is actually independent of the choice of $\mathfrak{s}$ (these appear in the notion of a *2-framing* on 3-manifolds, which Atiyah and Witten have used for some of their QFT studies). As a result, the "restriction" $\mathfrak{S}\_0|\_V$ on $M$ is ill-defined.  

*[proof of claim of canonical spin structure (learned from conversation with Rob Kirby): the spin structure fixes a trivialization over the 1-skeleton, and over circles there are two trivializations, so changing a trivialization of $\mathfrak{s}$ is doubled in $\mathfrak{s}\oplus\mathfrak{s}$ which modulo-2 is no change.]*


This also implies that the "restriction" to each $TM$-summand is ill-defined. (What I know that works: the collar-neighborhood theorem does allow an induced spin structure on $T(\partial X$) from a spin structure on $TX$ thanks to the splitting $TX|\_\partial=T(\partial X)\oplus\underline{\mathbb{R}}$ near the boundary.)


