Suppose I have a (d+1)-dimensional manifold with metric gμν. In it I have an embedded codimension-1 surface, Γ, with induced metric γab. Is Ricci scalar defined in terms of γab, R(γ), a covariant quantity with respect to (d+1)-dimensional diffeomorphisms?
Additionally, is there a way to relate extrinsic curvature, K, to R(γ) in arbitrary dimensions? I know that is possible in 2-dimensions, but I am interested in the more general case.
This post imported from StackExchange Physics at 2019-05-05 13:09 (UTC), posted by SE-user nGlacTOwnS