# Is intrinsic curvature of an embedded surface a covariant quantity from the embedding space point of view?

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.