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Let $M$ be a smooth $n$-manifold (which is not assumed to be orientable), and write $o(TM)\to M$ for its orientation bundle. Equivalently, it is the top exterior bundle $\Lambda^n(TM)\to M$. In any case, it is a real line bundle and is flat, where the locally constant transition functions are given by the sign of the Jacobian matrix of the transition functions of $TM\to M$. Let $\nabla^{o(TM)}$ be the corresponding flat connection.

My questions are:

  1. Is the flat connection on $o(TM)\to M$ unique (in some/any sense)? I saw in somewhere the following sentence "Let $\nabla^{o(TM)}$ be the natural flat connection on $o(TM)\to M$". I don't understand what natural means there.
  2. Is the flat connection $\nabla^{o(TM)}$ induced by some other connection? Let say I put a Riemannian metric on $M$, and denote by $\nabla^{TM}$ its Levi-Civita connection. What is the difference $\nabla^{o(TM)}-\nabla^{\Lambda^n(TM)}$?

Any reference related to the orientation bundle and its flat connection will be appreciated.

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There is a different construction of orientation bundles. One considers the $\{\pm1\}$-principal bundle $o(TM)$ of fibrewise orientations of $TM$. The associated real line bundle $o(TM)\times_{\pm 1}\mathbb R$ carries a flat connection which is natural under local diffeomorphisms.

This bundle can be described as having the signs of the determinants of Jacobi matrices as transition functions. On the other hand, the bundles $\Lambda^{\max}TM$ and $\Lambda^{\max}T^*M$ have as transition functions the actual determinant of the Jacobi matrix (or its inverse, respectively).

A choice of a volume density defines an isomorphism $\Lambda^{\max}T^*M\cong o(TM)\times_{\pm1}\mathbb R$. If the volume density comes from a Riemannian metric, this isomorphism will identify the connection above with the one induced from the Levi-Civita connection.

A different choice of volume density leads to a different identification, so in this sense, the connection on $\Lambda^{\max}T^*M$ is only natural with respect to local diffeomorphisms that preserve the volume density.

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  • $\begingroup$ Thank you for the correction, and the answer relating the flat connection with the one induced by the Levi-Civita connection. It clears some of my confusions. $\endgroup$
    – Ho Man-Ho
    Commented May 23, 2023 at 15:33

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