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This comes from O'Neill's Semi-Riemannian Geometry, in the proof of Proposition 11.22. Given a reductive coset manifold $M = G/H$ of a Lie Group $G$ with Lie subspace $m$, if you fix the differential of the projection map $d\pi$ to be an isometry, then there is to be a one-to-one correspondence between $Ad(H)$ invariant scalar products on m and G-invariant metrics on $M$.

The proof outlines a construction of the metric given an Ad(H) invariant scalar product on m by transferring the scalar product to the tangent space of M at the origin and then deriving it on the rest of $M$ via the differential maps of the induced translation maps.

The part I'm struggling with is how to show the smoothness of this metric. The book implies the way to see this is through local sections that exist because $\pi$ is a submersion, but I'm struggling to see how this is applied...

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