Two definitions of spin structure

Question

There is a say standard definition of spin structure for an oriented manifold $M$: it is a principial $spin(n)$ bundle $Spin(M)$ together with the equivariant map $\eta:Spin(M) \to SO(M)$ where $SO(M)$ is orthonormal frame bundle (of oriented frames) and equivariance is understood via the concrete map $\rho: spin(n) \to SO(n)$ being $2:1$ covering map. However I met also the following definition:
Spin structure on oriented $M$ is a principial $spin(n)$ bundle $P$ together with an explicit isomorphism $P \times_{\rho} \mathbb{R}^n \cong TM$ (left hand side is associated bundle).

I would like to understand how this definition implies the standard one.

Here are some my attempts:
1. There is always the isomorphism $O(M) \times_{id} \mathbb{R}^n \cong TM$ where $O(M)$ is orthonormal frame bundle. Once $M$ is oriented we can replace $O(M)$ by $SO(M)$.
2. Therefore our assumption gives us isomorphism $P \times_{\rho} \mathbb{R}^n \cong SO(M) \times_{id} \mathbb{R}^n$ so we get that the two associated bundles are isomorphic. Does it follows from this that we have an equivariant map $\eta:P \to SO(M)$? It seems to me that it should be true or at least well known (since this is general question, what one can say about the existence of maps between principial bundles, having some assumptions for the existence of isomorphism between the associated vector bundles).

Answer 1

Your attempts go in the right direction, but you are missing one step: If you have a principal $SO(n)$-bundle $Q\to M$ such that $Q\times_{SO(n)}\mathbb R^n\cong TM$ (as oriented bundles endowed with a metric), then $Q\cong SO(M)$. This can be seen directly, since to a point $y\in Q$ over $x\in M$, you can associate a linear isomorphism $\mathbb R^n\to T_xM$ by mapping $v\in\mathbb R^n$ to the class of $(y,v)$ in the associated bundle. Compatibility with the metric and the orientation shows that this map is orthogonal and orientation preserving.

Now given a principal $Spin(n)$-bundle $P\to M$, you can always quotient by $\mathbb Z_2$ to get a principal $SO(n)$-bundle $Q=P/{\mathbb Z_2}\to M$. By definition of the action of $Spin(n)$ on $\mathbb R^n$, you see that $P\times_{Spin(n)}\mathbb R^n\cong Q\times_{SO(n)}\mathbb R^n$. Hence form the above argument you get an isomorphism $Q\to SO(M)$ and composing this with the canonical projection $P\to Q$, you see that $P$ defines a Spin-sturcture.