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Nov 28 at 19:02 comment added B.Hueber @IgorKhavkine Thank you very much! That answers my question.
Nov 28 at 16:29 comment added Igor Khavkine Your condition on $\overline{g}$ (requiring $i_{t}^{\ast}\overline{g}$ to be a Riemannian metric) is weaker than the condition imposed on $h_t$ in the result of Bernal & Sanchez. Their condition is $h_t(\partial_t, \cdot) = 0$, where $\partial_t$ is the vector field pointing along the $\mathbb{R}$-fibers of the product $\mathbb{R} \times \Sigma$. It is this stronger condition that implies the absence of cross-terms in coordinates adapted to the product.
Oct 30 at 16:57 history edited B.Hueber CC BY-SA 4.0
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Oct 30 at 16:49 history asked B.Hueber CC BY-SA 4.0