Suppose I have a $(d+1)$-dimensional manifold with metric $g_{\mu\nu}$. In it I have an embedded codimension-$1$ surface, $\Gamma$, with induced metric $\gamma_{ab}$. Is Ricci scalar defined in terms of $\gamma_{ab}$, $R^{(\gamma)}$, a covariant quantity with respect to $(d+1)$-dimensional diffeomorphisms?

Additionally, is there a way to relate extrinsic curvature, $\mathcal K$, to $R^{(\gamma)}$ in arbitrary dimensions? I know that is possible in $2$-dimensions, but I am interested in the more general case.



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